--- a/src/HOL/Divides.thy Wed Oct 12 21:53:30 2016 +0200
+++ b/src/HOL/Divides.thy Wed Oct 12 22:06:06 2016 +0200
@@ -11,9 +11,8 @@
subsection \<open>Abstract division in commutative semirings.\<close>
-class semiring_div = semidom + modulo +
- assumes mod_div_equality: "a div b * b + a mod b = a"
- and div_by_0: "a div 0 = 0"
+class semiring_div = semidom + semiring_modulo +
+ assumes div_by_0: "a div 0 = 0"
and div_0: "0 div a = 0"
and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
@@ -41,14 +40,6 @@
text \<open>@{const divide} and @{const modulo}\<close>
-lemma mod_div_equality2: "b * (a div b) + a mod b = a"
- unfolding mult.commute [of b]
- by (rule mod_div_equality)
-
-lemma mod_div_equality': "a mod b + a div b * b = a"
- using mod_div_equality [of a b]
- by (simp only: ac_simps)
-
lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
by (simp add: mod_div_equality)
@@ -143,17 +134,6 @@
"(a + b) mod b = a mod b"
using mod_mult_self1 [of a 1 b] by simp
-lemma mod_div_decomp:
- fixes a b
- obtains q r where "q = a div b" and "r = a mod b"
- and "a = q * b + r"
-proof -
- from mod_div_equality have "a = a div b * b + a mod b" by simp
- moreover have "a div b = a div b" ..
- moreover have "a mod b = a mod b" ..
- note that ultimately show thesis by blast
-qed
-
lemma dvd_imp_mod_0 [simp]:
assumes "a dvd b"
shows "b mod a = 0"
@@ -189,7 +169,7 @@
hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
by (rule div_mult_self1 [symmetric])
also have "\<dots> = a div b"
- by (simp only: mod_div_equality')
+ by (simp only: mod_div_equality3)
also have "\<dots> = a div b + 0"
by simp
finally show ?thesis
@@ -202,7 +182,7 @@
have "a mod b mod b = (a mod b + a div b * b) mod b"
by (simp only: mod_mult_self1)
also have "\<dots> = a mod b"
- by (simp only: mod_div_equality')
+ by (simp only: mod_div_equality3)
finally show ?thesis .
qed
--- a/src/HOL/Library/Polynomial_Factorial.thy Wed Oct 12 21:53:30 2016 +0200
+++ b/src/HOL/Library/Polynomial_Factorial.thy Wed Oct 12 22:06:06 2016 +0200
@@ -1018,8 +1018,12 @@
by (intro ext) (simp_all add: dvd.dvd_def dvd_def)
interpretation field_poly:
- euclidean_ring "op div" "op *" "op mod" "op +" "op -" 0 "1 :: 'a :: field poly"
- normalize_field_poly unit_factor_field_poly euclidean_size_field_poly uminus
+ euclidean_ring where zero = "0 :: 'a :: field poly"
+ and one = 1 and plus = plus and uminus = uminus and minus = minus
+ and times = times
+ and normalize = normalize_field_poly and unit_factor = unit_factor_field_poly
+ and euclidean_size = euclidean_size_field_poly
+ and divide = divide and modulo = modulo
proof (standard, unfold dvd_field_poly)
fix p :: "'a poly"
show "unit_factor_field_poly p * normalize_field_poly p = p"
@@ -1034,7 +1038,7 @@
thus "is_unit (unit_factor_field_poly p)"
by (simp add: unit_factor_field_poly_def lead_coeff_nonzero is_unit_pCons_iff)
qed (auto simp: unit_factor_field_poly_def normalize_field_poly_def lead_coeff_mult
- euclidean_size_field_poly_def intro!: degree_mod_less' degree_mult_right_le)
+ euclidean_size_field_poly_def Rings.mod_div_equality intro!: degree_mod_less' degree_mult_right_le)
lemma field_poly_irreducible_imp_prime:
assumes "irreducible (p :: 'a :: field poly)"
@@ -1352,7 +1356,7 @@
"euclidean_size_poly p = (if p = 0 then 0 else 2 ^ degree p)"
instance
- by standard (auto simp: euclidean_size_poly_def intro!: degree_mod_less' degree_mult_right_le)
+ by standard (auto simp: euclidean_size_poly_def Rings.mod_div_equality intro!: degree_mod_less' degree_mult_right_le)
end
@@ -1423,7 +1427,7 @@
by auto
termination
by (relation "measure ((\<lambda>p. if p = 0 then 0 else Suc (degree p)) \<circ> snd)")
- (auto simp: degree_primitive_part degree_pseudo_mod_less)
+ (auto simp: degree_pseudo_mod_less)
declare gcd_poly_code_aux.simps [simp del]
--- a/src/HOL/Number_Theory/Euclidean_Algorithm.thy Wed Oct 12 21:53:30 2016 +0200
+++ b/src/HOL/Number_Theory/Euclidean_Algorithm.thy Wed Oct 12 22:06:06 2016 +0200
@@ -17,7 +17,7 @@
The existence of these functions makes it possible to derive gcd and lcm functions
for any Euclidean semiring.
\<close>
-class euclidean_semiring = semiring_div + normalization_semidom +
+class euclidean_semiring = semiring_modulo + normalization_semidom +
fixes euclidean_size :: "'a \<Rightarrow> nat"
assumes size_0 [simp]: "euclidean_size 0 = 0"
assumes mod_size_less:
@@ -26,6 +26,30 @@
"b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
begin
+lemma zero_mod_left [simp]: "0 mod a = 0"
+ using mod_div_equality [of 0 a] by simp
+
+lemma dvd_mod_iff:
+ assumes "k dvd n"
+ shows "(k dvd m mod n) = (k dvd m)"
+proof -
+ from assms have "(k dvd m mod n) \<longleftrightarrow> (k dvd ((m div n) * n + m mod n))"
+ by (simp add: dvd_add_right_iff)
+ also have "(m div n) * n + m mod n = m"
+ using mod_div_equality [of m n] by simp
+ finally show ?thesis .
+qed
+
+lemma mod_0_imp_dvd:
+ assumes "a mod b = 0"
+ shows "b dvd a"
+proof -
+ have "b dvd ((a div b) * b)" by simp
+ also have "(a div b) * b = a"
+ using mod_div_equality [of a b] by (simp add: assms)
+ finally show ?thesis .
+qed
+
lemma euclidean_size_normalize [simp]:
"euclidean_size (normalize a) = euclidean_size a"
proof (cases "a = 0")
@@ -48,7 +72,7 @@
obtains s and t where "a = s * b + t"
and "euclidean_size t < euclidean_size b"
proof -
- from div_mod_equality [of a b 0]
+ from mod_div_equality [of a b]
have "a = a div b * b + a mod b" by simp
with that and assms show ?thesis by (auto simp add: mod_size_less)
qed
@@ -58,6 +82,7 @@
shows "a dvd b"
proof (rule ccontr)
assume "\<not> a dvd b"
+ hence "b mod a \<noteq> 0" using mod_0_imp_dvd[of b a] by blast
then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
@@ -434,8 +459,6 @@
class euclidean_ring = euclidean_semiring + idom
begin
-subclass ring_div ..
-
function euclid_ext_aux :: "'a \<Rightarrow> _" where
"euclid_ext_aux r' r s' s t' t = (
if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')
@@ -484,7 +507,7 @@
(s' * x + t' * y) - r' div r * (s * x + t * y)" by (simp add: algebra_simps)
also have "s' * x + t' * y = r'" by fact
also have "s * x + t * y = r" by fact
- also have "r' - r' div r * r = r' mod r" using mod_div_equality[of r' r]
+ also have "r' - r' div r * r = r' mod r" using mod_div_equality [of r' r]
by (simp add: algebra_simps)
finally show "(s' - r' div r * s) * x + (t' - r' div r * t) * y = r' mod r" .
qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality')
--- a/src/HOL/Rings.thy Wed Oct 12 21:53:30 2016 +0200
+++ b/src/HOL/Rings.thy Wed Oct 12 22:06:06 2016 +0200
@@ -571,11 +571,6 @@
setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
-text \<open>Syntactic division remainder operator\<close>
-
-class modulo = dvd + divide +
- fixes modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
-
text \<open>Algebraic classes with division\<close>
class semidom_divide = semidom + divide +
@@ -1286,6 +1281,53 @@
end
+
+text \<open>Syntactic division remainder operator\<close>
+
+class modulo = dvd + divide +
+ fixes modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
+
+text \<open>Arbitrary quotient and remainder partitions\<close>
+
+class semiring_modulo = comm_semiring_1_cancel + divide + modulo +
+ assumes mod_div_equality: "a div b * b + a mod b = a"
+begin
+
+lemma mod_div_decomp:
+ fixes a b
+ obtains q r where "q = a div b" and "r = a mod b"
+ and "a = q * b + r"
+proof -
+ from mod_div_equality have "a = a div b * b + a mod b" by simp
+ moreover have "a div b = a div b" ..
+ moreover have "a mod b = a mod b" ..
+ note that ultimately show thesis by blast
+qed
+
+lemma mod_div_equality2: "b * (a div b) + a mod b = a"
+ using mod_div_equality [of a b] by (simp add: ac_simps)
+
+lemma mod_div_equality3: "a mod b + a div b * b = a"
+ using mod_div_equality [of a b] by (simp add: ac_simps)
+
+lemma mod_div_equality4: "a mod b + b * (a div b) = a"
+ using mod_div_equality [of a b] by (simp add: ac_simps)
+
+lemma minus_div_eq_mod: "a - a div b * b = a mod b"
+ by (rule add_implies_diff [symmetric]) (fact mod_div_equality3)
+
+lemma minus_div_eq_mod2: "a - b * (a div b) = a mod b"
+ by (rule add_implies_diff [symmetric]) (fact mod_div_equality4)
+
+lemma minus_mod_eq_div: "a - a mod b = a div b * b"
+ by (rule add_implies_diff [symmetric]) (fact mod_div_equality)
+
+lemma minus_mod_eq_div2: "a - a mod b = b * (a div b)"
+ by (rule add_implies_diff [symmetric]) (fact mod_div_equality2)
+
+end
+
+
class ordered_semiring = semiring + ordered_comm_monoid_add +
assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"