--- a/src/HOL/Euclidean_Division.thy Tue Apr 09 16:59:00 2019 +0000
+++ b/src/HOL/Euclidean_Division.thy Tue Apr 09 16:59:00 2019 +0000
@@ -165,25 +165,31 @@
subsection \<open>Euclidean (semi)rings with cancel rules\<close>
-class euclidean_semiring_cancel = euclidean_semiring +
- assumes 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"
+class euclidean_semiring_cancel = euclidean_semiring + semidom_divide_cancel
begin
+context
+ fixes b
+ assumes "b \<noteq> 0"
+begin
+
+lemma div_mult_self1 [simp]:
+ "(a + c * b) div b = c + a div b"
+ using \<open>b \<noteq> 0\<close> by (rule div_mult_self1)
+
lemma div_mult_self2 [simp]:
- assumes "b \<noteq> 0"
- shows "(a + b * c) div b = c + a div b"
- using assms div_mult_self1 [of b a c] by (simp add: mult.commute)
+ "(a + b * c) div b = c + a div b"
+ using \<open>b \<noteq> 0\<close> by (rule div_mult_self2)
lemma div_mult_self3 [simp]:
- assumes "b \<noteq> 0"
- shows "(c * b + a) div b = c + a div b"
- using assms by (simp add: add.commute)
+ "(c * b + a) div b = c + a div b"
+ using \<open>b \<noteq> 0\<close> by (rule div_mult_self3)
lemma div_mult_self4 [simp]:
- assumes "b \<noteq> 0"
- shows "(b * c + a) div b = c + a div b"
- using assms by (simp add: add.commute)
+ "(b * c + a) div b = c + a div b"
+ using \<open>b \<noteq> 0\<close> by (rule div_mult_self4)
+
+end
lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
proof (cases "b = 0")
@@ -194,7 +200,7 @@
by (simp add: div_mult_mod_eq)
also from False div_mult_self1 [of b a c] have
"\<dots> = (c + a div b) * b + (a + c * b) mod b"
- by (simp add: algebra_simps)
+ by (simp add: algebra_simps)
finally have "a = a div b * b + (a + c * b) mod b"
by (simp add: add.commute [of a] add.assoc distrib_right)
then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
@@ -222,16 +228,6 @@
"a * b mod b = 0"
using mod_mult_self1 [of 0 a b] by simp
-lemma div_add_self1:
- assumes "b \<noteq> 0"
- shows "(b + a) div b = a div b + 1"
- using assms div_mult_self1 [of b a 1] by (simp add: add.commute)
-
-lemma div_add_self2:
- assumes "b \<noteq> 0"
- shows "(a + b) div b = a div b + 1"
- using assms div_add_self1 [of b a] by (simp add: add.commute)
-
lemma mod_add_self1 [simp]:
"(b + a) mod b = a mod b"
using mod_mult_self1 [of a 1 b] by (simp add: add.commute)
@@ -284,14 +280,6 @@
finally show ?thesis .
qed
-lemma div_mult_mult2 [simp]:
- "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
- by (drule div_mult_mult1) (simp add: mult.commute)
-
-lemma div_mult_mult1_if [simp]:
- "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
- by simp_all
-
lemma mod_mult_mult1:
"(c * a) mod (c * b) = c * (a mod b)"
proof (cases "c = 0")
@@ -448,23 +436,14 @@
class euclidean_ring_cancel = euclidean_ring + euclidean_semiring_cancel
begin
-subclass idom_divide ..
-
-lemma div_minus_minus [simp]: "(- a) div (- b) = a div b"
- using div_mult_mult1 [of "- 1" a b] by simp
+subclass idom_divide_cancel ..
lemma mod_minus_minus [simp]: "(- a) mod (- b) = - (a mod b)"
using mod_mult_mult1 [of "- 1" a b] by simp
-lemma div_minus_right: "a div (- b) = (- a) div b"
- using div_minus_minus [of "- a" b] by simp
-
lemma mod_minus_right: "a mod (- b) = - ((- a) mod b)"
using mod_minus_minus [of "- a" b] by simp
-lemma div_minus1_right [simp]: "a div (- 1) = - a"
- using div_minus_right [of a 1] by simp
-
lemma mod_minus1_right [simp]: "a mod (- 1) = 0"
using mod_minus_right [of a 1] by simp
--- a/src/HOL/Fields.thy Tue Apr 09 16:59:00 2019 +0000
+++ b/src/HOL/Fields.thy Tue Apr 09 16:59:00 2019 +0000
@@ -378,30 +378,42 @@
by (simp add: divide_inverse)
qed
+subclass idom_divide_cancel
+proof
+ fix a b c
+ assume [simp]: "c \<noteq> 0"
+ show "(c * a) / (c * b) = a / b"
+ proof (cases "b = 0")
+ case True then show ?thesis
+ by simp
+ next
+ case False
+ then have "(c * a) / (c * b) = c * a * (inverse b * inverse c)"
+ by (simp add: divide_inverse nonzero_inverse_mult_distrib)
+ also have "... = a * inverse b * (inverse c * c)"
+ by (simp only: ac_simps)
+ also have "... = a * inverse b" by simp
+ finally show ?thesis by (simp add: divide_inverse)
+ qed
+next
+ fix a b c
+ assume "b \<noteq> 0"
+ have "((a * inverse b) * b + c * b) = (c + a * inverse b) * b"
+ using distrib [of c "a * inverse b" b] by (simp add: ac_simps)
+ also have "(a * inverse b) * b = a"
+ using \<open>b \<noteq> 0\<close> by (simp add: ac_simps)
+ finally show "(a + c * b) / b = c + a / b"
+ using \<open>b \<noteq> 0\<close> by (simp add: ac_simps divide_inverse [symmetric])
+qed
+
+lemmas nonzero_mult_divide_mult_cancel_left = div_mult_mult1 \<comment> \<open>duplicate\<close>
+lemmas nonzero_mult_divide_mult_cancel_right = div_mult_mult2 \<comment> \<open>duplicate\<close>
+
text\<open>There is no slick version using division by zero.\<close>
lemma inverse_add:
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = (a + b) * inverse a * inverse b"
by (simp add: division_ring_inverse_add ac_simps)
-lemma nonzero_mult_divide_mult_cancel_left [simp]:
- assumes [simp]: "c \<noteq> 0"
- shows "(c * a) / (c * b) = a / b"
-proof (cases "b = 0")
- case True then show ?thesis by simp
-next
- case False
- then have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
- by (simp add: divide_inverse nonzero_inverse_mult_distrib)
- also have "... = a * inverse b * (inverse c * c)"
- by (simp only: ac_simps)
- also have "... = a * inverse b" by simp
- finally show ?thesis by (simp add: divide_inverse)
-qed
-
-lemma nonzero_mult_divide_mult_cancel_right [simp]:
- "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
- using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps)
-
lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"
by (simp add: divide_inverse ac_simps)
--- a/src/HOL/Rings.thy Tue Apr 09 16:59:00 2019 +0000
+++ b/src/HOL/Rings.thy Tue Apr 09 16:59:00 2019 +0000
@@ -1699,6 +1699,69 @@
end
+text \<open>Integral (semi)domains with cancellation rules\<close>
+
+class semidom_divide_cancel = semidom_divide +
+ assumes div_mult_self1: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
+ and div_mult_mult1: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
+begin
+
+context
+ fixes b
+ assumes "b \<noteq> 0"
+begin
+
+lemma div_mult_self2:
+ "(a + b * c) div b = c + a div b"
+ using \<open>b \<noteq> 0\<close> div_mult_self1 [of b a c] by (simp add: ac_simps)
+
+lemma div_mult_self3:
+ "(c * b + a) div b = c + a div b"
+ using \<open>b \<noteq> 0\<close> div_mult_self1 [of b a c] by (simp add: ac_simps)
+
+lemma div_mult_self4:
+ "(b * c + a) div b = c + a div b"
+ using \<open>b \<noteq> 0\<close> div_mult_self1 [of b a c] by (simp add: ac_simps)
+
+lemma div_add_self1:
+ "(b + a) div b = a div b + 1"
+ using \<open>b \<noteq> 0\<close> div_mult_self1 [of b a 1] by (simp add: ac_simps)
+
+lemma div_add_self2:
+ "(a + b) div b = a div b + 1"
+ using \<open>b \<noteq> 0\<close> div_add_self1 [of a] by (simp add: ac_simps)
+
+end
+
+lemma div_mult_mult2:
+ "(a * c) div (b * c) = a div b" if "c \<noteq> 0"
+ using that div_mult_mult1 [of c a b] by (simp add: ac_simps)
+
+lemma div_mult_mult1_if [simp]:
+ "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
+ by (simp add: div_mult_mult1)
+
+lemma div_mult_mult2_if [simp]:
+ "(a * c) div (b * c) = (if c = 0 then 0 else a div b)"
+ using div_mult_mult1_if [of c a b] by (simp add: ac_simps)
+
+end
+
+class idom_divide_cancel = idom_divide + semidom_divide_cancel
+begin
+
+lemma div_minus_minus [simp]: "(- a) div (- b) = a div b"
+ using div_mult_mult1 [of "- 1" a b] by simp
+
+lemma div_minus_right: "a div (- b) = (- a) div b"
+ using div_minus_minus [of "- a" b] by simp
+
+lemma div_minus1_right [simp]: "a div (- 1) = - a"
+ using div_minus_right [of a 1] by simp
+
+end
+
+
text \<open>Quotient and remainder in integral domains\<close>
class semidom_modulo = algebraic_semidom + semiring_modulo