src/HOL/Rings.thy

   with False \<open>a = b * c\<close> show ?thesis
     by simp
 qed
 
 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
-
-
-subsection \<open>Basic notions following from divisibility\<close>
 
 class algebraic_semidom = semidom_divide
 begin
 
 text \<open>