--- a/src/HOL/Fields.thy Fri Oct 24 15:07:49 2014 +0200
+++ b/src/HOL/Fields.thy Fri Oct 24 15:07:51 2014 +0200
@@ -23,11 +23,37 @@
fixes inverse :: "'a \<Rightarrow> 'a"
and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "'/" 70)
+setup {* Sign.add_const_constraint
+ (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
+
+
+context semiring
+begin
+
+lemma [field_simps]:
+ shows distrib_left_NO_MATCH: "NO_MATCH a (x / y) \<Longrightarrow> a * (b + c) = a * b + a * c"
+ and distrib_right_NO_MATCH: "NO_MATCH c (x / y) \<Longrightarrow> (a + b) * c = a * c + b * c"
+ by (rule distrib_left distrib_right)+
+
+end
+
+context ring
+begin
+
+lemma [field_simps]:
+ shows left_diff_distrib_NO_MATCH: "NO_MATCH c (x / y) \<Longrightarrow> (a - b) * c = a * c - b * c"
+ and right_diff_distrib_NO_MATCH: "NO_MATCH a (x / y) \<Longrightarrow> a * (b - c) = a * b - a * c"
+ by (rule left_diff_distrib right_diff_distrib)+
+
+end
+
+setup {* Sign.add_const_constraint
+ (@{const_name "divide"}, SOME @{typ "'a::inverse \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
+
text{* Lemmas @{text divide_simps} move division to the outside and eliminates them on (in)equalities. *}
named_theorems divide_simps "rewrite rules to eliminate divisions"
-
class division_ring = ring_1 + inverse +
assumes left_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"