author chaieb Sun, 17 Jun 2007 13:39:29 +0200 changeset 23406 167b53019d6f parent 23405 8993b3144358 child 23407 0e4452fcbeb8
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
```--- a/src/HOL/Ring_and_Field.thy	Sun Jun 17 13:39:27 2007 +0200
+++ b/src/HOL/Ring_and_Field.thy	Sun Jun 17 13:39:29 2007 +0200
@@ -1007,7 +1007,6 @@
apply (cases "b = 0")
done
-
lemma divide_1 [simp]: "a/1 = (a::'a::field)"

@@ -1040,6 +1039,25 @@
done

+lemma nonzero_mult_divide_cancel_right':
+  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
+proof -
+  assume b: "b \<noteq> 0"
+  have "a * b / b = a * (b / b)" by (simp add: times_divide_eq_right)
+  also have "\<dots> = a" by (simp add: divide_self b)
+  finally show "a * b / b = a" .
+qed
+
+lemma nonzero_mult_divide_cancel_left':
+  "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
+proof -
+  assume b: "a \<noteq> 0"
+  have "a * b / a = b * a / a" by (simp add: mult_commute)
+  also have "\<dots> = b * (a / a)" by (simp add: times_divide_eq_right)
+  also have "\<dots> = b" by (simp add: divide_self b)
+  finally show "a * b / a = b" .
+qed
+
subsubsection{*Special Cancellation Simprules for Division*}

(* FIXME need not be a simp-rule once "divide_cancel_factor" has been fixed *)
@@ -1217,6 +1235,23 @@
"(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"

+lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
+proof
+  fix x::'a
+  have m1: "- (1::'a) < 0" by simp
+  from add_strict_right_mono[OF m1, where c=x]
+  have "(- 1) + x < x" by simp
+  thus "\<exists>y. y < x" by blast
+qed
+
+lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
+proof
+  fix x::'a
+  have m1: " (1::'a) > 0" by simp
+  from add_strict_right_mono[OF m1, where c=x]
+  have "1 + x > x" by simp
+  thus "\<exists>y. y > x" by blast
+qed

subsection{*Anti-Monotonicity of @{term inverse}*}
```