author huffman Mon, 14 May 2007 17:45:42 +0200 changeset 22967 0651b224528a parent 22966 9dc4f5048353 child 22968 7134874437ac
 src/HOL/Real/RealPow.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/Real/RealPow.thy	Mon May 14 17:37:31 2007 +0200
+++ b/src/HOL/Real/RealPow.thy	Mon May 14 17:45:42 2007 +0200
@@ -118,6 +118,75 @@
declare power_real_number_of [of _ "number_of w", standard, simp]

+subsection {* Properties of Squares *}
+
+lemma sum_squares_ge_zero:
+  fixes x y :: "'a::ordered_ring_strict"
+  shows "0 \<le> x * x + y * y"
+
+lemma not_sum_squares_lt_zero:
+  fixes x y :: "'a::ordered_ring_strict"
+  shows "\<not> x * x + y * y < 0"
+
+lemma sum_nonneg_eq_zero_iff:
+  fixes x y :: "'a::pordered_ab_group_add"
+  assumes x: "0 \<le> x" and y: "0 \<le> y"
+  shows "(x + y = 0) = (x = 0 \<and> y = 0)"
+proof (auto)
+  from y have "x + 0 \<le> x + y" by (rule add_left_mono)
+  also assume "x + y = 0"
+  finally have "x \<le> 0" by simp
+  thus "x = 0" using x by (rule order_antisym)
+next
+  from x have "0 + y \<le> x + y" by (rule add_right_mono)
+  also assume "x + y = 0"
+  finally have "y \<le> 0" by simp
+  thus "y = 0" using y by (rule order_antisym)
+qed
+
+lemma sum_squares_eq_zero_iff:
+  fixes x y :: "'a::ordered_ring_strict"
+  shows "(x * x + y * y = 0) = (x = 0 \<and> y = 0)"
+
+lemma sum_squares_le_zero_iff:
+  fixes x y :: "'a::ordered_ring_strict"
+  shows "(x * x + y * y \<le> 0) = (x = 0 \<and> y = 0)"
+by (simp add: order_le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
+
+lemma sum_squares_gt_zero_iff:
+  fixes x y :: "'a::ordered_ring_strict"
+  shows "(0 < x * x + y * y) = (x \<noteq> 0 \<or> y \<noteq> 0)"
+by (simp add: order_less_le sum_squares_ge_zero sum_squares_eq_zero_iff)
+
+lemma sum_power2_ge_zero:
+  fixes x y :: "'a::{ordered_idom,recpower}"
+  shows "0 \<le> x\<twosuperior> + y\<twosuperior>"
+unfolding power2_eq_square by (rule sum_squares_ge_zero)
+
+lemma not_sum_power2_lt_zero:
+  fixes x y :: "'a::{ordered_idom,recpower}"
+  shows "\<not> x\<twosuperior> + y\<twosuperior> < 0"
+unfolding power2_eq_square by (rule not_sum_squares_lt_zero)
+
+lemma sum_power2_eq_zero_iff:
+  fixes x y :: "'a::{ordered_idom,recpower}"
+  shows "(x\<twosuperior> + y\<twosuperior> = 0) = (x = 0 \<and> y = 0)"
+unfolding power2_eq_square by (rule sum_squares_eq_zero_iff)
+
+lemma sum_power2_le_zero_iff:
+  fixes x y :: "'a::{ordered_idom,recpower}"
+  shows "(x\<twosuperior> + y\<twosuperior> \<le> 0) = (x = 0 \<and> y = 0)"
+unfolding power2_eq_square by (rule sum_squares_le_zero_iff)
+
+lemma sum_power2_gt_zero_iff:
+  fixes x y :: "'a::{ordered_idom,recpower}"
+  shows "(0 < x\<twosuperior> + y\<twosuperior>) = (x \<noteq> 0 \<or> y \<noteq> 0)"
+unfolding power2_eq_square by (rule sum_squares_gt_zero_iff)
+
+
subsection{*Various Other Theorems*}

lemma real_sum_squares_cancel_a: "x * x = -(y * y) ==> x = (0::real) & y=0"```