author haftmann Tue Apr 28 15:50:30 2009 +0200 (2009-04-28) changeset 31016 e1309df633c6 parent 31015 555f4033cd97 child 31017 2c227493ea56
lemma sum_nonneg_eq_zero_iff
```     1.1 --- a/src/HOL/OrderedGroup.thy	Tue Apr 28 15:50:29 2009 +0200
1.2 +++ b/src/HOL/OrderedGroup.thy	Tue Apr 28 15:50:30 2009 +0200
1.3 @@ -637,6 +637,27 @@
1.4  lemma le_iff_diff_le_0: "a \<le> b \<longleftrightarrow> a - b \<le> 0"
1.5  by (simp add: algebra_simps)
1.6
1.7 +lemma sum_nonneg_eq_zero_iff:
1.8 +  assumes x: "0 \<le> x" and y: "0 \<le> y"
1.9 +  shows "(x + y = 0) = (x = 0 \<and> y = 0)"
1.10 +proof -
1.11 +  have "x + y = 0 \<Longrightarrow> x = 0"
1.12 +  proof -
1.13 +    from y have "x + 0 \<le> x + y" by (rule add_left_mono)
1.14 +    also assume "x + y = 0"
1.15 +    finally have "x \<le> 0" by simp
1.16 +    then show "x = 0" using x by (rule antisym)
1.17 +  qed
1.18 +  moreover have "x + y = 0 \<Longrightarrow> y = 0"
1.19 +  proof -
1.20 +    from x have "0 + y \<le> x + y" by (rule add_right_mono)
1.21 +    also assume "x + y = 0"
1.22 +    finally have "y \<le> 0" by simp
1.23 +    then show "y = 0" using y by (rule antisym)
1.24 +  qed
1.25 +  ultimately show ?thesis by auto
1.26 +qed
1.27 +
1.28  text{*Legacy - use @{text algebra_simps} *}
1.29  lemmas group_simps[noatp] = algebra_simps
1.30
```