equal
deleted
inserted
replaced
1144 \<bar>x\<bar> / y = \<bar>x / y\<bar>" |
1144 \<bar>x\<bar> / y = \<bar>x / y\<bar>" |
1145 apply (subst abs_divide) |
1145 apply (subst abs_divide) |
1146 apply (simp add: order_less_imp_le) |
1146 apply (simp add: order_less_imp_le) |
1147 done |
1147 done |
1148 |
1148 |
1149 lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / abs b) = (0 \<le> a | b = 0)" |
1149 lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / \<bar>b\<bar>) = (0 \<le> a | b = 0)" |
1150 by (auto simp: zero_le_divide_iff) |
1150 by (auto simp: zero_le_divide_iff) |
1151 |
1151 |
1152 lemma divide_le_0_abs_iff [simp]: "(a / abs b \<le> 0) = (a \<le> 0 | b = 0)" |
1152 lemma divide_le_0_abs_iff [simp]: "(a / \<bar>b\<bar> \<le> 0) = (a \<le> 0 | b = 0)" |
1153 by (auto simp: divide_le_0_iff) |
1153 by (auto simp: divide_le_0_iff) |
1154 |
1154 |
1155 lemma field_le_mult_one_interval: |
1155 lemma field_le_mult_one_interval: |
1156 assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y" |
1156 assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y" |
1157 shows "x \<le> y" |
1157 shows "x \<le> y" |