--- a/src/HOL/Fields.thy Sun Nov 26 13:19:52 2017 +0100
+++ b/src/HOL/Fields.thy Sun Nov 26 21:08:32 2017 +0100
@@ -400,10 +400,10 @@
lemma inverse_mult_distrib [simp]:
"inverse (a * b) = inverse a * inverse b"
proof cases
- assume "a \<noteq> 0 & b \<noteq> 0"
+ assume "a \<noteq> 0 \<and> b \<noteq> 0"
thus ?thesis by (simp add: nonzero_inverse_mult_distrib ac_simps)
next
- assume "~ (a \<noteq> 0 & b \<noteq> 0)"
+ assume "\<not> (a \<noteq> 0 \<and> b \<noteq> 0)"
thus ?thesis by force
qed
@@ -596,7 +596,7 @@
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
shows "b \<le> a"
proof (rule classical)
- assume "~ b \<le> a"
+ assume "\<not> b \<le> a"
hence "a < b" by (simp add: linorder_not_le)
hence bpos: "0 < b" by (blast intro: apos less_trans)
hence "a * inverse a \<le> a * inverse b"
@@ -649,9 +649,9 @@
assumes less: "a < b" and apos: "0 < a"
shows "inverse b < inverse a"
proof (rule ccontr)
- assume "~ inverse b < inverse a"
+ assume "\<not> inverse b < inverse a"
hence "inverse a \<le> inverse b" by simp
- hence "~ (a < b)"
+ hence "\<not> (a < b)"
by (simp add: not_less inverse_le_imp_le [OF _ apos])
thus False by (rule notE [OF _ less])
qed
@@ -1142,19 +1142,19 @@
text\<open>Simplify quotients that are compared with the value 1.\<close>
lemma le_divide_eq_1:
- "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
+ "(1 \<le> b / a) = ((0 < a \<and> a \<le> b) \<or> (a < 0 \<and> b \<le> a))"
by (auto simp add: le_divide_eq)
lemma divide_le_eq_1:
- "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
+ "(b / a \<le> 1) = ((0 < a \<and> b \<le> a) \<or> (a < 0 \<and> a \<le> b) \<or> a=0)"
by (auto simp add: divide_le_eq)
lemma less_divide_eq_1:
- "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
+ "(1 < b / a) = ((0 < a \<and> a < b) \<or> (a < 0 \<and> b < a))"
by (auto simp add: less_divide_eq)
lemma divide_less_eq_1:
- "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
+ "(b / a < 1) = ((0 < a \<and> b < a) \<or> (a < 0 \<and> a < b) \<or> a=0)"
by (auto simp add: divide_less_eq)
lemma divide_nonneg_nonneg [simp]:
@@ -1208,11 +1208,11 @@
by (auto simp add: divide_less_eq)
lemma eq_divide_eq_1 [simp]:
- "(1 = b/a) = ((a \<noteq> 0 & a = b))"
+ "(1 = b/a) = ((a \<noteq> 0 \<and> a = b))"
by (auto simp add: eq_divide_eq)
lemma divide_eq_eq_1 [simp]:
- "(b/a = 1) = ((a \<noteq> 0 & a = b))"
+ "(b/a = 1) = ((a \<noteq> 0 \<and> a = b))"
by (auto simp add: divide_eq_eq)
lemma abs_div_pos: "0 < y ==>
@@ -1221,10 +1221,10 @@
apply (simp add: order_less_imp_le)
done
-lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / \<bar>b\<bar>) = (0 \<le> a | b = 0)"
+lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / \<bar>b\<bar>) = (0 \<le> a \<or> b = 0)"
by (auto simp: zero_le_divide_iff)
-lemma divide_le_0_abs_iff [simp]: "(a / \<bar>b\<bar> \<le> 0) = (a \<le> 0 | b = 0)"
+lemma divide_le_0_abs_iff [simp]: "(a / \<bar>b\<bar> \<le> 0) = (a \<le> 0 \<or> b = 0)"
by (auto simp: divide_le_0_iff)
lemma field_le_mult_one_interval: