equal
deleted
inserted
replaced
132 |
132 |
133 lemma less_irrefl [iff]: "\<not> x < x" |
133 lemma less_irrefl [iff]: "\<not> x < x" |
134 by (simp add: less_le_not_le) |
134 by (simp add: less_le_not_le) |
135 |
135 |
136 lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y" |
136 lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y" |
137 unfolding less_le_not_le by blast |
137 by (simp add: less_le_not_le) |
138 |
138 |
139 |
139 |
140 text \<open>Asymmetry.\<close> |
140 text \<open>Asymmetry.\<close> |
141 |
141 |
142 lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)" |
142 lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)" |
200 lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y" |
200 lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y" |
201 \<comment> \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close> |
201 \<comment> \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close> |
202 by (fact order.order_iff_strict) |
202 by (fact order.order_iff_strict) |
203 |
203 |
204 lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y" |
204 lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y" |
205 unfolding less_le by blast |
205 by (simp add: less_le) |
206 |
206 |
207 |
207 |
208 text \<open>Useful for simplification, but too risky to include by default.\<close> |
208 text \<open>Useful for simplification, but too risky to include by default.\<close> |
209 |
209 |
210 lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False" |
210 lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False" |