equal
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inserted
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162 then obtain a b where "\<forall>x\<in>A. a \<le> x" "\<forall>x\<in>B. b \<le> x" unfolding bdd_below_def by auto |
162 then obtain a b where "\<forall>x\<in>A. a \<le> x" "\<forall>x\<in>B. b \<le> x" unfolding bdd_below_def by auto |
163 hence "\<forall>x \<in> A \<union> B. inf a b \<le> x" by (auto intro: Un_iff le_infI1 le_infI2) |
163 hence "\<forall>x \<in> A \<union> B. inf a b \<le> x" by (auto intro: Un_iff le_infI1 le_infI2) |
164 thus "bdd_below (A \<union> B)" unfolding bdd_below_def .. |
164 thus "bdd_below (A \<union> B)" unfolding bdd_below_def .. |
165 qed |
165 qed |
166 |
166 |
167 lemma bdd_above_sup[simp]: "bdd_above ((\<lambda>x. sup (f x) (g x)) ` A) \<longleftrightarrow> bdd_above (f`A) \<and> bdd_above (g`A)" |
167 lemma bdd_above_image_sup[simp]: |
168 by (auto simp: bdd_above_def intro: le_supI1 le_supI2) |
168 "bdd_above ((\<lambda>x. sup (f x) (g x)) ` A) \<longleftrightarrow> bdd_above (f`A) \<and> bdd_above (g`A)" |
169 |
169 by (auto simp: bdd_above_def intro: le_supI1 le_supI2) |
170 lemma bdd_below_inf[simp]: "bdd_below ((\<lambda>x. inf (f x) (g x)) ` A) \<longleftrightarrow> bdd_below (f`A) \<and> bdd_below (g`A)" |
170 |
171 by (auto simp: bdd_below_def intro: le_infI1 le_infI2) |
171 lemma bdd_below_image_inf[simp]: |
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172 "bdd_below ((\<lambda>x. inf (f x) (g x)) ` A) \<longleftrightarrow> bdd_below (f`A) \<and> bdd_below (g`A)" |
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173 by (auto simp: bdd_below_def intro: le_infI1 le_infI2) |
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174 |
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175 lemma bdd_below_UN[simp]: "finite I \<Longrightarrow> bdd_below (\<Union>i\<in>I. A i) = (\<forall>i \<in> I. bdd_below (A i))" |
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176 by (induction I rule: finite.induct) auto |
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177 |
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178 lemma bdd_above_UN[simp]: "finite I \<Longrightarrow> bdd_above (\<Union>i\<in>I. A i) = (\<forall>i \<in> I. bdd_above (A i))" |
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179 by (induction I rule: finite.induct) auto |
172 |
180 |
173 end |
181 end |
174 |
182 |
175 |
183 |
176 text \<open> |
184 text \<open> |