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| theory X imports Complex_Main begin | |
| consts a :: "nat => real" | |
| theorem | |
| "k > 0 ==> | |
| (ALL e. e > 0 --> (EX N1. (ALL n. n > N1 --> abs (a n - b) < e))) | |
| = (ALL e. e > 0 --> (EX N2. (ALL n. n > N2 --> abs (a n - b) < k * e)))" | |
| apply(rule iffI) | |
| apply(rule allI) | |
| apply(rule impI) | |
| apply(drule_tac x="k * e" in spec) | |
| apply(drule mp) | |
| apply(simp) | |
| apply(erule exE) | |
| apply(rule_tac x="N1" in exI) | |
| apply(rule allI) | |
| apply(rule impI) | |
| apply(drule_tac x="n" in spec) | |
| apply(drule mp) | |
| apply(simp) | |
| apply(simp) | |
| (**) | |
| apply(rule allI) | |
| apply(rule impI) | |
| apply(drule_tac x="e / k" in spec) | |
| apply(drule mp) | |
| apply(simp) | |
| apply(erule exE) | |
| apply(rule_tac x="N2" in exI) | |
| apply(rule allI) | |
| apply(rule impI) | |
| apply(drule_tac x="n" in spec) | |
| apply(drule mp) | |
| apply(simp) | |
| apply(simp) | |
| done | |
| theorem | |
| "k > 0 ==> | |
| (ALL e. e > 0 --> (EX N1. n > N1 --> abs (a n - b) < e)) | |
| = (ALL e. e > 0 --> (EX N2. n > N2 --> abs (a n - b) < k * e))" | |
| apply(rule iffI) | |
| apply(rule allI) | |
| apply(rule impI) | |
| apply(drule_tac x="k * e" in spec) | |
| apply(drule mp) | |
| apply(simp) | |
| apply(erule exE) | |
| apply(rule_tac x="N1" in exI) | |
| apply(rule impI) | |
| apply(drule mp) | |
| apply(simp) | |
| apply(simp) | |
| (**) | |
| apply(rule allI) | |
| apply(rule impI) | |
| apply(drule_tac x="e / k" in spec) | |
| apply(drule mp) | |
| apply(simp) | |
| apply(erule exE) | |
| apply(rule_tac x="N2" in exI) | |
| apply(rule impI) | |
| apply(drule mp) | |
| apply(simp) | |
| apply(simp) | |
| done |
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