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March 4, 2020 15:11
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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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