|
| "This is a contradiction." AND "This contradicts the fact that" AND "The converse is obvious." | 21 |
| "This contradicts the fact that" AND "This completes the proof." AND "The converse is obvious." | 19 |
| "We proceed by induction." AND "We begin by observing that" AND "This is clear." | 18 |
| "it is well known that" AND "it is not yet known whether" AND "a central problem in" | 13 |
| "under the additional assumption that" AND "it is well known that" AND "it has long been known that" | 12 |
| "This is trivial." AND "This is straightforward." AND "This is simple." | 11 |
| "it is well known that" AND "it has long been known that" AND "a central problem in" | 10 |
| "was a milestone in" AND "the main result was the" AND "have raised the question of whether" | 10 |
| "was a milestone in" AND "recently, there has been much interest in the" AND "have raised the question of whether" | 9 |
| "the main result was the" AND "recently, there has been much interest in the" AND "have raised the question of whether" | 9 |
| "was a milestone in" AND "the main result was the" AND "recently, there has been much interest in the" | 9 |
| "recently, there has been much interest in the" AND "it would be interesting to apply the techniques of" AND "have raised the question of whether" | 8 |
| "was a milestone in" AND "the main result was the" AND "in this setting, the ability to" | 8 |
| "was a milestone in" AND "have raised the question of whether" AND "every student is aware that" | 8 |
| "was a milestone in" AND "the main result was the" AND "it would be interesting to apply the techniques of" | 8 |
| "under the additional assumption that" AND "recently, there has been much interest in the" AND "have raised the question of whether" | 8 |
| "recently, there has been much interest in the" AND "it would be interesting to apply the techniques of" AND "every student is aware that" | 8 |
| "under the additional assumption that" AND "the main result was the" AND "have raised the question of whether" | 8 |
| "was a milestone in" AND "recently, there has been much interest in the" AND "it would be interesting to apply the techniques of" | 8 |
| "was a milestone in" AND "under the additional assumption that" AND "it is well known that" | 8 |
| "was a milestone in" AND "the groundbreaking work of" AND "have raised the question of whether" | 8 |
| "the main result was the" AND "recently, there has been much interest in the" AND "every student is aware that" | 8 |
| "the main result was the" AND "it would be interesting to apply the techniques of" AND "every student is aware that" | 8 |
| "we wish to extend the results of" AND "the main result was the" AND "have raised the question of whether" | 8 |
| "we wish to extend the results of" AND "it has long been known that" AND "have raised the question of whether" | 8 |
| "was a milestone in" AND "the main result was the" AND "it has long been known that" | 8 |
| "was a milestone in" AND "under the additional assumption that" AND "recently, there has been much interest in the" | 8 |
| "the main result was the" AND "in this setting, the ability to" AND "have raised the question of whether" | 8 |
| "the main result was the" AND "recently, there has been much interest in the" AND "it would be interesting to apply the techniques of" | 8 |
| "We show the contrapositive." AND "We follow" AND "This is obvious." | 8 |
| "was a milestone in" AND "the main result was the" AND "the groundbreaking work of" | 8 |
| "we wish to extend the results of" AND "was a milestone in" AND "the main result was the" | 8 |
| "was a milestone in" AND "it would be interesting to apply the techniques of" AND "every student is aware that" | 8 |
| "was a milestone in" AND "recently, there has been much interest in the" AND "every student is aware that" | 8 |
| "was a milestone in" AND "it would be interesting to apply the techniques of" AND "have raised the question of whether" | 8 |
| "was a milestone in" AND "it has long been known that" AND "have raised the question of whether" | 8 |
| "we wish to extend the results of" AND "unfortunately, we cannot assume that" AND "recently, there has been much interest in the" | 8 |
| "was a milestone in" AND "the main result was the" AND "every student is aware that" | 8 |
| "the main result was the" AND "have raised the question of whether" AND "every student is aware that" | 8 |
| "was a milestone in" AND "was a major advance." AND "the groundbreaking work of" | 8 |
| "we wish to extend the results of" AND "unfortunately, we cannot assume that" AND "in this setting, the ability to" | 8 |
| "we wish to extend the results of" AND "was a major advance." AND "this reduces the results of" | 8 |
| "we wish to extend the results of" AND "this reduces the results of" AND "recently, there has been much interest in the" | 8 |
| "the groundbreaking work of" AND "it is well known that" AND "a central problem in" | 8 |
| "we wish to extend the results of" AND "was a major advance." AND "the groundbreaking work of" | 8 |
| "we wish to extend the results of" AND "was a milestone in" AND "it has long been known that" | 8 |
| "This is trivial." AND "This is simple." AND "This is clear." | 8 |
| "the main result was the" AND "it would be interesting to apply the techniques of" AND "have raised the question of whether" | 8 |
| "we wish to extend the results of" AND "was a milestone in" AND "have raised the question of whether" | 8 |
| "We proceed by induction." AND "This is trivial." AND "Suppose the contrary." | 8 |