| 1. | The second recursion theorem can be applied to any total recursive function.
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| 2. | The second part of the first recursion theorem follows from the first part.
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| 3. | To apply the first recursion theorem, the recursion equations must first be recast as a recursive operator.
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| 4. | A classic example using the second recursion theorem is the function Q ( x, y ) = x.
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| 5. | Quines are possible in any Turing complete programming language, as a direct consequence of Kleene's recursion theorem.
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| 6. | Like the second recursion theorem, the first recursion theorem can be used to obtain functions satisfying systems of recursion equations.
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| 7. | Like the second recursion theorem, the first recursion theorem can be used to obtain functions satisfying systems of recursion equations.
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| 8. | The same definition of recursive function can be given, in computability theory, by applying Kleene's recursion theorem.
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| 9. | Compared to the second recursion theorem, the first recursion theorem produces a stronger conclusion but only when narrower hypotheses are satisfied.
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| 10. | Compared to the second recursion theorem, the first recursion theorem produces a stronger conclusion but only when narrower hypotheses are satisfied.
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