The idea is to use some appropriate coding. A primitive recursive encoding of pairs is an encoding $\mathbb{N}^2 \to \mathbb{N}$, denoted by $\langle x,y \rangle$, such that the following functions are recursive: $e(x,y) = \langle x,y \rangle$, $p_1(\langle x,y \rangle) = x$, $p_2(\langle x,y \ra...
thanks Dr. +1. I Works on these to understand it. thanks Dr.
Dear Dr. I'm sorry for wasting your time. I try more links and do with myself but the second one is very hard. I read this article researchgate.net/profile/Varmo_Vene/publication/… but I couldn't get it. would you please provide hint for second one? I think it's similar to first but couldn't solve it up to yet.
I try more after +15 hours. nice solution is mentioned by you for first type. anyway thanks. maybe the second one is completely different, thanks again.
The second one requires more sophisticated encoding, which can be done in many ways. The question already assumes some encoding of sequences $[\cdots]$, and you need one that supports some simple operations that will allow you to implement the course-of-values recursion. Instead of keeping track of a pair $\langle f_1(x), f_1(x+1) \rangle$ as in (1), you keep track of the entire sequence $[f_2(0),\ldots,f_2(x)]$.
so because I didn't see any example I couldent understand it. It's very hard. I have a three proof on my note about it, but really your teaching is so interesting, I get first one completely, is it possible complete for type two?
The second one is very similar. You should be able to work it out yourself. Unless you solve exercises on your own you will never really understand the material.
Dr, you are right. I'm a TA and know it. but this complicated one is boring for me for lot's of effort to understand. thanks anyway, I accept it but If you prefer please add some direct hint for me. thanks for your valuable and value to answer me, Dr.
Infact my trouble is how we get show entire sequence is P.R?
I know f(n)=[f(0), f(1),...f(n-1)] n!=0 for f(n)=g(n,f(n)) is P.R. but I didn't understand how take it for type 2/
maybe I think it's solve for $f_2(0)=c$ but not for this mentioned in the type 2.
Dr Would you please explain more for me? How I can overcome to this challenge, specially when comes to entire sequence?
The solution to part 2 is very similar to the solution to part 1, the only difference being that the encoding in part 2 is more complicated. There are no conceptual difficulties, only technical ones.
Dr This is not assignment, I see another answer but it's not satisfy me, because you solve the first one very nice and creative, two days I didn't sleep to learn it. I'm a TA on university. Just I want to learn. the purpose of this site is to learning not doing assignment. I try more but I confused. please kindly learn more to me. thanks again Dr. for wasting your time.
If it's not an assignment then there's certainly no reason for me to spell out the solution. If you care so much about a proof, I'm sure it can be found in textbooks. Also, the solution on math.se is the same as mine and also includes a solution for part 2.
Dr. I do lots of search on some books, the topic course of value recursion is not have an enough example. I don't say give me a solution, I say your hint is not enough to reach the solution for me. the other link on MATH mentioned the solution, If I want solution, I didn't try to ask you for getting more hint to reach solution.
Discussion between Yuval Filmus and L…
Discussion between Yuval Filmus and L…