# [isabelle-dev] Of lazy lists and friendly corecs

Manuel Eberl eberlm at in.tum.de
Fri Oct 7 14:22:19 CEST 2016

```Hallo,

I've been playing around with corec and friends in combination with the
lazy lists from "\$AFP/Coinductive/Coinductive_List" and encountered the
following problem:

Suppose I want a function that takes two (implicitly sorted) lazy lists
and merges them in the following fashion:

primcorec lmerge :: "int llist ⇒ int llist ⇒ int llist" where
"lmerge xs ys = (if lnull xs then ys
else if lnull ys then xs
else if lhd xs ≤ lhd ys then LCons (lhd xs) (lmerge
(ltl xs) ys)
else LCons (lhd ys) (lmerge xs (ltl ys)))"

I've still not fully understood what ‘friendly’ means, but I should be
very surprised if this function were not friendly. So I tried this:

friend_of_corec lmerge where
"lmerge xs ys = (if lnull xs ∧ lnull ys then LNil
else if lnull xs then LCons (lhd ys) (ltl ys)
else if lnull ys then LCons (lhd xs) (ltl xs)
else if lhd xs ≤ lhd ys then LCons (lhd xs) (lmerge
(ltl xs) ys)
else LCons (lhd ys) (lmerge xs (ltl ys)))"

Unfortunately, I get the following error:

exception TYPE raised (line 168 of "consts.ML"): Illegal type for
constant "Coinductive_List.llist.LNil" :: 'b

I now have the following questions:
1. Something like "if lnull xs then ys" works in primcorec, but in corec
and friend_of_corec I have to do odd things like "LCons (lhd ys) (ltl
ys)". Is that a known limitation?
2. What is the relationship between primcorec and friendliness? If I
have a function defined by primcorec, is the additional step with
friend_of_corec fundamentally necessary or is that just a technical
issue? (I would have naïvely assumed that primitively-corecursive
functions are always friendly or something like that)
3. What causes the above error and how do I get around it?

Cheers,

Manuel

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