scala - Immutable Covariant Priority Queues with Higher-kinded Types -

i define different implementations of immutable covariant priority queues. first of all, of implementations provide following interface:

trait pqlike[t, this[_]] {   def isempty : boolean   def enqueue[e >: t](x : e)(implicit ord : ordering[e]) : this[e]   def dequeue[t] : this[t]   def first : t } 

so this stands implementation's type constructor , t type of elements stored in queue. instance, if linearpriorityqueue[t] implementation of priority queue, want method enqueue linearpriorityqueue return linearpriorityqueue.

i define priorityqueue[t] trait (or abstract class) implementations of priority queues can typed (in addition specific type) priorityqueues.

i tried this:

trait priorityqueue[+t] extends pqlike[t, _ <: pqlike[_,_]]  trait linearpriorityqueue[+t] extends priorityqueue[t] pqlike[t,linearpriorityqueue]      object epq extends linearpriorityqueue[nothing] {   def isempty = true   def first = throw new priorityqueueexception("first on empty queue")   def dequeue = throw new priorityqueueexception("dequeue on empty queue")   def enqueue[t](x : t) = npq(x,this) }  case class npq[+t](x : t, q : linearpriorityqueue[t]) extends linearpriorityqueue[t] {   def isempty = false   def first = x   def dequeue = q   def enqueue[e >: t](y : e) = npq(y,this) // not real implementation } 

but i'm getting type errors related kinds cannot fix.

thanks

edit: ok, have made progress. compiling , working expected, although don't know if solution simplified further (note have done renamings):

trait ispriorityqueue[+t, +repr[+x] <: ispriorityqueue[x,repr]] {   def isempty : boolean   def first : t   def enqueue[e >: t](x : e)(implicit ord : ordering[e]) : repr[e]   def dequeue : repr[t] }  trait priorityqueue[+t] extends ispriorityqueue[t, priorityqueue]  trait linearpriorityqueue[+t] extends priorityqueue[t] ispriorityqueue[t,linearpriorityqueue]  object emptylpq extends linearpriorityqueue[nothing] {   def isempty = true   def first = throw new priorityqueueexception("first on empty queue")   def dequeue = throw new priorityqueueexception("dequeue on empty queue")   def enqueue[e](x : e)(implicit ord : ordering[e]) = nodelpq(x,this) }  case class nodelpq[+t](hd : t, tl : linearpriorityqueue[t]) extends linearpriorityqueue[t] {   def isempty = false   def first = hd   def dequeue = tl   def enqueue[e >: t](x : e)(implicit ord : ordering[e]) =     if(ord.compare(x,hd)<0)       nodelpq(x,this)     else       nodelpq(hd,tl.enqueue(x)) } 

i comments on code , possible simplifications. thinking maybe trait priorityqueue defined using existential type (a priorityqueue extends ispriorityqueue representation) haven't been able express that.

edit: 1 have comments on solution?