YES
We show the termination of the TRS R:

  average(s(x),y) -> average(x,s(y))
  average(x,s(s(s(y)))) -> s(average(s(x),y))
  average(|0|(),|0|()) -> |0|()
  average(|0|(),s(|0|())) -> |0|()
  average(|0|(),s(s(|0|()))) -> s(|0|())


-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: average#(s(x),y) -> average#(x,s(y))
p2: average#(x,s(s(s(y)))) -> average#(s(x),y)

and R consists of:

r1: average(s(x),y) -> average(x,s(y))
r2: average(x,s(s(s(y)))) -> s(average(s(x),y))
r3: average(|0|(),|0|()) -> |0|()
r4: average(|0|(),s(|0|())) -> |0|()
r5: average(|0|(),s(s(|0|()))) -> s(|0|())

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: average#(s(x),y) -> average#(x,s(y))
p2: average#(x,s(s(s(y)))) -> average#(s(x),y)

and R consists of:

r1: average(s(x),y) -> average(x,s(y))
r2: average(x,s(s(s(y)))) -> s(average(s(x),y))
r3: average(|0|(),|0|()) -> |0|()
r4: average(|0|(),s(|0|())) -> |0|()
r5: average(|0|(),s(s(|0|()))) -> s(|0|())

The set of usable rules consists of

  r3, r4, r5

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          average#_A(x1,x2) = x1 + x2 + 3
          s_A(x1) = x1 + 1
          average_A(x1,x2) = x1 + 3
          |0|_A() = 1
  
    base order 2:
  
      precedence:
      
        average# > s = average > |0|
      
  
    partial status:
  
      pi(average#) = [1]
      pi(s) = []
      pi(average) = [1]
      pi(|0|) = []

The following rules are removed:

  p1



-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: average#(x,s(s(s(y)))) -> average#(s(x),y)

and R consists of:

r1: average(s(x),y) -> average(x,s(y))
r2: average(x,s(s(s(y)))) -> s(average(s(x),y))
r3: average(|0|(),|0|()) -> |0|()
r4: average(|0|(),s(|0|())) -> |0|()
r5: average(|0|(),s(s(|0|()))) -> s(|0|())

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: average#(x,s(s(s(y)))) -> average#(s(x),y)

and R consists of:

r1: average(s(x),y) -> average(x,s(y))
r2: average(x,s(s(s(y)))) -> s(average(s(x),y))
r3: average(|0|(),|0|()) -> |0|()
r4: average(|0|(),s(|0|())) -> |0|()
r5: average(|0|(),s(s(|0|()))) -> s(|0|())

The set of usable rules consists of

  r3, r4, r5

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          average#_A(x1,x2) = x1 + x2 + 4
          s_A(x1) = x1 + 3
          average_A(x1,x2) = 5
          |0|_A() = 1
  
    base order 2:
  
      precedence:
      
        average# = s > average > |0|
      
  
    partial status:
  
      pi(average#) = []
      pi(s) = [1]
      pi(average) = []
      pi(|0|) = []

The following rules are removed:

  p1

Then no dependency pair remains.