YES

We show the termination of the TRS R:

  bin(x,|0|()) -> s(|0|())
  bin(|0|(),s(y)) -> |0|()
  bin(s(x),s(y)) -> +(bin(x,s(y)),bin(x,y))

-- SCC decomposition.

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

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

and R consists of:

r1: bin(x,|0|()) -> s(|0|())
r2: bin(|0|(),s(y)) -> |0|()
r3: bin(s(x),s(y)) -> +(bin(x,s(y)),bin(x,y))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

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

and R consists of:

r1: bin(x,|0|()) -> s(|0|())
r2: bin(|0|(),s(y)) -> |0|()
r3: bin(s(x),s(y)) -> +(bin(x,s(y)),bin(x,y))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        bin#_A(x1,x2) = x1 + x2
        s_A(x1) = x1 + 1
    
    2. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        bin#_A(x1,x2) = x1
        s_A(x1) = 1
    

The next rules are strictly ordered:

  p1, p2

We remove them from the problem.  Then no dependency pair remains.