YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

p1: +#(x,s(y)) -> s#(+(x,y))
p2: +#(x,s(y)) -> +#(x,y)
p3: s#(+(|0|(),y)) -> s#(y)

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p2}
  {p3}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            +#_A(x1,x2) = ((0,0),(1,0)) x2 + (1,2)
            s_A(x1) = x1 + (2,1)
    
      precedence:
      
        +# = s
      
      partial status:
    
        pi(+#) = []
        pi(s) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            +#_A(x1,x2) = (0,0)
            s_A(x1) = (1,1)
    
      precedence:
      
        s > +#
      
      partial status:
    
        pi(+#) = []
        pi(s) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: s#(+(|0|(),y)) -> s#(y)

and R consists of:

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

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            s#_A(x1) = x1 + (2,2)
            +_A(x1,x2) = ((1,0),(0,0)) x2 + (3,1)
            |0|_A() = (1,1)
    
      precedence:
      
        |0| > + > s#
      
      partial status:
    
        pi(s#) = [1]
        pi(+) = []
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            s#_A(x1) = (1,1)
            +_A(x1,x2) = (2,2)
            |0|_A() = (2,2)
    
      precedence:
      
        |0| > s# > +
      
      partial status:
    
        pi(s#) = []
        pi(+) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p1

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