YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

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

and R consists of:

r1: +(|0|(),y) -> y
r2: +(s(x),y) -> s(+(x,y))
r3: -(|0|(),y) -> |0|()
r4: -(x,|0|()) -> x
r5: -(s(x),s(y)) -> -(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^2
      order: standard order
      interpretations:
        +#_A(x1,x2) = ((1,1),(1,1)) x1
        s_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        +#_A(x1,x2) = x1
        s_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        +#_A(x1,x2) = ((1,1),(1,1)) x1
        s_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    

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(x),s(y)) -> -#(x,y)

and R consists of:

r1: +(|0|(),y) -> y
r2: +(s(x),y) -> s(+(x,y))
r3: -(|0|(),y) -> |0|()
r4: -(x,|0|()) -> x
r5: -(s(x),s(y)) -> -(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^2
      order: standard order
      interpretations:
        -#_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,0),(1,1)) x2
        s_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        -#_A(x1,x2) = ((1,0),(1,1)) x1
        s_A(x1) = ((1,1),(0,1)) x1 + (0,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        -#_A(x1,x2) = ((1,0),(1,1)) x1
        s_A(x1) = ((1,1),(1,0)) x1 + (0,1)
    

The next rules are strictly ordered:

  p1

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