YES

We show the termination of the TRS R:

  p(m,n,s(r)) -> p(m,r,n)
  p(m,s(n),|0|()) -> p(|0|(),n,m)
  p(m,|0|(),|0|()) -> m

-- SCC decomposition.

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

p1: p#(m,n,s(r)) -> p#(m,r,n)
p2: p#(m,s(n),|0|()) -> p#(|0|(),n,m)

and R consists of:

r1: p(m,n,s(r)) -> p(m,r,n)
r2: p(m,s(n),|0|()) -> p(|0|(),n,m)
r3: p(m,|0|(),|0|()) -> m

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: p#(m,n,s(r)) -> p#(m,r,n)
p2: p#(m,s(n),|0|()) -> p#(|0|(),n,m)

and R consists of:

r1: p(m,n,s(r)) -> p(m,r,n)
r2: p(m,s(n),|0|()) -> p(|0|(),n,m)
r3: p(m,|0|(),|0|()) -> m

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          p#_A(x1,x2,x3) = max{x1 + 2, x2 - 1, x3 - 1}
          s_A(x1) = x1 + 4
          |0|_A = 0
    
      precedence:
      
        p# = s = |0|
      
      partial status:
    
        pi(p#) = []
        pi(s) = []
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          p#_A(x1,x2,x3) = 0
          s_A(x1) = x1 + 3
          |0|_A = 2
    
      precedence:
      
        s = |0| > p#
      
      partial status:
    
        pi(p#) = []
        pi(s) = [1]
        pi(|0|) = []
    

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: p#(m,n,s(r)) -> p#(m,r,n)

and R consists of:

r1: p(m,n,s(r)) -> p(m,r,n)
r2: p(m,s(n),|0|()) -> p(|0|(),n,m)
r3: p(m,|0|(),|0|()) -> m

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: p#(m,n,s(r)) -> p#(m,r,n)

and R consists of:

r1: p(m,n,s(r)) -> p(m,r,n)
r2: p(m,s(n),|0|()) -> p(|0|(),n,m)
r3: p(m,|0|(),|0|()) -> m

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          p#_A(x1,x2,x3) = max{x2 + 2, x3 + 1}
          s_A(x1) = max{3, x1 + 2}
    
      precedence:
      
        p# = s
      
      partial status:
    
        pi(p#) = [3]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          p#_A(x1,x2,x3) = max{0, x3 - 1}
          s_A(x1) = x1
    
      precedence:
      
        p# = s
      
      partial status:
    
        pi(p#) = []
        pi(s) = [1]
    

The next rules are strictly ordered:

  p1

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