YES

We show the termination of the relative TRS R/S:

  R:
  f(x,c(y)) -> f(x,s(f(y,y)))
  f(s(x),y) -> f(x,s(c(y)))

  S:
  rand(x) -> x
  rand(x) -> rand(s(x))

-- SCC decomposition.

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

p1: f#(x,c(y)) -> f#(x,s(f(y,y)))
p2: f#(x,c(y)) -> f#(y,y)
p3: f#(s(x),y) -> f#(x,s(c(y)))

and R consists of:

r1: f(x,c(y)) -> f(x,s(f(y,y)))
r2: f(s(x),y) -> f(x,s(c(y)))
r3: rand(x) -> x
r4: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  {p2}
  {p3}


-- Reduction pair.

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

p1: f#(x,c(y)) -> f#(y,y)

and R consists of:

r1: f(x,c(y)) -> f(x,s(f(y,y)))
r2: f(s(x),y) -> f(x,s(c(y)))
r3: rand(x) -> x
r4: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          f#_A(x1,x2) = ((1,0),(1,1)) x2 + (1,1)
          c_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          f_A(x1,x2) = ((1,0),(0,0)) x2 + (1,2)
          s_A(x1) = (0,1)
          rand_A(x1) = ((1,0),(1,0)) x1 + (1,2)
  
    precedence:
    
      c = rand > f# = f > s
    
    partial status:
  
      pi(f#) = [2]
      pi(c) = []
      pi(f) = []
      pi(s) = []
      pi(rand) = []

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: f#(s(x),y) -> f#(x,s(c(y)))

and R consists of:

r1: f(x,c(y)) -> f(x,s(f(y,y)))
r2: f(s(x),y) -> f(x,s(c(y)))
r3: rand(x) -> x
r4: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          f#_A(x1,x2) = x1
          s_A(x1) = x1
          c_A(x1) = (1,0)
          f_A(x1,x2) = ((1,0),(1,1)) x1 + (2,1)
          rand_A(x1) = ((1,0),(1,1)) x1 + (1,1)
  
    precedence:
    
      f# = s = c = f > rand
    
    partial status:
  
      pi(f#) = [1]
      pi(s) = [1]
      pi(c) = []
      pi(f) = []
      pi(rand) = []

The next rules are strictly ordered:

  p1

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