YES
We show the termination of the TRS R:

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


-- SCC decomposition.

Consider the 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)))

The estimated dependency graph contains the following SCCs:

  {p2}
  {p3}


-- Reduction pair.

Consider the 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)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            f#_A(x1,x2) = max{x1, x2 - 3}
            c_A(x1) = max{14, x1 + 3}
            f#!_A(x1,x2) = max{1, x1 - 13, x2 - 15}
            c!_A(x1) = max{2, x1 - 39}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            f#_A(x1,x2) = max{x1, x2 - 7}
            c_A(x1) = x1 + 7
            f#!_A(x1,x2) = max{3, x1 - 1, x2 - 3}
            c!_A(x1) = max{2, x1 - 2}
        
  
    partial status:
  
      pi(f#) = []
      pi(c) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

Consider the 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)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            f#_A(x1,x2) = max{0, x1 - 13, x2 - 11}
            s_A(x1) = max{10, x1 + 7}
            c_A(x1) = max{2, x1 - 7}
            f#!_A(x1,x2) = max{4, x1 - 11, x2 - 15}
            s!_A(x1) = max{1, x1 - 18}
            c!_A(x1) = max{3, x1 - 33}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            f#_A(x1,x2) = max{0, x1 - 15, x2 - 14}
            s_A(x1) = max{11, x1 + 8}
            c_A(x1) = max{5, x1 - 8}
            f#!_A(x1,x2) = max{4, x1 - 13, x2 - 17}
            s!_A(x1) = max{1, x1 - 14}
            c!_A(x1) = max{3, x1 - 23}
        
  
    partial status:
  
      pi(f#) = [1]
      pi(s) = [1]
      pi(c) = []

The following rules are removed:

  p1

Then no dependency pair remains.