YES
We show the termination of the TRS R:

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


-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1}
  {p2}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        g#_A(x1) = max{5, x1}
        c_A(x1,x2) = max{5, x2}
        s_A(x1) = x1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          g#_A(x1) = max{0, x1 - 13}
          c_A(x1,x2) = max{12, x1 - 5, x2 - 3}
          s_A(x1) = max{6, x1}
          g#!_A(x1) = x1 + 7
          c!_A(x1,x2) = max{7, x1 - 15, x2 - 8}
          s!_A(x1) = max{5, x1 - 16}
  
    partial status:
  
      pi(g#) = [1]
      pi(c) = [2]
      pi(s) = [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#(c(s(x),y)) -> f#(c(x,s(y)))

and R consists of:

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

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f#_A(x1) = x1 + 7
        c_A(x1,x2) = max{1, x1 - 8}
        s_A(x1) = max{28, x1 + 11}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          f#_A(x1) = max{0, x1 - 22}
          c_A(x1,x2) = max{x1 + 21, x2 + 2}
          s_A(x1) = max{2, x1}
          f#!_A(x1) = max{1, x1 - 15}
          c!_A(x1,x2) = max{9, x1 - 10, x2 - 22}
          s!_A(x1) = max{10, x1 - 37}
  
    partial status:
  
      pi(f#) = [1]
      pi(c) = []
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.