YES
We show the termination of the TRS R:

  minus(x,|0|()) -> x
  minus(s(x),s(y)) -> minus(x,y)
  f(|0|()) -> s(|0|())
  f(s(x)) -> minus(s(x),g(f(x)))
  g(|0|()) -> |0|()
  g(s(x)) -> minus(s(x),f(g(x)))


-- SCC decomposition.

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

p1: minus#(s(x),s(y)) -> minus#(x,y)
p2: f#(s(x)) -> minus#(s(x),g(f(x)))
p3: f#(s(x)) -> g#(f(x))
p4: f#(s(x)) -> f#(x)
p5: g#(s(x)) -> minus#(s(x),f(g(x)))
p6: g#(s(x)) -> f#(g(x))
p7: g#(s(x)) -> g#(x)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: f(|0|()) -> s(|0|())
r4: f(s(x)) -> minus(s(x),g(f(x)))
r5: g(|0|()) -> |0|()
r6: g(s(x)) -> minus(s(x),f(g(x)))

The estimated dependency graph contains the following SCCs:

  {p3, p4, p6, p7}
  {p1}


-- Reduction pair.

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

p1: g#(s(x)) -> g#(x)
p2: g#(s(x)) -> f#(g(x))
p3: f#(s(x)) -> f#(x)
p4: f#(s(x)) -> g#(f(x))

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: f(|0|()) -> s(|0|())
r4: f(s(x)) -> minus(s(x),g(f(x)))
r5: g(|0|()) -> |0|()
r6: g(s(x)) -> minus(s(x),f(g(x)))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        g#_A(x1) = max{1, x1 - 38}
        s_A(x1) = max{37, x1 + 28}
        f#_A(x1) = max{1, x1 - 26}
        g_A(x1) = max{24, x1 + 15}
        f_A(x1) = max{37, x1 + 21}
        minus_A(x1,x2) = max{50, x1 + 14, x2 + 5}
        |0|_A = 9
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          g#_A(x1) = max{0, x1 - 39}
          s_A(x1) = max{25, x1 + 23}
          f#_A(x1) = max{0, x1 - 27}
          g_A(x1) = max{14, x1 + 8}
          f_A(x1) = max{38, x1 + 7}
          minus_A(x1,x2) = max{33, x1 + 7, x2 - 47}
          |0|_A = 15
          g#!_A(x1) = max{1, x1 - 39}
          s!_A(x1) = max{2, x1 - 32}
          f#!_A(x1) = 0
          g!_A(x1) = max{2, x1 - 39}
          f!_A(x1) = max{2, x1 - 55}
          minus!_A(x1,x2) = max{3, x1 - 38, x2 - 31}
          |0|!_A = 0
  
    partial status:
  
      pi(g#) = []
      pi(s) = [1]
      pi(f#) = []
      pi(g) = [1]
      pi(f) = [1]
      pi(minus) = [1, 2]
      pi(|0|) = []

The following rules are removed:

  p2, p3, p4



-- SCC decomposition.

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

p1: g#(s(x)) -> g#(x)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: f(|0|()) -> s(|0|())
r4: f(s(x)) -> minus(s(x),g(f(x)))
r5: g(|0|()) -> |0|()
r6: g(s(x)) -> minus(s(x),f(g(x)))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: g#(s(x)) -> g#(x)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: f(|0|()) -> s(|0|())
r4: f(s(x)) -> minus(s(x),g(f(x)))
r5: g(|0|()) -> |0|()
r6: g(s(x)) -> minus(s(x),f(g(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{2, x1 - 2}
        s_A(x1) = max{6, x1 + 1}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          g#_A(x1) = 0
          s_A(x1) = max{10, x1 - 38}
          g#!_A(x1) = max{1, x1 - 13}
          s!_A(x1) = max{5, x1 - 12}
  
    partial status:
  
      pi(g#) = []
      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: minus#(s(x),s(y)) -> minus#(x,y)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: f(|0|()) -> s(|0|())
r4: f(s(x)) -> minus(s(x),g(f(x)))
r5: g(|0|()) -> |0|()
r6: g(s(x)) -> minus(s(x),f(g(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:
      
        minus#_A(x1,x2) = x1 + 9
        s_A(x1) = x1 + 10
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus#_A(x1,x2) = 0
          s_A(x1) = max{1, x1 - 13}
          minus#!_A(x1,x2) = x2 + 17
          s!_A(x1) = max{2, x1 - 3}
  
    partial status:
  
      pi(minus#) = [1]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.