YES

We show the termination of the TRS R:

  active(c()) -> mark(f(g(c())))
  active(f(g(X))) -> mark(g(X))
  mark(c()) -> active(c())
  mark(f(X)) -> active(f(X))
  mark(g(X)) -> active(g(X))
  f(mark(X)) -> f(X)
  f(active(X)) -> f(X)
  g(mark(X)) -> g(X)
  g(active(X)) -> g(X)

-- SCC decomposition.

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

p1: active#(c()) -> mark#(f(g(c())))
p2: active#(c()) -> f#(g(c()))
p3: active#(c()) -> g#(c())
p4: active#(f(g(X))) -> mark#(g(X))
p5: mark#(c()) -> active#(c())
p6: mark#(f(X)) -> active#(f(X))
p7: mark#(g(X)) -> active#(g(X))
p8: f#(mark(X)) -> f#(X)
p9: f#(active(X)) -> f#(X)
p10: g#(mark(X)) -> g#(X)
p11: g#(active(X)) -> g#(X)

and R consists of:

r1: active(c()) -> mark(f(g(c())))
r2: active(f(g(X))) -> mark(g(X))
r3: mark(c()) -> active(c())
r4: mark(f(X)) -> active(f(X))
r5: mark(g(X)) -> active(g(X))
r6: f(mark(X)) -> f(X)
r7: f(active(X)) -> f(X)
r8: g(mark(X)) -> g(X)
r9: g(active(X)) -> g(X)

The estimated dependency graph contains the following SCCs:

  {p1, p4, p5, p6, p7}
  {p8, p9}
  {p10, p11}


-- Reduction pair.

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

p1: active#(c()) -> mark#(f(g(c())))
p2: mark#(f(X)) -> active#(f(X))
p3: active#(f(g(X))) -> mark#(g(X))
p4: mark#(g(X)) -> active#(g(X))
p5: mark#(c()) -> active#(c())

and R consists of:

r1: active(c()) -> mark(f(g(c())))
r2: active(f(g(X))) -> mark(g(X))
r3: mark(c()) -> active(c())
r4: mark(f(X)) -> active(f(X))
r5: mark(g(X)) -> active(g(X))
r6: f(mark(X)) -> f(X)
r7: f(active(X)) -> f(X)
r8: g(mark(X)) -> g(X)
r9: g(active(X)) -> g(X)

The set of usable rules consists of

  r6, r7, r8, r9

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        active#_A(x1) = ((0,1),(1,1)) x1
        c_A() = (5,7)
        mark#_A(x1) = ((1,0),(1,1)) x1 + (3,4)
        f_A(x1) = ((0,1),(0,1)) x1 + (0,2)
        g_A(x1) = (1,3)
        mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
        active_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        active#_A(x1) = (1,1)
        c_A() = (1,1)
        mark#_A(x1) = (0,0)
        f_A(x1) = (1,0)
        g_A(x1) = (0,1)
        mark_A(x1) = ((1,1),(1,1)) x1
        active_A(x1) = ((1,1),(1,1)) x1
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5

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

-- Reduction pair.

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

p1: f#(mark(X)) -> f#(X)
p2: f#(active(X)) -> f#(X)

and R consists of:

r1: active(c()) -> mark(f(g(c())))
r2: active(f(g(X))) -> mark(g(X))
r3: mark(c()) -> active(c())
r4: mark(f(X)) -> active(f(X))
r5: mark(g(X)) -> active(g(X))
r6: f(mark(X)) -> f(X)
r7: f(active(X)) -> f(X)
r8: g(mark(X)) -> g(X)
r9: g(active(X)) -> g(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        f#_A(x1) = ((1,1),(1,1)) x1
        mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
        active_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        f#_A(x1) = ((1,1),(0,1)) x1
        mark_A(x1) = x1 + (1,1)
        active_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    

The next rules are strictly ordered:

  p1, p2
  r1, r2, r3, r4, r5, r6, r7, r8, r9

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

-- Reduction pair.

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

p1: g#(mark(X)) -> g#(X)
p2: g#(active(X)) -> g#(X)

and R consists of:

r1: active(c()) -> mark(f(g(c())))
r2: active(f(g(X))) -> mark(g(X))
r3: mark(c()) -> active(c())
r4: mark(f(X)) -> active(f(X))
r5: mark(g(X)) -> active(g(X))
r6: f(mark(X)) -> f(X)
r7: f(active(X)) -> f(X)
r8: g(mark(X)) -> g(X)
r9: g(active(X)) -> g(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        g#_A(x1) = ((0,1),(1,0)) x1
        mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
        active_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        g#_A(x1) = (0,0)
        mark_A(x1) = (1,1)
        active_A(x1) = (1,1)
    

The next rules are strictly ordered:

  p1, p2

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