YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

p1: active#(f(f(X))) -> mark#(c(f(g(f(X)))))
p2: active#(f(f(X))) -> c#(f(g(f(X))))
p3: active#(f(f(X))) -> f#(g(f(X)))
p4: active#(f(f(X))) -> g#(f(X))
p5: active#(c(X)) -> mark#(d(X))
p6: active#(c(X)) -> d#(X)
p7: active#(h(X)) -> mark#(c(d(X)))
p8: active#(h(X)) -> c#(d(X))
p9: active#(h(X)) -> d#(X)
p10: mark#(f(X)) -> active#(f(mark(X)))
p11: mark#(f(X)) -> f#(mark(X))
p12: mark#(f(X)) -> mark#(X)
p13: mark#(c(X)) -> active#(c(X))
p14: mark#(g(X)) -> active#(g(X))
p15: mark#(d(X)) -> active#(d(X))
p16: mark#(h(X)) -> active#(h(mark(X)))
p17: mark#(h(X)) -> h#(mark(X))
p18: mark#(h(X)) -> mark#(X)
p19: f#(mark(X)) -> f#(X)
p20: f#(active(X)) -> f#(X)
p21: c#(mark(X)) -> c#(X)
p22: c#(active(X)) -> c#(X)
p23: g#(mark(X)) -> g#(X)
p24: g#(active(X)) -> g#(X)
p25: d#(mark(X)) -> d#(X)
p26: d#(active(X)) -> d#(X)
p27: h#(mark(X)) -> h#(X)
p28: h#(active(X)) -> h#(X)

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1, p5, p7, p10, p12, p13, p14, p15, p16, p18}
  {p21, p22}
  {p19, p20}
  {p23, p24}
  {p25, p26}
  {p27, p28}


-- Reduction pair.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1
          f_A(x1) = x1 + 1
          mark#_A(x1) = x1
          c_A(x1) = x1
          g_A(x1) = x1
          h_A(x1) = x1 + 1
          mark_A(x1) = x1
          d_A(x1) = x1
          active_A(x1) = x1
  
    precedence:
    
      active# = f = mark# = c = g = mark = active > h = d
    
    partial status:
  
      pi(active#) = []
      pi(f) = []
      pi(mark#) = []
      pi(c) = []
      pi(g) = []
      pi(h) = [1]
      pi(mark) = []
      pi(d) = []
      pi(active) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1 + 1
          f_A(x1) = x1 + 2
          mark#_A(x1) = x1 + 1
          c_A(x1) = x1
          g_A(x1) = x1
          mark_A(x1) = x1
          d_A(x1) = x1
          h_A(x1) = x1 + 2
          active_A(x1) = x1
  
    precedence:
    
      active# = f = mark# = g = d > h > c = mark = active
    
    partial status:
  
      pi(active#) = []
      pi(f) = []
      pi(mark#) = []
      pi(c) = []
      pi(g) = []
      pi(mark) = []
      pi(d) = []
      pi(h) = []
      pi(active) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8}


-- Reduction pair.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = 0
          f_A(x1) = 0
          mark#_A(x1) = x1
          c_A(x1) = x1
          g_A(x1) = 1
          h_A(x1) = 3
          mark_A(x1) = x1 + 2
          d_A(x1) = 0
          active_A(x1) = x1 + 2
  
    precedence:
    
      f > c = h > active# = mark# = g = mark = d = active
    
    partial status:
  
      pi(active#) = []
      pi(f) = []
      pi(mark#) = []
      pi(c) = [1]
      pi(g) = []
      pi(h) = []
      pi(mark) = []
      pi(d) = []
      pi(active) = []

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

p1: active#(f(f(X))) -> mark#(c(f(g(f(X)))))
p2: mark#(h(X)) -> active#(h(mark(X)))
p3: active#(h(X)) -> mark#(c(d(X)))
p4: mark#(d(X)) -> active#(d(X))
p5: active#(c(X)) -> mark#(d(X))
p6: mark#(c(X)) -> active#(c(X))
p7: mark#(f(X)) -> active#(f(mark(X)))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

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

p1: active#(f(f(X))) -> mark#(c(f(g(f(X)))))
p2: mark#(f(X)) -> active#(f(mark(X)))
p3: active#(c(X)) -> mark#(d(X))
p4: mark#(c(X)) -> active#(c(X))
p5: active#(h(X)) -> mark#(c(d(X)))
p6: mark#(d(X)) -> active#(d(X))
p7: mark#(h(X)) -> active#(h(mark(X)))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1
          f_A(x1) = 9
          mark#_A(x1) = x1 + 1
          c_A(x1) = 5
          g_A(x1) = 1
          mark_A(x1) = x1 + 3
          d_A(x1) = 1
          h_A(x1) = 9
          active_A(x1) = x1 + 2
  
    precedence:
    
      f = g = mark > mark# = c = d = h = active > active#
    
    partial status:
  
      pi(active#) = []
      pi(f) = []
      pi(mark#) = [1]
      pi(c) = []
      pi(g) = []
      pi(mark) = [1]
      pi(d) = []
      pi(h) = []
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: active#(f(f(X))) -> mark#(c(f(g(f(X)))))
p2: active#(c(X)) -> mark#(d(X))
p3: mark#(c(X)) -> active#(c(X))
p4: active#(h(X)) -> mark#(c(d(X)))
p5: mark#(d(X)) -> active#(d(X))
p6: mark#(h(X)) -> active#(h(mark(X)))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: active#(f(f(X))) -> mark#(c(f(g(f(X)))))
p2: mark#(h(X)) -> active#(h(mark(X)))
p3: active#(h(X)) -> mark#(c(d(X)))
p4: mark#(d(X)) -> active#(d(X))
p5: active#(c(X)) -> mark#(d(X))
p6: mark#(c(X)) -> active#(c(X))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1 + 3
          f_A(x1) = 3
          mark#_A(x1) = 5
          c_A(x1) = 2
          g_A(x1) = 2
          h_A(x1) = 2
          mark_A(x1) = 4
          d_A(x1) = 1
          active_A(x1) = 4
  
    precedence:
    
      active# = f = mark# > c = g = h = d > mark = active
    
    partial status:
  
      pi(active#) = []
      pi(f) = []
      pi(mark#) = []
      pi(c) = []
      pi(g) = []
      pi(h) = []
      pi(mark) = []
      pi(d) = []
      pi(active) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1
          f_A(x1) = x1 + 10
          mark#_A(x1) = x1 + 3
          c_A(x1) = x1 + 5
          g_A(x1) = 1
          d_A(x1) = x1 + 1
          h_A(x1) = x1 + 10
          mark_A(x1) = x1 + 2
          active_A(x1) = x1
  
    precedence:
    
      active# = f = mark# = c = g = d = h = mark = active
    
    partial status:
  
      pi(active#) = [1]
      pi(f) = []
      pi(mark#) = [1]
      pi(c) = []
      pi(g) = []
      pi(d) = []
      pi(h) = []
      pi(mark) = []
      pi(active) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(c(X)) -> active#(c(X))
p2: active#(c(X)) -> mark#(d(X))
p3: mark#(h(X)) -> active#(h(mark(X)))
p4: active#(h(X)) -> mark#(c(d(X)))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

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

p1: mark#(c(X)) -> active#(c(X))
p2: active#(h(X)) -> mark#(c(d(X)))
p3: mark#(h(X)) -> active#(h(mark(X)))
p4: active#(c(X)) -> mark#(d(X))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          c_A(x1) = 2
          active#_A(x1) = 2
          h_A(x1) = x1 + 4
          d_A(x1) = 0
          mark_A(x1) = x1 + 1
          active_A(x1) = x1
          f_A(x1) = 3
          g_A(x1) = x1
  
    precedence:
    
      g > h > mark# = c = active# = d = mark = active = f
    
    partial status:
  
      pi(mark#) = []
      pi(c) = []
      pi(active#) = []
      pi(h) = []
      pi(d) = []
      pi(mark) = []
      pi(active) = []
      pi(f) = []
      pi(g) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(c(X)) -> active#(c(X))
p2: active#(h(X)) -> mark#(c(d(X)))
p3: mark#(h(X)) -> active#(h(mark(X)))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: mark#(c(X)) -> active#(c(X))
p2: active#(h(X)) -> mark#(c(d(X)))
p3: mark#(h(X)) -> active#(h(mark(X)))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + 1
          c_A(x1) = x1
          active#_A(x1) = x1
          h_A(x1) = x1 + 2
          d_A(x1) = x1
          mark_A(x1) = x1
          active_A(x1) = x1
          f_A(x1) = x1 + 1
          g_A(x1) = x1
  
    precedence:
    
      mark# > active# = f > c > h = d = mark = active > g
    
    partial status:
  
      pi(mark#) = [1]
      pi(c) = []
      pi(active#) = []
      pi(h) = []
      pi(d) = []
      pi(mark) = [1]
      pi(active) = [1]
      pi(f) = []
      pi(g) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: active#(h(X)) -> mark#(c(d(X)))
p2: mark#(h(X)) -> active#(h(mark(X)))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(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: active#(h(X)) -> mark#(c(d(X)))
p2: mark#(h(X)) -> active#(h(mark(X)))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1
          h_A(x1) = 0
          mark#_A(x1) = x1
          c_A(x1) = 0
          d_A(x1) = 1
          mark_A(x1) = 0
          active_A(x1) = 0
          f_A(x1) = 0
          g_A(x1) = 0
  
    precedence:
    
      d = f > mark = active > h > mark# > active# > c = g
    
    partial status:
  
      pi(active#) = [1]
      pi(h) = []
      pi(mark#) = [1]
      pi(c) = []
      pi(d) = []
      pi(mark) = []
      pi(active) = []
      pi(f) = []
      pi(g) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(h(X)) -> active#(h(mark(X)))

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          c#_A(x1) = x1
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      c# = mark = active
    
    partial status:
  
      pi(c#) = [1]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: c#(active(X)) -> c#(X)

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: c#(active(X)) -> c#(X)

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          c#_A(x1) = x1
          active_A(x1) = x1 + 1
  
    precedence:
    
      c# = active
    
    partial status:
  
      pi(c#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

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(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          f#_A(x1) = x1
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      f# = mark = active
    
    partial status:
  
      pi(f#) = [1]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          f#_A(x1) = x1
          active_A(x1) = x1 + 1
  
    precedence:
    
      f# = active
    
    partial status:
  
      pi(f#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

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(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          g#_A(x1) = x1
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      g# = mark = active
    
    partial status:
  
      pi(g#) = [1]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(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#(active(X)) -> g#(X)

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          g#_A(x1) = x1
          active_A(x1) = x1 + 1
  
    precedence:
    
      g# = active
    
    partial status:
  
      pi(g#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

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: d#(mark(X)) -> d#(X)
p2: d#(active(X)) -> d#(X)

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          d#_A(x1) = x1
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      d# = mark = active
    
    partial status:
  
      pi(d#) = [1]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: d#(active(X)) -> d#(X)

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: d#(active(X)) -> d#(X)

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          d#_A(x1) = x1
          active_A(x1) = x1 + 1
  
    precedence:
    
      d# = active
    
    partial status:
  
      pi(d#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

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: h#(mark(X)) -> h#(X)
p2: h#(active(X)) -> h#(X)

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          h#_A(x1) = x1
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      h# = mark = active
    
    partial status:
  
      pi(h#) = [1]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: h#(active(X)) -> h#(X)

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: h#(active(X)) -> h#(X)

and R consists of:

r1: active(f(f(X))) -> mark(c(f(g(f(X)))))
r2: active(c(X)) -> mark(d(X))
r3: active(h(X)) -> mark(c(d(X)))
r4: mark(f(X)) -> active(f(mark(X)))
r5: mark(c(X)) -> active(c(X))
r6: mark(g(X)) -> active(g(X))
r7: mark(d(X)) -> active(d(X))
r8: mark(h(X)) -> active(h(mark(X)))
r9: f(mark(X)) -> f(X)
r10: f(active(X)) -> f(X)
r11: c(mark(X)) -> c(X)
r12: c(active(X)) -> c(X)
r13: g(mark(X)) -> g(X)
r14: g(active(X)) -> g(X)
r15: d(mark(X)) -> d(X)
r16: d(active(X)) -> d(X)
r17: h(mark(X)) -> h(X)
r18: h(active(X)) -> h(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          h#_A(x1) = x1
          active_A(x1) = x1 + 1
  
    precedence:
    
      h# = active
    
    partial status:
  
      pi(h#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

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