YES
We show the termination of the TRS R:

  active(f(x)) -> mark(x)
  top(active(c())) -> top(mark(c()))
  top(mark(x)) -> top(check(x))
  check(f(x)) -> f(check(x))
  check(x) -> start(match(f(X()),x))
  match(f(x),f(y)) -> f(match(x,y))
  match(X(),x) -> proper(x)
  proper(c()) -> ok(c())
  proper(f(x)) -> f(proper(x))
  f(ok(x)) -> ok(f(x))
  start(ok(x)) -> found(x)
  f(found(x)) -> found(f(x))
  top(found(x)) -> top(active(x))
  active(f(x)) -> f(active(x))
  f(mark(x)) -> mark(f(x))


-- SCC decomposition.

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

p1: top#(active(c())) -> top#(mark(c()))
p2: top#(mark(x)) -> top#(check(x))
p3: top#(mark(x)) -> check#(x)
p4: check#(f(x)) -> f#(check(x))
p5: check#(f(x)) -> check#(x)
p6: check#(x) -> start#(match(f(X()),x))
p7: check#(x) -> match#(f(X()),x)
p8: check#(x) -> f#(X())
p9: match#(f(x),f(y)) -> f#(match(x,y))
p10: match#(f(x),f(y)) -> match#(x,y)
p11: match#(X(),x) -> proper#(x)
p12: proper#(f(x)) -> f#(proper(x))
p13: proper#(f(x)) -> proper#(x)
p14: f#(ok(x)) -> f#(x)
p15: f#(found(x)) -> f#(x)
p16: top#(found(x)) -> top#(active(x))
p17: top#(found(x)) -> active#(x)
p18: active#(f(x)) -> f#(active(x))
p19: active#(f(x)) -> active#(x)
p20: f#(mark(x)) -> f#(x)

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p16}
  {p5}
  {p19}
  {p10}
  {p13}
  {p14, p15, p20}


-- Reduction pair.

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

p1: top#(active(c())) -> top#(mark(c()))
p2: top#(mark(x)) -> top#(check(x))
p3: top#(found(x)) -> top#(active(x))

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r10, r11, r12, r13, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          top#_A(x1) = x1
          active_A(x1) = x1
          c_A() = 4
          mark_A(x1) = 3
          check_A(x1) = 3
          found_A(x1) = x1
          f_A(x1) = 3
          top_A(x1) = 2
          start_A(x1) = x1
          match_A(x1,x2) = x1
          X_A() = 1
          ok_A(x1) = x1
  
    base order 2:
  
      precedence:
      
        c = check > f = start = match > top = ok > found > active = X > mark > top#
      
  
    partial status:
  
      pi(top#) = []
      pi(active) = [1]
      pi(c) = []
      pi(mark) = []
      pi(check) = []
      pi(found) = []
      pi(f) = []
      pi(top) = []
      pi(start) = [1]
      pi(match) = []
      pi(X) = []
      pi(ok) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: top#(mark(x)) -> top#(check(x))
p2: top#(found(x)) -> top#(active(x))

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(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: top#(mark(x)) -> top#(check(x))
p2: top#(found(x)) -> top#(active(x))

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The set of usable rules consists of

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

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          top#_A(x1) = x1
          mark_A(x1) = x1 + 9
          check_A(x1) = x1 + 7
          found_A(x1) = x1 + 3
          active_A(x1) = x1 + 2
          f_A(x1) = x1 + 8
          top_A(x1) = 0
          c_A() = 1
          start_A(x1) = x1
          match_A(x1,x2) = x2 + 6
          X_A() = 1
          proper_A(x1) = x1 + 5
          ok_A(x1) = x1 + 4
  
    base order 2:
  
      precedence:
      
        match > proper > mark = check = active = f = start > X = ok > found = c > top# = top
      
  
    partial status:
  
      pi(top#) = []
      pi(mark) = [1]
      pi(check) = [1]
      pi(found) = []
      pi(active) = [1]
      pi(f) = [1]
      pi(top) = []
      pi(c) = []
      pi(start) = []
      pi(match) = [2]
      pi(X) = []
      pi(proper) = [1]
      pi(ok) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: top#(found(x)) -> top#(active(x))

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: top#(found(x)) -> top#(active(x))

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The set of usable rules consists of

  r1, r2, r3, r10, r11, r12, r13, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          top#_A(x1) = x1
          found_A(x1) = 5
          active_A(x1) = 4
          f_A(x1) = x1
          mark_A(x1) = 3
          top_A(x1) = 1
          c_A() = 2
          check_A(x1) = x1
          ok_A(x1) = x1
          start_A(x1) = x1 + 6
  
    base order 2:
  
      precedence:
      
        c > active = f = mark = top = start > found = check > top# = ok
      
  
    partial status:
  
      pi(top#) = []
      pi(found) = []
      pi(active) = []
      pi(f) = []
      pi(mark) = []
      pi(top) = []
      pi(c) = []
      pi(check) = [1]
      pi(ok) = []
      pi(start) = [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: check#(f(x)) -> check#(x)

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The set of usable rules consists of

  r2, r3, r10, r11, r12, r13, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          check#_A(x1) = x1
          f_A(x1) = x1 + 5
          top_A(x1) = 0
          active_A(x1) = x1 + 4
          c_A() = 1
          mark_A(x1) = x1 + 6
          check_A(x1) = x1 + 4
          ok_A(x1) = x1 + 7
          start_A(x1) = x1
          found_A(x1) = x1 + 6
  
    base order 2:
  
      precedence:
      
        check# = active > f = mark = check > start > found > ok > c > top
      
  
    partial status:
  
      pi(check#) = [1]
      pi(f) = [1]
      pi(top) = []
      pi(active) = [1]
      pi(c) = []
      pi(mark) = [1]
      pi(check) = [1]
      pi(ok) = [1]
      pi(start) = []
      pi(found) = [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: active#(f(x)) -> active#(x)

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The set of usable rules consists of

  r2, r3, r10, r11, r12, r13, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          active#_A(x1) = x1
          f_A(x1) = x1 + 5
          top_A(x1) = 0
          active_A(x1) = x1 + 4
          c_A() = 1
          mark_A(x1) = x1 + 6
          check_A(x1) = x1 + 4
          ok_A(x1) = x1 + 7
          start_A(x1) = x1
          found_A(x1) = x1 + 6
  
    base order 2:
  
      precedence:
      
        active# = active > f = mark = check > start > found > ok > c > top
      
  
    partial status:
  
      pi(active#) = [1]
      pi(f) = [1]
      pi(top) = []
      pi(active) = [1]
      pi(c) = []
      pi(mark) = [1]
      pi(check) = [1]
      pi(ok) = [1]
      pi(start) = []
      pi(found) = [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: match#(f(x),f(y)) -> match#(x,y)

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The set of usable rules consists of

  r2, r3, r10, r11, r12, r13, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          match#_A(x1,x2) = x1 + x2 + 7
          f_A(x1) = x1 + 6
          top_A(x1) = 0
          active_A(x1) = x1 + 1
          c_A() = 1
          mark_A(x1) = x1 + 5
          check_A(x1) = x1 + 8
          ok_A(x1) = x1 + 3
          start_A(x1) = x1 + 2
          found_A(x1) = x1 + 2
  
    base order 2:
  
      precedence:
      
        match# = check > f = start > active = mark > ok > c = found > top
      
  
    partial status:
  
      pi(match#) = [2]
      pi(f) = []
      pi(top) = []
      pi(active) = [1]
      pi(c) = []
      pi(mark) = [1]
      pi(check) = [1]
      pi(ok) = [1]
      pi(start) = [1]
      pi(found) = [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: proper#(f(x)) -> proper#(x)

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The set of usable rules consists of

  r2, r3, r10, r11, r12, r13, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          proper#_A(x1) = x1
          f_A(x1) = x1 + 5
          top_A(x1) = 0
          active_A(x1) = x1 + 4
          c_A() = 1
          mark_A(x1) = x1 + 6
          check_A(x1) = x1 + 4
          ok_A(x1) = x1 + 7
          start_A(x1) = x1
          found_A(x1) = x1 + 6
  
    base order 2:
  
      precedence:
      
        proper# = active > f = mark = check > start > found > ok > c > top
      
  
    partial status:
  
      pi(proper#) = [1]
      pi(f) = [1]
      pi(top) = []
      pi(active) = [1]
      pi(c) = []
      pi(mark) = [1]
      pi(check) = [1]
      pi(ok) = [1]
      pi(start) = []
      pi(found) = [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#(ok(x)) -> f#(x)
p2: f#(mark(x)) -> f#(x)
p3: f#(found(x)) -> f#(x)

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The set of usable rules consists of

  r2, r3, r10, r11, r12, r13, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          f#_A(x1) = x1
          ok_A(x1) = x1 + 7
          mark_A(x1) = x1 + 6
          found_A(x1) = x1 + 6
          top_A(x1) = 2
          active_A(x1) = x1 + 3
          c_A() = 1
          check_A(x1) = x1 + 4
          f_A(x1) = x1 + 5
          start_A(x1) = x1
  
    base order 2:
  
      precedence:
      
        check > f# = active > mark = f = start > ok > found > top > c
      
  
    partial status:
  
      pi(f#) = [1]
      pi(ok) = [1]
      pi(mark) = [1]
      pi(found) = [1]
      pi(top) = []
      pi(active) = [1]
      pi(c) = []
      pi(check) = [1]
      pi(f) = [1]
      pi(start) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: f#(mark(x)) -> f#(x)
p2: f#(found(x)) -> f#(x)

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(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: f#(mark(x)) -> f#(x)
p2: f#(found(x)) -> f#(x)

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The set of usable rules consists of

  r10, r11, r12, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          f#_A(x1) = x1
          mark_A(x1) = x1 + 4
          found_A(x1) = x1 + 6
          f_A(x1) = x1 + 1
          ok_A(x1) = x1 + 7
          start_A(x1) = x1
  
    base order 2:
  
      precedence:
      
        start > f > mark = ok > found > f#
      
  
    partial status:
  
      pi(f#) = [1]
      pi(mark) = [1]
      pi(found) = [1]
      pi(f) = []
      pi(ok) = [1]
      pi(start) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: f#(found(x)) -> f#(x)

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(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#(found(x)) -> f#(x)

and R consists of:

r1: active(f(x)) -> mark(x)
r2: top(active(c())) -> top(mark(c()))
r3: top(mark(x)) -> top(check(x))
r4: check(f(x)) -> f(check(x))
r5: check(x) -> start(match(f(X()),x))
r6: match(f(x),f(y)) -> f(match(x,y))
r7: match(X(),x) -> proper(x)
r8: proper(c()) -> ok(c())
r9: proper(f(x)) -> f(proper(x))
r10: f(ok(x)) -> ok(f(x))
r11: start(ok(x)) -> found(x)
r12: f(found(x)) -> found(f(x))
r13: top(found(x)) -> top(active(x))
r14: active(f(x)) -> f(active(x))
r15: f(mark(x)) -> mark(f(x))

The set of usable rules consists of

  r2, r3, r10, r11, r12, r13, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          f#_A(x1) = x1
          found_A(x1) = x1 + 11
          top_A(x1) = 10
          active_A(x1) = x1 + 4
          c_A() = 1
          mark_A(x1) = x1 + 8
          check_A(x1) = x1 + 7
          f_A(x1) = x1 + 5
          ok_A(x1) = x1 + 12
          start_A(x1) = x1
  
    base order 2:
  
      precedence:
      
        f# = top = active > mark = f > check = ok > c = start > found
      
  
    partial status:
  
      pi(f#) = [1]
      pi(found) = [1]
      pi(top) = []
      pi(active) = [1]
      pi(c) = []
      pi(mark) = [1]
      pi(check) = []
      pi(f) = [1]
      pi(ok) = [1]
      pi(start) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.