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, r4, r5, r6, r7, r8, r9, r10, r11, r12, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        top#_A(x1) = max{9, x1 - 6}
        active_A(x1) = max{14, x1 - 52}
        c_A = 68
        mark_A(x1) = max{11, x1 - 106}
        check_A(x1) = 12
        found_A(x1) = max{7, x1 - 45}
        f_A(x1) = max{12, x1 - 53}
        start_A(x1) = max{8, x1 - 31}
        match_A(x1,x2) = x1 + 18
        X_A = 52
        proper_A(x1) = 69
        ok_A(x1) = max{11, x1 - 13}
  
    base order 2:
  
      precedence:
      
        check > match > active = proper > c = f = X > mark = ok > top# > found > start
      
  
    partial status:
  
      pi(top#) = []
      pi(active) = []
      pi(c) = []
      pi(mark) = []
      pi(check) = []
      pi(found) = []
      pi(f) = []
      pi(start) = []
      pi(match) = [1]
      pi(X) = []
      pi(proper) = []
      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, r4, r5, r6, r7, r8, r9, r10, r11, r12, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        top#_A(x1) = max{1, x1 - 53}
        mark_A(x1) = max{171, x1 + 88}
        check_A(x1) = max{140, x1 + 71}
        found_A(x1) = max{138, x1 + 55}
        active_A(x1) = max{107, x1 + 24}
        f_A(x1) = max{148, x1 + 65}
        start_A(x1) = max{139, x1 + 69}
        match_A(x1,x2) = max{3, x1 - 218, x2 + 1}
        X_A = 223
        proper_A(x1) = max{4, x1}
        c_A = 142
        ok_A(x1) = max{5, x1 - 14}
  
    base order 2:
  
      precedence:
      
        check > match > proper > X = c > active > f = start > found > top# = mark = ok
      
  
    partial status:
  
      pi(top#) = []
      pi(mark) = []
      pi(check) = []
      pi(found) = [1]
      pi(active) = []
      pi(f) = []
      pi(start) = []
      pi(match) = [2]
      pi(X) = []
      pi(proper) = [1]
      pi(c) = []
      pi(ok) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: top#(mark(x)) -> top#(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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: top#(mark(x)) -> top#(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

  r4, r5, r6, r7, r8, r9, r10, r11, r12, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        top#_A(x1) = x1 + 6
        mark_A(x1) = max{18, x1 + 7}
        check_A(x1) = max{5, x1}
        f_A(x1) = max{10, x1 + 3}
        start_A(x1) = max{1, x1 - 12}
        match_A(x1,x2) = max{11, x1 - 11, x2 + 5}
        X_A = 1
        proper_A(x1) = max{6, x1 + 4}
        c_A = 10
        ok_A(x1) = max{7, x1 - 11}
        found_A(x1) = max{0, x1 - 24}
  
    base order 2:
  
      precedence:
      
        match > proper > mark = f = found > top# = start = c > check = ok > X
      
  
    partial status:
  
      pi(top#) = []
      pi(mark) = []
      pi(check) = [1]
      pi(f) = []
      pi(start) = []
      pi(match) = [2]
      pi(X) = []
      pi(proper) = [1]
      pi(c) = []
      pi(ok) = []
      pi(found) = []

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

  r10, r11, r12, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        check#_A(x1) = max{1, x1 - 1}
        f_A(x1) = max{13, x1 + 3}
        ok_A(x1) = max{7, x1 - 14}
        start_A(x1) = max{7, x1 - 15}
        found_A(x1) = max{6, x1 - 30}
        mark_A(x1) = max{10, x1 - 20}
  
    base order 2:
  
      precedence:
      
        f > check# = mark > ok > found > start
      
  
    partial status:
  
      pi(check#) = []
      pi(f) = [1]
      pi(ok) = []
      pi(start) = []
      pi(found) = []
      pi(mark) = []

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

  r10, r11, r12, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        active#_A(x1) = max{1, x1 - 1}
        f_A(x1) = max{13, x1 + 3}
        ok_A(x1) = max{7, x1 - 14}
        start_A(x1) = max{7, x1 - 15}
        found_A(x1) = max{6, x1 - 30}
        mark_A(x1) = max{10, x1 - 20}
  
    base order 2:
  
      precedence:
      
        f > active# = mark > ok > found > start
      
  
    partial status:
  
      pi(active#) = []
      pi(f) = [1]
      pi(ok) = []
      pi(start) = []
      pi(found) = []
      pi(mark) = []

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, r11, r13

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        match#_A(x1,x2) = x2 + 5
        f_A(x1) = x1 + 6
        top_A(x1) = 8
        active_A(x1) = 5
        c_A = 7
        mark_A(x1) = max{9, x1 - 19}
        check_A(x1) = max{10, x1 - 5}
        start_A(x1) = max{9, x1 - 11}
        ok_A(x1) = max{10, x1 - 8}
        found_A(x1) = max{9, x1 - 19}
  
    base order 2:
  
      precedence:
      
        check > match# = start = ok > f = mark = found > top > c > active
      
  
    partial status:
  
      pi(match#) = [2]
      pi(f) = [1]
      pi(top) = []
      pi(active) = []
      pi(c) = []
      pi(mark) = []
      pi(check) = []
      pi(start) = []
      pi(ok) = []
      pi(found) = []

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

  r10, r11, r12, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        proper#_A(x1) = max{1, x1 - 1}
        f_A(x1) = max{13, x1 + 3}
        ok_A(x1) = max{7, x1 - 14}
        start_A(x1) = max{7, x1 - 15}
        found_A(x1) = max{6, x1 - 30}
        mark_A(x1) = max{9, x1 - 47}
  
    base order 2:
  
      precedence:
      
        f > proper# = mark > ok > found > start
      
  
    partial status:
  
      pi(proper#) = []
      pi(f) = [1]
      pi(ok) = []
      pi(start) = []
      pi(found) = []
      pi(mark) = []

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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f#_A(x1) = max{3, x1 + 2}
        ok_A(x1) = x1 + 3
        mark_A(x1) = x1 + 8
        found_A(x1) = x1 + 8
  
    base order 2:
  
      precedence:
      
        found > ok > f# > mark
      
  
    partial status:
  
      pi(f#) = [1]
      pi(ok) = [1]
      pi(mark) = [1]
      pi(found) = [1]

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: f#(ok(x)) -> f#(x)
p2: 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}


-- 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)

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

  r11

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f#_A(x1) = max{1, x1 - 1}
        ok_A(x1) = x1 + 8
        mark_A(x1) = x1 + 4
        start_A(x1) = max{7, x1 - 19}
        found_A(x1) = max{7, x1 - 11}
  
    base order 2:
  
      precedence:
      
        start > ok > mark = found > f#
      
  
    partial status:
  
      pi(f#) = []
      pi(ok) = [1]
      pi(mark) = [1]
      pi(start) = []
      pi(found) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: f#(ok(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#(ok(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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f#_A(x1) = max{1, x1 - 1}
        ok_A(x1) = max{13, x1 + 3}
  
    base order 2:
  
      precedence:
      
        ok > f#
      
  
    partial status:
  
      pi(f#) = []
      pi(ok) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.