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, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        top#_A(x1) = 1
        active_A(x1) = max{6, x1}
        c_A = 12
        mark_A(x1) = 5
        check_A(x1) = max{14, x1 + 8}
        found_A(x1) = 11
        f_A(x1) = max{15, x1 - 7}
        start_A(x1) = max{12, x1 - 24}
        match_A(x1,x2) = max{23, x1 - 16, x2 - 16}
        X_A = 6
        proper_A(x1) = max{15, x1 - 17}
        ok_A(x1) = 6
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          top#_A(x1) = max{0, x1 - 42}
          active_A(x1) = max{9, x1 - 33}
          c_A = 74
          mark_A(x1) = max{6, x1 - 65}
          check_A(x1) = 31
          found_A(x1) = max{2, x1 - 15}
          f_A(x1) = max{9, x1 - 32}
          start_A(x1) = max{1, x1 - 4}
          match_A(x1,x2) = max{3, x1 + 1}
          X_A = 66
          proper_A(x1) = 63
          ok_A(x1) = max{6, x1 - 11}
          top#!_A(x1) = max{47, x1 + 16}
          active!_A(x1) = 58
          c!_A = 45
          mark!_A(x1) = max{46, x1 - 82}
          check!_A(x1) = max{58, x1 - 50}
          found!_A(x1) = max{48, x1 - 10}
          f!_A(x1) = max{57, x1 - 7}
          start!_A(x1) = max{1, x1 - 86}
          match!_A(x1,x2) = max{95, x1 + 27, x2 - 14}
          X!_A = 1
          proper!_A(x1) = max{94, x1 - 75}
          ok!_A(x1) = 1
  
    partial status:
  
      pi(top#) = []
      pi(active) = [1]
      pi(c) = []
      pi(mark) = []
      pi(check) = [1]
      pi(found) = []
      pi(f) = []
      pi(start) = []
      pi(match) = []
      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, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        top#_A(x1) = max{1, x1 - 8}
        mark_A(x1) = max{45, x1 + 27}
        check_A(x1) = x1 + 18
        found_A(x1) = max{2, x1}
        active_A(x1) = max{9, x1}
        f_A(x1) = max{46, x1 + 29}
        start_A(x1) = max{3, x1 - 14}
        match_A(x1,x2) = max{x1 - 47, x2 + 31}
        X_A = 5
        proper_A(x1) = max{30, x1 + 16}
        c_A = 63
        ok_A(x1) = max{62, x1 + 15}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          top#_A(x1) = max{0, x1 - 4}
          mark_A(x1) = max{2, x1 - 39}
          check_A(x1) = max{2, x1 - 75}
          found_A(x1) = max{2, x1 - 12}
          active_A(x1) = max{3, x1 - 26}
          f_A(x1) = max{2, x1 - 13}
          start_A(x1) = max{2, x1 - 9}
          match_A(x1,x2) = max{8, x1 - 56, x2 - 67}
          X_A = 4
          proper_A(x1) = max{2, x1 - 67}
          c_A = 2
          ok_A(x1) = max{0, x1 - 3}
          top#!_A(x1) = max{43, x1 + 40}
          mark!_A(x1) = max{1, x1 - 45}
          check!_A(x1) = max{44, x1 - 44}
          found!_A(x1) = 0
          active!_A(x1) = max{40, x1 - 17}
          f!_A(x1) = max{2, x1 - 5}
          start!_A(x1) = x1 + 22
          match!_A(x1,x2) = max{43, x1 - 47, x2 - 45}
          X!_A = 1
          proper!_A(x1) = max{42, x1 - 46}
          c!_A = 42
          ok!_A(x1) = max{1, x1 - 8}
  
    partial status:
  
      pi(top#) = []
      pi(mark) = []
      pi(check) = []
      pi(found) = []
      pi(active) = [1]
      pi(f) = [1]
      pi(start) = []
      pi(match) = []
      pi(X) = []
      pi(proper) = [1]
      pi(c) = []
      pi(ok) = []

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, r10, r12, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        top#_A(x1) = max{3, x1 - 10}
        found_A(x1) = x1 + 19
        active_A(x1) = x1 + 9
        f_A(x1) = x1 + 6
        mark_A(x1) = max{4, x1 - 28}
        ok_A(x1) = max{1, x1 - 17}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          top#_A(x1) = max{0, x1 - 5}
          found_A(x1) = max{3, x1 - 4}
          active_A(x1) = max{4, x1 - 28}
          f_A(x1) = max{3, x1 - 5}
          mark_A(x1) = max{3, x1 - 37}
          ok_A(x1) = max{3, x1 - 11}
          top#!_A(x1) = x1 + 27
          found!_A(x1) = 16
          active!_A(x1) = max{22, x1 - 4}
          f!_A(x1) = max{17, x1 + 5}
          mark!_A(x1) = max{16, x1 - 34}
          ok!_A(x1) = max{0, x1 - 1}
  
    partial status:
  
      pi(top#) = []
      pi(found) = [1]
      pi(active) = []
      pi(f) = [1]
      pi(mark) = []
      pi(ok) = []

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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        check#_A(x1) = max{23, x1 + 13}
        f_A(x1) = max{23, x1 + 13}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          check#_A(x1) = 0
          f_A(x1) = max{3, x1 - 18}
          check#!_A(x1) = 0
          f!_A(x1) = max{8, x1 - 18}
  
    partial status:
  
      pi(check#) = [1]
      pi(f) = [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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        active#_A(x1) = max{12, x1}
        f_A(x1) = max{12, x1 + 1}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          active#_A(x1) = 0
          f_A(x1) = max{7, x1 - 18}
          active#!_A(x1) = 0
          f!_A(x1) = max{4, x1 - 28}
  
    partial status:
  
      pi(active#) = [1]
      pi(f) = [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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        match#_A(x1,x2) = max{23, x2}
        f_A(x1) = x1 + 23
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          match#_A(x1,x2) = 0
          f_A(x1) = max{20, x1 - 17}
          match#!_A(x1,x2) = x2 + 19
          f!_A(x1) = max{31, x1 + 20}
  
    partial status:
  
      pi(match#) = [2]
      pi(f) = [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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        proper#_A(x1) = max{4, x1 + 2}
        f_A(x1) = max{26, x1 + 3}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          proper#_A(x1) = 0
          f_A(x1) = max{21, x1 - 57}
          proper#!_A(x1) = 1
          f!_A(x1) = max{8, x1 - 87}
  
    partial status:
  
      pi(proper#) = [1]
      pi(f) = [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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f#_A(x1) = max{14, x1 + 7}
        ok_A(x1) = max{14, x1 + 10}
        mark_A(x1) = max{79, x1 + 8}
        found_A(x1) = max{14, x1 + 10}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          f#_A(x1) = 0
          ok_A(x1) = max{6, x1 - 6}
          mark_A(x1) = max{7, x1 - 6}
          found_A(x1) = max{8, x1 - 22}
          f#!_A(x1) = 0
          ok!_A(x1) = max{4, x1 - 29}
          mark!_A(x1) = max{2, x1 - 33}
          found!_A(x1) = max{3, x1 - 30}
  
    partial status:
  
      pi(f#) = [1]
      pi(ok) = []
      pi(mark) = [1]
      pi(found) = []

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.