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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            top#_A(x1) = max{27, x1 - 25}
            active_A(x1) = max{22, x1 - 1}
            c_A = 25
            mark_A(x1) = max{20, x1 - 2}
            check_A(x1) = 52
            found_A(x1) = max{45, x1 + 26}
            f_A(x1) = max{19, x1}
            start_A(x1) = x1 + 27
            match_A(x1,x2) = x1 + 1
            X_A = 24
            proper_A(x1) = 25
            ok_A(x1) = max{18, x1}
            top#!_A(x1) = max{69, x1 + 17}
            active!_A(x1) = x1 + 10
            c!_A = 38
            mark!_A(x1) = 8
            check!_A(x1) = max{14, x1 + 12}
            found!_A(x1) = 10
            f!_A(x1) = 14
            start!_A(x1) = max{11, x1 - 12}
            match!_A(x1,x2) = max{11, x1 - 1}
            X!_A = 15
            proper!_A(x1) = 14
            ok!_A(x1) = max{15, x1 - 26}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            top#_A(x1) = 0
            active_A(x1) = max{16, x1 - 18}
            c_A = 36
            mark_A(x1) = 16
            check_A(x1) = 16
            found_A(x1) = max{16, x1 - 12}
            f_A(x1) = max{16, x1 - 17}
            start_A(x1) = max{16, x1 - 12}
            match_A(x1,x2) = max{15, x1 + 12}
            X_A = 24
            proper_A(x1) = 36
            ok_A(x1) = max{16, x1}
            top#!_A(x1) = x1 + 14
            active!_A(x1) = 0
            c!_A = 4
            mark!_A(x1) = 0
            check!_A(x1) = 0
            found!_A(x1) = 2
            f!_A(x1) = max{11, x1 - 25}
            start!_A(x1) = 0
            match!_A(x1,x2) = 0
            X!_A = 1
            proper!_A(x1) = 11
            ok!_A(x1) = 3
        
  
    partial status:
  
      pi(top#) = []
      pi(active) = [1]
      pi(c) = []
      pi(mark) = []
      pi(check) = [1]
      pi(found) = []
      pi(f) = [1]
      pi(start) = []
      pi(match) = [2]
      pi(X) = []
      pi(proper) = [1]
      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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            top#_A(x1) = 0
            mark_A(x1) = 19
            check_A(x1) = x1 + 42
            found_A(x1) = 24
            active_A(x1) = x1 + 23
            f_A(x1) = max{25, x1 - 22}
            start_A(x1) = max{42, x1 - 12}
            match_A(x1,x2) = max{x1 - 86, x2 + 23}
            X_A = 111
            proper_A(x1) = 25
            c_A = 66
            ok_A(x1) = 13
            top#!_A(x1) = 0
            mark!_A(x1) = 1
            check!_A(x1) = x1 + 51
            found!_A(x1) = 9
            active!_A(x1) = x1 + 32
            f!_A(x1) = max{6, x1 - 14}
            start!_A(x1) = max{3, x1 - 4}
            match!_A(x1,x2) = max{75, x1 - 37, x2 + 32}
            X!_A = 50
            proper!_A(x1) = max{75, x1 + 23}
            c!_A = 2
            ok!_A(x1) = 2
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            top#_A(x1) = max{0, x1 - 10}
            mark_A(x1) = max{53, x1 + 24}
            check_A(x1) = x1 + 23
            found_A(x1) = max{23, x1 + 9}
            active_A(x1) = max{4, x1 + 1}
            f_A(x1) = max{52, x1 + 39}
            start_A(x1) = max{23, x1 - 3}
            match_A(x1,x2) = x2 + 26
            X_A = 13
            proper_A(x1) = max{13, x1 + 12}
            c_A = 16
            ok_A(x1) = max{25, x1 + 12}
            top#!_A(x1) = max{3, x1}
            mark!_A(x1) = 3
            check!_A(x1) = max{69, x1 + 15}
            found!_A(x1) = max{3, x1 - 53}
            active!_A(x1) = max{69, x1 - 54}
            f!_A(x1) = 68
            start!_A(x1) = max{16, x1 - 26}
            match!_A(x1,x2) = max{70, x1 - 38, x2 - 53}
            X!_A = 50
            proper!_A(x1) = max{49, x1 - 54}
            c!_A = 49
            ok!_A(x1) = 0
        
  
    partial status:
  
      pi(top#) = []
      pi(mark) = []
      pi(check) = [1]
      pi(found) = []
      pi(active) = []
      pi(f) = []
      pi(start) = []
      pi(match) = [1]
      pi(X) = []
      pi(proper) = []
      pi(c) = []
      pi(ok) = []

The following rules are removed:

  p1, p2

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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            check#_A(x1) = 0
            f_A(x1) = max{7, x1 - 21}
            check#!_A(x1) = max{4, x1 - 41}
            f!_A(x1) = max{4, x1 - 41}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            check#_A(x1) = 0
            f_A(x1) = max{9, x1 - 15}
            check#!_A(x1) = max{1, x1 - 45}
            f!_A(x1) = max{5, x1 - 45}
        
  
    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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            active#_A(x1) = 0
            f_A(x1) = max{6, x1 - 15}
            active#!_A(x1) = max{2, x1 - 34}
            f!_A(x1) = max{2, x1 - 34}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            active#_A(x1) = 0
            f_A(x1) = max{7, x1 - 20}
            active#!_A(x1) = max{2, x1 - 67}
            f!_A(x1) = max{2, x1 - 66}
        
  
    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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            match#_A(x1,x2) = 0
            f_A(x1) = max{7, x1 - 18}
            match#!_A(x1,x2) = max{0, x2 - 41}
            f!_A(x1) = max{2, x1 - 40}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            match#_A(x1,x2) = 0
            f_A(x1) = max{7, x1 - 11}
            match#!_A(x1,x2) = max{3, x1 - 39, x2 - 8}
            f!_A(x1) = max{2, x1 - 38}
        
  
    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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            proper#_A(x1) = 0
            f_A(x1) = max{6, x1 - 15}
            proper#!_A(x1) = max{2, x1 - 55}
            f!_A(x1) = max{2, x1 - 55}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            proper#_A(x1) = 0
            f_A(x1) = max{7, x1 - 12}
            proper#!_A(x1) = max{2, x1 - 64}
            f!_A(x1) = max{2, x1 - 63}
        
  
    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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            f#_A(x1) = 0
            ok_A(x1) = max{4, x1 - 7}
            mark_A(x1) = max{4, x1 - 12}
            found_A(x1) = max{7, x1 - 7}
            f#!_A(x1) = 14
            ok!_A(x1) = max{3, x1 - 40}
            mark!_A(x1) = max{7, x1 - 44}
            found!_A(x1) = max{13, x1 - 39}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            f#_A(x1) = 0
            ok_A(x1) = max{13, x1 - 12}
            mark_A(x1) = max{12, x1 - 13}
            found_A(x1) = max{3, x1 - 22}
            f#!_A(x1) = 0
            ok!_A(x1) = max{3, x1 - 49}
            mark!_A(x1) = max{1, x1 - 51}
            found!_A(x1) = max{5, x1 - 25}
        
  
    partial status:
  
      pi(f#) = [1]
      pi(ok) = [1]
      pi(mark) = [1]
      pi(found) = [1]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.