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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            top#_A(x1) = ((1,0),(0,0)) x1 + (1,2)
            active_A(x1) = x1 + (0,3)
            c_A() = (2,1)
            mark_A(x1) = (0,0)
            check_A(x1) = (0,0)
            found_A(x1) = ((1,0),(0,0)) x1
            proper_A(x1) = (3,2)
            ok_A(x1) = ((1,0),(0,0)) x1
            f_A(x1) = (0,0)
            match_A(x1,x2) = ((1,0),(0,0)) x1 + (0,3)
            X_A() = (4,3)
            start_A(x1) = ((1,0),(0,0)) x1
    
      precedence:
      
        check > f > top# = mark > active > c = match = X = start > proper > found = ok
      
      partial status:
    
        pi(top#) = []
        pi(active) = [1]
        pi(c) = []
        pi(mark) = []
        pi(check) = []
        pi(found) = []
        pi(proper) = []
        pi(ok) = []
        pi(f) = []
        pi(match) = []
        pi(X) = []
        pi(start) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            top#_A(x1) = (2,2)
            active_A(x1) = x1 + (3,5)
            c_A() = (0,1)
            mark_A(x1) = (1,0)
            check_A(x1) = (6,5)
            found_A(x1) = (4,3)
            proper_A(x1) = (3,2)
            ok_A(x1) = (1,0)
            f_A(x1) = (5,4)
            match_A(x1,x2) = (4,6)
            X_A() = (3,3)
            start_A(x1) = (5,4)
    
      precedence:
      
        check > proper = ok = f = match > active = mark > start > found > top# = c > X
      
      partial status:
    
        pi(top#) = []
        pi(active) = [1]
        pi(c) = []
        pi(mark) = []
        pi(check) = []
        pi(found) = []
        pi(proper) = []
        pi(ok) = []
        pi(f) = []
        pi(match) = []
        pi(X) = []
        pi(start) = []
    

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: 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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            top#_A(x1) = ((0,0),(1,0)) x1 + (2,6)
            mark_A(x1) = ((1,0),(0,0)) x1 + (3,1)
            check_A(x1) = ((1,0),(1,1)) x1 + (1,5)
            found_A(x1) = ((1,0),(0,0)) x1 + (0,3)
            active_A(x1) = ((1,0),(1,0)) x1 + (0,5)
            proper_A(x1) = ((1,0),(1,0)) x1 + (0,1)
            c_A() = (2,1)
            ok_A(x1) = ((1,0),(0,0)) x1 + (0,2)
            f_A(x1) = x1 + (5,4)
            match_A(x1,x2) = ((1,0),(1,1)) x2 + (0,2)
            X_A() = (2,2)
            start_A(x1) = x1 + (1,2)
    
      precedence:
      
        active = proper = match > check = f > X > top# = mark = found = ok = start > c
      
      partial status:
    
        pi(top#) = []
        pi(mark) = []
        pi(check) = [1]
        pi(found) = []
        pi(active) = []
        pi(proper) = []
        pi(c) = []
        pi(ok) = []
        pi(f) = []
        pi(match) = [2]
        pi(X) = []
        pi(start) = [1]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            top#_A(x1) = (0,0)
            mark_A(x1) = (0,0)
            check_A(x1) = ((0,0),(1,0)) x1 + (6,0)
            found_A(x1) = (1,0)
            active_A(x1) = (4,5)
            proper_A(x1) = (4,5)
            c_A() = (0,0)
            ok_A(x1) = (2,3)
            f_A(x1) = (3,4)
            match_A(x1,x2) = (5,6)
            X_A() = (4,6)
            start_A(x1) = ((1,0),(1,1)) x1
    
      precedence:
      
        start > c > proper = match > ok > active > f > X > found > top# = mark = check
      
      partial status:
    
        pi(top#) = []
        pi(mark) = []
        pi(check) = []
        pi(found) = []
        pi(active) = []
        pi(proper) = []
        pi(c) = []
        pi(ok) = []
        pi(f) = []
        pi(match) = []
        pi(X) = []
        pi(start) = [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: 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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            top#_A(x1) = x1 + (1,1)
            found_A(x1) = ((1,0),(1,1)) x1 + (4,3)
            active_A(x1) = ((1,0),(1,1)) x1 + (2,1)
            f_A(x1) = ((1,0),(1,1)) x1 + (3,2)
            ok_A(x1) = x1 + (4,1)
            mark_A(x1) = (4,3)
    
      precedence:
      
        top# > found = active = f = mark > ok
      
      partial status:
    
        pi(top#) = [1]
        pi(found) = [1]
        pi(active) = [1]
        pi(f) = [1]
        pi(ok) = [1]
        pi(mark) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            top#_A(x1) = (0,0)
            found_A(x1) = ((1,0),(1,1)) x1 + (3,3)
            active_A(x1) = ((1,0),(1,1)) x1 + (1,1)
            f_A(x1) = ((1,0),(1,1)) x1 + (2,2)
            ok_A(x1) = x1 + (1,1)
            mark_A(x1) = (0,0)
    
      precedence:
      
        top# > found = active = f = ok = mark
      
      partial status:
    
        pi(top#) = []
        pi(found) = [1]
        pi(active) = [1]
        pi(f) = []
        pi(ok) = []
        pi(mark) = []
    

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: 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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            check#_A(x1) = ((1,0),(1,1)) x1 + (2,2)
            f_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        check# = f
      
      partial status:
    
        pi(check#) = []
        pi(f) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            check#_A(x1) = ((1,0),(0,0)) x1
            f_A(x1) = (1,1)
    
      precedence:
      
        f > check#
      
      partial status:
    
        pi(check#) = []
        pi(f) = []
    

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: 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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            active#_A(x1) = ((1,0),(1,1)) x1 + (2,2)
            f_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        active# = f
      
      partial status:
    
        pi(active#) = []
        pi(f) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            active#_A(x1) = ((1,0),(0,0)) x1
            f_A(x1) = (1,1)
    
      precedence:
      
        f > active#
      
      partial status:
    
        pi(active#) = []
        pi(f) = []
    

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: 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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            match#_A(x1,x2) = ((1,0),(0,0)) x1
            f_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        match# > f
      
      partial status:
    
        pi(match#) = []
        pi(f) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            match#_A(x1,x2) = (0,0)
            f_A(x1) = (1,1)
    
      precedence:
      
        match# = f
      
      partial status:
    
        pi(match#) = []
        pi(f) = []
    

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: 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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            proper#_A(x1) = ((1,0),(1,1)) x1 + (2,2)
            f_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        proper# = f
      
      partial status:
    
        pi(proper#) = []
        pi(f) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            proper#_A(x1) = ((1,0),(0,0)) x1
            f_A(x1) = (1,1)
    
      precedence:
      
        f > proper#
      
      partial status:
    
        pi(proper#) = []
        pi(f) = []
    

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#(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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            f#_A(x1) = ((1,0),(0,0)) x1
            ok_A(x1) = ((1,0),(0,0)) x1 + (1,1)
            mark_A(x1) = x1 + (1,1)
            found_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        ok > mark = found > f#
      
      partial status:
    
        pi(f#) = []
        pi(ok) = []
        pi(mark) = [1]
        pi(found) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            f#_A(x1) = (0,2)
            ok_A(x1) = (1,1)
            mark_A(x1) = ((1,0),(0,0)) x1 + (1,1)
            found_A(x1) = (1,1)
    
      precedence:
      
        f# = ok = mark = found
      
      partial status:
    
        pi(f#) = []
        pi(ok) = []
        pi(mark) = []
        pi(found) = []
    

The next rules are strictly ordered:

  p1, p2, p3

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