YES

We show the termination of the TRS R:

  active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
  active(sqr(|0|())) -> mark(|0|())
  active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
  active(dbl(|0|())) -> mark(|0|())
  active(dbl(s(X))) -> mark(s(s(dbl(X))))
  active(add(|0|(),X)) -> mark(X)
  active(add(s(X),Y)) -> mark(s(add(X,Y)))
  active(first(|0|(),X)) -> mark(nil())
  active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
  mark(terms(X)) -> active(terms(mark(X)))
  mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
  mark(recip(X)) -> active(recip(mark(X)))
  mark(sqr(X)) -> active(sqr(mark(X)))
  mark(s(X)) -> active(s(X))
  mark(|0|()) -> active(|0|())
  mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
  mark(dbl(X)) -> active(dbl(mark(X)))
  mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
  mark(nil()) -> active(nil())
  terms(mark(X)) -> terms(X)
  terms(active(X)) -> terms(X)
  cons(mark(X1),X2) -> cons(X1,X2)
  cons(X1,mark(X2)) -> cons(X1,X2)
  cons(active(X1),X2) -> cons(X1,X2)
  cons(X1,active(X2)) -> cons(X1,X2)
  recip(mark(X)) -> recip(X)
  recip(active(X)) -> recip(X)
  sqr(mark(X)) -> sqr(X)
  sqr(active(X)) -> sqr(X)
  s(mark(X)) -> s(X)
  s(active(X)) -> s(X)
  add(mark(X1),X2) -> add(X1,X2)
  add(X1,mark(X2)) -> add(X1,X2)
  add(active(X1),X2) -> add(X1,X2)
  add(X1,active(X2)) -> add(X1,X2)
  dbl(mark(X)) -> dbl(X)
  dbl(active(X)) -> dbl(X)
  first(mark(X1),X2) -> first(X1,X2)
  first(X1,mark(X2)) -> first(X1,X2)
  first(active(X1),X2) -> first(X1,X2)
  first(X1,active(X2)) -> first(X1,X2)

-- SCC decomposition.

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

p1: active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N))))
p2: active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
p3: active#(terms(N)) -> recip#(sqr(N))
p4: active#(terms(N)) -> sqr#(N)
p5: active#(terms(N)) -> terms#(s(N))
p6: active#(terms(N)) -> s#(N)
p7: active#(sqr(|0|())) -> mark#(|0|())
p8: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p9: active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
p10: active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
p11: active#(sqr(s(X))) -> sqr#(X)
p12: active#(sqr(s(X))) -> dbl#(X)
p13: active#(dbl(|0|())) -> mark#(|0|())
p14: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p15: active#(dbl(s(X))) -> s#(s(dbl(X)))
p16: active#(dbl(s(X))) -> s#(dbl(X))
p17: active#(dbl(s(X))) -> dbl#(X)
p18: active#(add(|0|(),X)) -> mark#(X)
p19: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p20: active#(add(s(X),Y)) -> s#(add(X,Y))
p21: active#(add(s(X),Y)) -> add#(X,Y)
p22: active#(first(|0|(),X)) -> mark#(nil())
p23: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p24: active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
p25: active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
p26: mark#(terms(X)) -> active#(terms(mark(X)))
p27: mark#(terms(X)) -> terms#(mark(X))
p28: mark#(terms(X)) -> mark#(X)
p29: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p30: mark#(cons(X1,X2)) -> cons#(mark(X1),X2)
p31: mark#(cons(X1,X2)) -> mark#(X1)
p32: mark#(recip(X)) -> active#(recip(mark(X)))
p33: mark#(recip(X)) -> recip#(mark(X))
p34: mark#(recip(X)) -> mark#(X)
p35: mark#(sqr(X)) -> active#(sqr(mark(X)))
p36: mark#(sqr(X)) -> sqr#(mark(X))
p37: mark#(sqr(X)) -> mark#(X)
p38: mark#(s(X)) -> active#(s(X))
p39: mark#(|0|()) -> active#(|0|())
p40: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p41: mark#(add(X1,X2)) -> add#(mark(X1),mark(X2))
p42: mark#(add(X1,X2)) -> mark#(X1)
p43: mark#(add(X1,X2)) -> mark#(X2)
p44: mark#(dbl(X)) -> active#(dbl(mark(X)))
p45: mark#(dbl(X)) -> dbl#(mark(X))
p46: mark#(dbl(X)) -> mark#(X)
p47: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p48: mark#(first(X1,X2)) -> first#(mark(X1),mark(X2))
p49: mark#(first(X1,X2)) -> mark#(X1)
p50: mark#(first(X1,X2)) -> mark#(X2)
p51: mark#(nil()) -> active#(nil())
p52: terms#(mark(X)) -> terms#(X)
p53: terms#(active(X)) -> terms#(X)
p54: cons#(mark(X1),X2) -> cons#(X1,X2)
p55: cons#(X1,mark(X2)) -> cons#(X1,X2)
p56: cons#(active(X1),X2) -> cons#(X1,X2)
p57: cons#(X1,active(X2)) -> cons#(X1,X2)
p58: recip#(mark(X)) -> recip#(X)
p59: recip#(active(X)) -> recip#(X)
p60: sqr#(mark(X)) -> sqr#(X)
p61: sqr#(active(X)) -> sqr#(X)
p62: s#(mark(X)) -> s#(X)
p63: s#(active(X)) -> s#(X)
p64: add#(mark(X1),X2) -> add#(X1,X2)
p65: add#(X1,mark(X2)) -> add#(X1,X2)
p66: add#(active(X1),X2) -> add#(X1,X2)
p67: add#(X1,active(X2)) -> add#(X1,X2)
p68: dbl#(mark(X)) -> dbl#(X)
p69: dbl#(active(X)) -> dbl#(X)
p70: first#(mark(X1),X2) -> first#(X1,X2)
p71: first#(X1,mark(X2)) -> first#(X1,X2)
p72: first#(active(X1),X2) -> first#(X1,X2)
p73: first#(X1,active(X2)) -> first#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p8, p14, p18, p19, p23, p26, p28, p29, p31, p32, p34, p35, p37, p38, p40, p42, p43, p44, p46, p47, p49, p50}
  {p54, p55, p56, p57}
  {p58, p59}
  {p60, p61}
  {p52, p53}
  {p62, p63}
  {p64, p65, p66, p67}
  {p68, p69}
  {p70, p71, p72, p73}


-- Reduction pair.

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

p1: active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N))))
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> mark#(X1)
p4: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p5: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p6: mark#(dbl(X)) -> mark#(X)
p7: mark#(dbl(X)) -> active#(dbl(mark(X)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(add(X1,X2)) -> mark#(X2)
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p12: active#(add(|0|(),X)) -> mark#(X)
p13: mark#(s(X)) -> active#(s(X))
p14: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p15: mark#(sqr(X)) -> mark#(X)
p16: mark#(sqr(X)) -> active#(sqr(mark(X)))
p17: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p18: mark#(recip(X)) -> mark#(X)
p19: mark#(recip(X)) -> active#(recip(mark(X)))
p20: mark#(cons(X1,X2)) -> mark#(X1)
p21: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p22: mark#(terms(X)) -> mark#(X)
p23: mark#(terms(X)) -> active#(terms(mark(X)))

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          active#_A(x1) = max{120, x1 + 4}
          terms_A(x1) = max{658, x1 + 396}
          mark#_A(x1) = max{136, x1 + 36}
          cons_A(x1,x2) = x1 + 109
          recip_A(x1) = x1 + 136
          sqr_A(x1) = max{119, x1 + 88}
          s_A(x1) = 118
          first_A(x1,x2) = max{x1 + 323, x2 + 328}
          mark_A(x1) = x1 + 32
          dbl_A(x1) = x1 + 151
          add_A(x1,x2) = max{x1 + 99, x2 + 62}
          |0|_A = 85
          active_A(x1) = max{117, x1}
          nil_A = 297
    
      precedence:
      
        mark > s = dbl > sqr = add > terms = |0| > mark# > active# > cons = recip = first = active = nil
      
      partial status:
    
        pi(active#) = [1]
        pi(terms) = [1]
        pi(mark#) = [1]
        pi(cons) = [1]
        pi(recip) = [1]
        pi(sqr) = [1]
        pi(s) = []
        pi(first) = [1, 2]
        pi(mark) = []
        pi(dbl) = [1]
        pi(add) = [1, 2]
        pi(|0|) = []
        pi(active) = [1]
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          active#_A(x1) = max{2, x1 - 700}
          terms_A(x1) = max{1904, x1 + 1869}
          mark#_A(x1) = max{504, x1 - 96}
          cons_A(x1,x2) = x1 + 601
          recip_A(x1) = max{604, x1 + 1}
          sqr_A(x1) = max{662, x1 + 316}
          s_A(x1) = 443
          first_A(x1,x2) = max{x1 + 333, x2 + 702}
          mark_A(x1) = 404
          dbl_A(x1) = x1 + 694
          add_A(x1,x2) = max{x1 + 62, x2 + 1046}
          |0|_A = 441
          active_A(x1) = 403
          nil_A = 775
    
      precedence:
      
        nil > active# = terms = mark# = cons > recip > sqr = s = first = mark > dbl = add = |0| = active
      
      partial status:
    
        pi(active#) = []
        pi(terms) = [1]
        pi(mark#) = []
        pi(cons) = []
        pi(recip) = []
        pi(sqr) = [1]
        pi(s) = []
        pi(first) = [2]
        pi(mark) = []
        pi(dbl) = [1]
        pi(add) = [2]
        pi(|0|) = []
        pi(active) = []
        pi(nil) = []
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23

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: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(X1,active(X2)) -> cons#(X1,X2)
p3: cons#(active(X1),X2) -> cons#(X1,X2)
p4: cons#(X1,mark(X2)) -> cons#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          cons#_A(x1,x2) = max{2, x1 + 1, x2 + 1}
          mark_A(x1) = max{1, x1}
          active_A(x1) = max{1, x1}
    
      precedence:
      
        cons# = mark = active
      
      partial status:
    
        pi(cons#) = [1, 2]
        pi(mark) = [1]
        pi(active) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          cons#_A(x1,x2) = max{x1 + 1, x2 - 1}
          mark_A(x1) = x1
          active_A(x1) = x1
    
      precedence:
      
        cons# = mark = active
      
      partial status:
    
        pi(cons#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    

The next rules are strictly ordered:

  p1, p2, p3, p4

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: recip#(mark(X)) -> recip#(X)
p2: recip#(active(X)) -> recip#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          recip#_A(x1) = max{6, x1 + 4}
          mark_A(x1) = max{4, x1 + 3}
          active_A(x1) = max{2, x1 + 1}
    
      precedence:
      
        recip# = mark = active
      
      partial status:
    
        pi(recip#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          recip#_A(x1) = x1 + 1
          mark_A(x1) = x1
          active_A(x1) = x1
    
      precedence:
      
        recip# = mark = active
      
      partial status:
    
        pi(recip#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    

The next rules are strictly ordered:

  p1, p2

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: sqr#(mark(X)) -> sqr#(X)
p2: sqr#(active(X)) -> sqr#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          sqr#_A(x1) = max{6, x1 + 4}
          mark_A(x1) = max{4, x1 + 3}
          active_A(x1) = max{2, x1 + 1}
    
      precedence:
      
        sqr# = mark = active
      
      partial status:
    
        pi(sqr#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          sqr#_A(x1) = x1 + 1
          mark_A(x1) = x1
          active_A(x1) = x1
    
      precedence:
      
        sqr# = mark = active
      
      partial status:
    
        pi(sqr#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    

The next rules are strictly ordered:

  p1, p2

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: terms#(mark(X)) -> terms#(X)
p2: terms#(active(X)) -> terms#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          terms#_A(x1) = max{6, x1 + 4}
          mark_A(x1) = max{4, x1 + 3}
          active_A(x1) = max{2, x1 + 1}
    
      precedence:
      
        terms# = mark = active
      
      partial status:
    
        pi(terms#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          terms#_A(x1) = x1 + 1
          mark_A(x1) = x1
          active_A(x1) = x1
    
      precedence:
      
        terms# = mark = active
      
      partial status:
    
        pi(terms#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    

The next rules are strictly ordered:

  p1, p2

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: s#(mark(X)) -> s#(X)
p2: s#(active(X)) -> s#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          s#_A(x1) = max{6, x1 + 4}
          mark_A(x1) = max{4, x1 + 3}
          active_A(x1) = max{2, x1 + 1}
    
      precedence:
      
        s# = mark = active
      
      partial status:
    
        pi(s#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          s#_A(x1) = x1 + 1
          mark_A(x1) = x1
          active_A(x1) = x1
    
      precedence:
      
        s# = mark = active
      
      partial status:
    
        pi(s#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    

The next rules are strictly ordered:

  p1, p2

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: add#(mark(X1),X2) -> add#(X1,X2)
p2: add#(X1,active(X2)) -> add#(X1,X2)
p3: add#(active(X1),X2) -> add#(X1,X2)
p4: add#(X1,mark(X2)) -> add#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          add#_A(x1,x2) = max{2, x1 + 1, x2 + 1}
          mark_A(x1) = max{1, x1}
          active_A(x1) = max{1, x1}
    
      precedence:
      
        add# = mark = active
      
      partial status:
    
        pi(add#) = [1, 2]
        pi(mark) = [1]
        pi(active) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          add#_A(x1,x2) = max{x1 + 1, x2 - 1}
          mark_A(x1) = x1
          active_A(x1) = x1
    
      precedence:
      
        add# = mark = active
      
      partial status:
    
        pi(add#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    

The next rules are strictly ordered:

  p1, p2, p3, p4

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: dbl#(mark(X)) -> dbl#(X)
p2: dbl#(active(X)) -> dbl#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          dbl#_A(x1) = x1 + 2
          mark_A(x1) = x1
          active_A(x1) = x1 + 2
    
      precedence:
      
        mark > active > dbl#
      
      partial status:
    
        pi(dbl#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          dbl#_A(x1) = x1 + 1
          mark_A(x1) = x1
          active_A(x1) = x1
    
      precedence:
      
        dbl# = mark = active
      
      partial status:
    
        pi(dbl#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    

The next rules are strictly ordered:

  p1, p2

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: first#(mark(X1),X2) -> first#(X1,X2)
p2: first#(X1,active(X2)) -> first#(X1,X2)
p3: first#(active(X1),X2) -> first#(X1,X2)
p4: first#(X1,mark(X2)) -> first#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          first#_A(x1,x2) = max{2, x1 + 1, x2 + 1}
          mark_A(x1) = max{1, x1}
          active_A(x1) = max{1, x1}
    
      precedence:
      
        first# = mark = active
      
      partial status:
    
        pi(first#) = [1, 2]
        pi(mark) = [1]
        pi(active) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          first#_A(x1,x2) = max{x1 + 1, x2 - 1}
          mark_A(x1) = x1
          active_A(x1) = x1
    
      precedence:
      
        first# = mark = active
      
      partial status:
    
        pi(first#) = [1]
        pi(mark) = [1]
        pi(active) = [1]
    

The next rules are strictly ordered:

  p1, p2, p3, p4

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