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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          active#_A(x1) = ((1,0),(0,0)) x1 + (2,2)
          terms_A(x1) = (2,2)
          mark#_A(x1) = (4,2)
          cons_A(x1,x2) = (1,1)
          recip_A(x1) = (1,3)
          sqr_A(x1) = (2,7)
          s_A(x1) = (1,3)
          first_A(x1,x2) = (2,5)
          mark_A(x1) = (3,6)
          dbl_A(x1) = (2,1)
          add_A(x1,x2) = (2,1)
          |0|_A() = (0,3)
          active_A(x1) = (3,6)
          nil_A() = (1,4)
  
    precedence:
    
      terms = s > sqr = first = mark = dbl = add = |0| = active = nil > cons > active# = mark# > recip
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = []
      pi(dbl) = []
      pi(add) = []
      pi(|0|) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p19

We remove them from the problem.

-- 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: 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#(cons(X1,X2)) -> mark#(X1)
p20: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p21: mark#(terms(X)) -> mark#(X)
p22: 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 estimated dependency graph contains the following SCCs:

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


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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          active#_A(x1) = ((0,0),(1,0)) x1 + (4,3)
          terms_A(x1) = (3,7)
          mark#_A(x1) = (4,6)
          cons_A(x1,x2) = (2,4)
          recip_A(x1) = (5,0)
          sqr_A(x1) = (3,1)
          s_A(x1) = (0,0)
          mark_A(x1) = (3,2)
          add_A(x1,x2) = (3,1)
          dbl_A(x1) = (3,3)
          |0|_A() = (2,0)
          first_A(x1,x2) = (3,3)
          active_A(x1) = (3,2)
          nil_A() = (1,3)
  
    precedence:
    
      terms = recip = sqr = s > cons > active# = mark# = mark = add = dbl = |0| = first = active = nil
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(mark) = []
      pi(add) = []
      pi(dbl) = []
      pi(|0|) = []
      pi(first) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p5

We remove them from the problem.

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


-- 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#(cons(X1,X2)) -> mark#(X1)
p20: mark#(terms(X)) -> mark#(X)
p21: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          active#_A(x1) = ((0,0),(1,0)) x1 + (2,1)
          terms_A(x1) = (5,0)
          mark#_A(x1) = (2,6)
          cons_A(x1,x2) = (3,7)
          recip_A(x1) = (1,7)
          sqr_A(x1) = (5,5)
          s_A(x1) = (4,0)
          first_A(x1,x2) = (5,8)
          mark_A(x1) = (7,9)
          dbl_A(x1) = (5,0)
          add_A(x1,x2) = (5,0)
          |0|_A() = (6,0)
          active_A(x1) = (7,9)
          nil_A() = (0,1)
  
    precedence:
    
      terms = recip = sqr = dbl > cons > active# = mark# > first = mark = |0| = active > s = add = nil
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = []
      pi(dbl) = []
      pi(add) = []
      pi(|0|) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p13

We remove them from the problem.

-- 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: 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: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p14: mark#(sqr(X)) -> mark#(X)
p15: mark#(sqr(X)) -> active#(sqr(mark(X)))
p16: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p17: mark#(recip(X)) -> mark#(X)
p18: mark#(cons(X1,X2)) -> mark#(X1)
p19: mark#(terms(X)) -> mark#(X)
p20: 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 estimated dependency graph contains the following SCCs:

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


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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1 + (32,5)
          terms_A(x1) = x1 + (32,21)
          mark#_A(x1) = ((1,0),(1,0)) x1 + (33,2)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + (6,7)
          recip_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          sqr_A(x1) = x1 + (7,1)
          s_A(x1) = (2,6)
          mark_A(x1) = ((1,0),(1,1)) x1 + (0,7)
          add_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (34,0)
          dbl_A(x1) = ((1,0),(0,0)) x1 + (34,11)
          |0|_A() = (34,35)
          first_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (34,0)
          active_A(x1) = x1 + (0,7)
          nil_A() = (68,1)
  
    precedence:
    
      recip = sqr > terms > cons > active# = mark# > dbl = |0| = first > s = mark = add = active = nil
    
    partial status:
  
      pi(active#) = []
      pi(terms) = [1]
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(mark) = []
      pi(add) = []
      pi(dbl) = []
      pi(|0|) = []
      pi(first) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

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


-- 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#(sqr(X)) -> mark#(X)
p14: mark#(sqr(X)) -> active#(sqr(mark(X)))
p15: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p16: mark#(recip(X)) -> mark#(X)
p17: mark#(cons(X1,X2)) -> mark#(X1)
p18: mark#(terms(X)) -> mark#(X)
p19: active#(sqr(s(X))) -> mark#(s(add(sqr(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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          active#_A(x1) = ((1,0),(0,0)) x1 + (2,19)
          terms_A(x1) = ((1,0),(0,0)) x1 + (46,2)
          mark#_A(x1) = x1 + (9,11)
          cons_A(x1,x2) = x1 + (25,1)
          recip_A(x1) = x1 + (3,2)
          sqr_A(x1) = ((1,0),(1,1)) x1 + (10,1)
          s_A(x1) = (0,7)
          first_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (26,20)
          mark_A(x1) = ((1,0),(1,1)) x1 + (0,5)
          dbl_A(x1) = x1 + (11,10)
          add_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (10,9)
          |0|_A() = (8,1)
          active_A(x1) = x1 + (0,3)
          nil_A() = (9,1)
  
    precedence:
    
      first = |0| > dbl > s > mark = add = active = nil > terms = mark# = recip = sqr > active# > cons
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = [1]
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = [1]
      pi(s) = []
      pi(first) = [2]
      pi(mark) = []
      pi(dbl) = []
      pi(add) = []
      pi(|0|) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- 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: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p4: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p5: mark#(dbl(X)) -> mark#(X)
p6: mark#(dbl(X)) -> active#(dbl(mark(X)))
p7: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: active#(add(|0|(),X)) -> mark#(X)
p12: mark#(sqr(X)) -> mark#(X)
p13: mark#(sqr(X)) -> active#(sqr(mark(X)))
p14: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p15: mark#(recip(X)) -> mark#(X)
p16: mark#(cons(X1,X2)) -> mark#(X1)
p17: mark#(terms(X)) -> mark#(X)
p18: active#(sqr(s(X))) -> mark#(s(add(sqr(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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18}


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

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          active#_A(x1) = ((1,0),(0,0)) x1 + (3,23)
          terms_A(x1) = ((1,0),(0,0)) x1 + (18,2)
          mark#_A(x1) = ((1,0),(0,0)) x1 + (3,23)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + (4,0)
          recip_A(x1) = ((1,0),(0,0)) x1 + (4,5)
          sqr_A(x1) = ((1,0),(1,0)) x1 + (6,0)
          s_A(x1) = (22,6)
          mark_A(x1) = x1 + (0,4)
          add_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2
          dbl_A(x1) = ((1,0),(0,0)) x1 + (1,24)
          |0|_A() = (2,0)
          first_A(x1,x2) = ((1,0),(0,0)) x2 + (5,24)
          active_A(x1) = x1 + (0,3)
          nil_A() = (5,0)
  
    precedence:
    
      active# = terms = mark# = sqr = s = add > |0| = first > cons > nil > recip = mark = dbl = active
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(mark) = []
      pi(add) = []
      pi(dbl) = []
      pi(|0|) = []
      pi(first) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p13

We remove them from the problem.

-- 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: mark#(terms(X)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(recip(X)) -> mark#(X)
p5: mark#(sqr(X)) -> active#(sqr(mark(X)))
p6: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p7: mark#(sqr(X)) -> mark#(X)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p9: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(add(X1,X2)) -> mark#(X2)
p12: mark#(dbl(X)) -> active#(dbl(mark(X)))
p13: mark#(dbl(X)) -> mark#(X)
p14: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p15: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p16: mark#(first(X1,X2)) -> mark#(X2)
p17: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))

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, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17}


-- 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)) -> active#(first(mark(X1),mark(X2)))
p4: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p5: mark#(dbl(X)) -> mark#(X)
p6: mark#(dbl(X)) -> active#(dbl(mark(X)))
p7: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p12: mark#(sqr(X)) -> mark#(X)
p13: mark#(sqr(X)) -> active#(sqr(mark(X)))
p14: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p15: mark#(recip(X)) -> mark#(X)
p16: mark#(cons(X1,X2)) -> mark#(X1)
p17: mark#(terms(X)) -> 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          active#_A(x1) = ((1,0),(0,0)) x1 + (0,11)
          terms_A(x1) = ((1,0),(0,0)) x1 + (9,10)
          mark#_A(x1) = x1 + (2,4)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + (3,2)
          recip_A(x1) = ((1,0),(0,0)) x1 + (1,2)
          sqr_A(x1) = ((1,0),(0,0)) x1 + (3,9)
          s_A(x1) = (4,3)
          first_A(x1,x2) = ((1,0),(0,0)) x2 + (5,12)
          mark_A(x1) = x1
          dbl_A(x1) = ((1,0),(1,0)) x1 + (8,12)
          add_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,8)
          active_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          |0|_A() = (2,2)
          nil_A() = (1,1)
  
    precedence:
    
      terms = recip = s = add > sqr > active# = dbl > cons > mark# = first = mark = active = |0| = nil
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = [1]
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = []
      pi(dbl) = []
      pi(add) = []
      pi(active) = []
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- 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: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p4: mark#(dbl(X)) -> mark#(X)
p5: mark#(dbl(X)) -> active#(dbl(mark(X)))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(add(X1,X2)) -> mark#(X2)
p8: mark#(add(X1,X2)) -> mark#(X1)
p9: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p10: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p11: mark#(sqr(X)) -> mark#(X)
p12: mark#(sqr(X)) -> active#(sqr(mark(X)))
p13: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p14: mark#(recip(X)) -> mark#(X)
p15: mark#(cons(X1,X2)) -> mark#(X1)
p16: mark#(terms(X)) -> 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16}


-- 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#(terms(X)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(recip(X)) -> mark#(X)
p5: mark#(sqr(X)) -> active#(sqr(mark(X)))
p6: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p7: mark#(sqr(X)) -> mark#(X)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p9: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(add(X1,X2)) -> mark#(X2)
p12: mark#(dbl(X)) -> active#(dbl(mark(X)))
p13: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p14: mark#(dbl(X)) -> mark#(X)
p15: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p16: mark#(first(X1,X2)) -> mark#(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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          active#_A(x1) = ((0,0),(1,0)) x1 + (5,1)
          terms_A(x1) = (6,3)
          mark#_A(x1) = (5,4)
          cons_A(x1,x2) = (1,2)
          recip_A(x1) = (0,0)
          sqr_A(x1) = (3,0)
          s_A(x1) = (2,1)
          mark_A(x1) = (5,3)
          add_A(x1,x2) = (3,2)
          dbl_A(x1) = (3,0)
          first_A(x1,x2) = (1,0)
          active_A(x1) = (5,3)
          |0|_A() = (4,0)
          nil_A() = (6,0)
  
    precedence:
    
      active# = mark# = sqr > terms = cons = recip > s = mark = add = dbl = first = active = |0| = nil
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(mark) = []
      pi(add) = []
      pi(dbl) = []
      pi(first) = []
      pi(active) = []
      pi(|0|) = []
      pi(nil) = []

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: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> active#(sqr(mark(X)))
p5: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> mark#(X2)
p11: mark#(dbl(X)) -> active#(dbl(mark(X)))
p12: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p13: mark#(dbl(X)) -> mark#(X)
p14: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p15: mark#(first(X1,X2)) -> mark#(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, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p4: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p5: mark#(dbl(X)) -> mark#(X)
p6: mark#(dbl(X)) -> active#(dbl(mark(X)))
p7: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p12: mark#(sqr(X)) -> mark#(X)
p13: mark#(sqr(X)) -> active#(sqr(mark(X)))
p14: mark#(recip(X)) -> mark#(X)
p15: mark#(cons(X1,X2)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,24)
          terms_A(x1) = ((1,0),(1,1)) x1 + (21,47)
          first_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (23,25)
          active#_A(x1) = ((1,0),(1,1)) x1
          mark_A(x1) = ((1,0),(1,1)) x1 + (0,5)
          add_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (16,11)
          s_A(x1) = (15,1)
          dbl_A(x1) = x1 + (18,0)
          sqr_A(x1) = ((1,0),(1,0)) x1 + (2,0)
          recip_A(x1) = ((1,0),(0,0)) x1 + (2,6)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + (16,21)
          active_A(x1) = x1 + (0,5)
          |0|_A() = (0,4)
          nil_A() = (22,3)
  
    precedence:
    
      dbl = sqr > s > mark = add = |0| > mark# = terms = active# = recip = cons = active = nil > first
    
    partial status:
  
      pi(mark#) = [1]
      pi(terms) = [1]
      pi(first) = []
      pi(active#) = []
      pi(mark) = []
      pi(add) = []
      pi(s) = []
      pi(dbl) = []
      pi(sqr) = []
      pi(recip) = []
      pi(cons) = []
      pi(active) = [1]
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p11

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p4: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p5: mark#(dbl(X)) -> mark#(X)
p6: mark#(dbl(X)) -> active#(dbl(mark(X)))
p7: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: mark#(sqr(X)) -> mark#(X)
p12: mark#(sqr(X)) -> active#(sqr(mark(X)))
p13: mark#(recip(X)) -> mark#(X)
p14: mark#(cons(X1,X2)) -> mark#(X1)

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, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> active#(sqr(mark(X)))
p5: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> mark#(X2)
p11: mark#(dbl(X)) -> active#(dbl(mark(X)))
p12: mark#(dbl(X)) -> mark#(X)
p13: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p14: mark#(first(X1,X2)) -> mark#(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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (2,6)
          terms_A(x1) = ((1,0),(1,0)) x1 + (20,21)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + (15,4)
          recip_A(x1) = ((1,0),(0,0)) x1 + (3,4)
          sqr_A(x1) = ((1,0),(0,0)) x1 + (1,10)
          active#_A(x1) = x1 + (0,4)
          mark_A(x1) = ((1,0),(1,0)) x1
          dbl_A(x1) = ((1,0),(0,0)) x1 + (3,0)
          s_A(x1) = (9,3)
          add_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (3,1)
          first_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (16,5)
          active_A(x1) = ((1,0),(0,0)) x1
          |0|_A() = (1,11)
          nil_A() = (0,0)
  
    precedence:
    
      cons > recip > terms > dbl > mark# = active# > sqr = s = |0| > mark = first = active = nil > add
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(active#) = []
      pi(mark) = []
      pi(dbl) = []
      pi(s) = []
      pi(add) = []
      pi(first) = []
      pi(active) = []
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p12

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> active#(sqr(mark(X)))
p5: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> mark#(X2)
p11: mark#(dbl(X)) -> active#(dbl(mark(X)))
p12: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p13: mark#(first(X1,X2)) -> mark#(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, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p4: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p5: mark#(dbl(X)) -> active#(dbl(mark(X)))
p6: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p7: mark#(add(X1,X2)) -> mark#(X2)
p8: mark#(add(X1,X2)) -> mark#(X1)
p9: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p10: mark#(sqr(X)) -> mark#(X)
p11: mark#(sqr(X)) -> active#(sqr(mark(X)))
p12: mark#(recip(X)) -> mark#(X)
p13: mark#(cons(X1,X2)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = (10,10)
          terms_A(x1) = (4,0)
          first_A(x1,x2) = (1,1)
          active#_A(x1) = ((0,0),(1,0)) x1 + (10,1)
          mark_A(x1) = (6,2)
          add_A(x1,x2) = (9,1)
          s_A(x1) = (3,0)
          dbl_A(x1) = (9,0)
          sqr_A(x1) = (8,0)
          recip_A(x1) = (5,0)
          cons_A(x1,x2) = (2,0)
          active_A(x1) = (6,2)
          |0|_A() = (7,0)
          nil_A() = (8,0)
  
    precedence:
    
      mark# = active# = add = dbl > terms = first = mark = s = sqr = recip = cons = active = |0| = nil
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(first) = []
      pi(active#) = []
      pi(mark) = []
      pi(add) = []
      pi(s) = []
      pi(dbl) = []
      pi(sqr) = []
      pi(recip) = []
      pi(cons) = []
      pi(active) = []
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p4: mark#(dbl(X)) -> active#(dbl(mark(X)))
p5: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p6: mark#(add(X1,X2)) -> mark#(X2)
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p9: mark#(sqr(X)) -> mark#(X)
p10: mark#(sqr(X)) -> active#(sqr(mark(X)))
p11: mark#(recip(X)) -> mark#(X)
p12: mark#(cons(X1,X2)) -> mark#(X1)

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, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> active#(sqr(mark(X)))
p5: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> mark#(X2)
p11: mark#(dbl(X)) -> active#(dbl(mark(X)))
p12: mark#(first(X1,X2)) -> mark#(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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = (5,2)
          terms_A(x1) = (0,0)
          cons_A(x1,x2) = (3,0)
          recip_A(x1) = (5,2)
          sqr_A(x1) = (0,0)
          active#_A(x1) = ((0,0),(1,0)) x1 + (5,1)
          mark_A(x1) = (0,0)
          dbl_A(x1) = (1,0)
          s_A(x1) = (4,0)
          add_A(x1,x2) = (1,1)
          first_A(x1,x2) = (2,0)
          active_A(x1) = (0,0)
          |0|_A() = (2,0)
          nil_A() = (1,0)
  
    precedence:
    
      cons = recip = sqr > mark# = terms = active# = mark = dbl = first = active > add > s = |0| = nil
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(active#) = []
      pi(mark) = []
      pi(dbl) = []
      pi(s) = []
      pi(add) = []
      pi(first) = []
      pi(active) = []
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p5: mark#(sqr(X)) -> mark#(X)
p6: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p7: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p8: mark#(add(X1,X2)) -> mark#(X1)
p9: mark#(add(X1,X2)) -> mark#(X2)
p10: mark#(dbl(X)) -> active#(dbl(mark(X)))
p11: mark#(first(X1,X2)) -> mark#(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, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(dbl(X)) -> active#(dbl(mark(X)))
p4: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p5: mark#(add(X1,X2)) -> mark#(X2)
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p9: mark#(sqr(X)) -> mark#(X)
p10: mark#(recip(X)) -> mark#(X)
p11: mark#(cons(X1,X2)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,1)) x1 + (0,2)
          terms_A(x1) = x1 + (4,1)
          first_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,1)
          dbl_A(x1) = ((0,0),(1,0)) x1 + (0,4)
          active#_A(x1) = (0,3)
          mark_A(x1) = ((1,0),(1,1)) x1
          add_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (1,4)
          s_A(x1) = (0,0)
          sqr_A(x1) = ((1,0),(1,1)) x1 + (1,9)
          recip_A(x1) = ((1,0),(1,1)) x1 + (1,3)
          cons_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (1,3)
          active_A(x1) = x1
          |0|_A() = (0,1)
          nil_A() = (0,0)
  
    precedence:
    
      add > mark = sqr = recip = cons = |0| > terms = nil > first = s = active > dbl > active# > mark#
    
    partial status:
  
      pi(mark#) = [1]
      pi(terms) = [1]
      pi(first) = []
      pi(dbl) = []
      pi(active#) = []
      pi(mark) = [1]
      pi(add) = [2]
      pi(s) = []
      pi(sqr) = [1]
      pi(recip) = []
      pi(cons) = [1]
      pi(active) = [1]
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p8

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(dbl(X)) -> active#(dbl(mark(X)))
p4: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p5: mark#(add(X1,X2)) -> mark#(X2)
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: mark#(sqr(X)) -> mark#(X)
p9: mark#(recip(X)) -> mark#(X)
p10: mark#(cons(X1,X2)) -> mark#(X1)

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, p2, p3, p4, p5, p6, p7, p8, p9, p10}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> mark#(X)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(dbl(X)) -> active#(dbl(mark(X)))
p10: mark#(first(X1,X2)) -> mark#(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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,14)
          terms_A(x1) = x1 + (28,13)
          cons_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (9,15)
          recip_A(x1) = x1 + (9,13)
          sqr_A(x1) = ((1,0),(1,1)) x1 + (9,12)
          add_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (9,11)
          active#_A(x1) = ((1,0),(1,0)) x1 + (0,1)
          mark_A(x1) = ((1,0),(1,1)) x1 + (0,12)
          s_A(x1) = (6,1)
          dbl_A(x1) = (7,18)
          first_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (31,7)
          active_A(x1) = x1 + (0,8)
          |0|_A() = (0,0)
          nil_A() = (1,1)
  
    precedence:
    
      dbl > terms = add > mark > recip > mark# = cons = active# = first = |0| = nil > sqr = s = active
    
    partial status:
  
      pi(mark#) = [1]
      pi(terms) = [1]
      pi(cons) = [1]
      pi(recip) = [1]
      pi(sqr) = []
      pi(add) = [1]
      pi(active#) = []
      pi(mark) = []
      pi(s) = []
      pi(dbl) = []
      pi(first) = []
      pi(active) = [1]
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p9

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> mark#(X)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(first(X1,X2)) -> mark#(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, p2, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(add(X1,X2)) -> mark#(X2)
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(sqr(X)) -> mark#(X)
p8: mark#(recip(X)) -> mark#(X)
p9: mark#(cons(X1,X2)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (0,1)
          terms_A(x1) = ((1,0),(1,1)) x1 + (18,11)
          first_A(x1,x2) = x1 + x2 + (35,18)
          add_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (20,13)
          active#_A(x1) = (5,14)
          mark_A(x1) = ((1,0),(1,1)) x1 + (0,22)
          s_A(x1) = ((0,0),(1,0)) x1 + (4,6)
          sqr_A(x1) = ((1,0),(1,0)) x1 + (3,19)
          recip_A(x1) = ((1,0),(1,0)) x1 + (1,21)
          cons_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (13,23)
          active_A(x1) = x1 + (0,12)
          |0|_A() = (2,10)
          dbl_A(x1) = ((1,0),(1,0)) x1 + (20,21)
          nil_A() = (1,1)
  
    precedence:
    
      terms = first = s = recip = dbl > mark# = add = active# = mark = sqr > cons > active = |0| = nil
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(first) = []
      pi(add) = [2]
      pi(active#) = []
      pi(mark) = []
      pi(s) = []
      pi(sqr) = []
      pi(recip) = []
      pi(cons) = []
      pi(active) = [1]
      pi(|0|) = []
      pi(dbl) = []
      pi(nil) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(add(X1,X2)) -> mark#(X2)
p3: mark#(add(X1,X2)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p5: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(recip(X)) -> mark#(X)
p8: mark#(cons(X1,X2)) -> mark#(X1)

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, p2, p3, p4, p5, p6, p7, p8}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> mark#(X)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> mark#(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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,1)) x1 + (1,1)
          terms_A(x1) = ((1,0),(1,1)) x1 + (8,11)
          cons_A(x1,x2) = ((1,0),(1,1)) x1 + (2,1)
          recip_A(x1) = x1 + (2,2)
          sqr_A(x1) = ((1,0),(1,1)) x1 + (3,2)
          add_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (5,17)
          active#_A(x1) = (4,16)
          mark_A(x1) = x1
          s_A(x1) = (2,12)
          active_A(x1) = x1
          |0|_A() = (1,0)
          dbl_A(x1) = x1 + (3,0)
          first_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,13)
          nil_A() = (2,13)
  
    precedence:
    
      mark = s = active = nil > terms > mark# = cons = recip = sqr = add = active# = |0| = dbl = first
    
    partial status:
  
      pi(mark#) = [1]
      pi(terms) = [1]
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(add) = []
      pi(active#) = []
      pi(mark) = []
      pi(s) = []
      pi(active) = []
      pi(|0|) = []
      pi(dbl) = [1]
      pi(first) = []
      pi(nil) = []

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> mark#(X)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> mark#(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, p2, p3, p4, p6, p7}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(add(X1,X2)) -> mark#(X2)
p3: mark#(add(X1,X2)) -> mark#(X1)
p4: mark#(sqr(X)) -> mark#(X)
p5: mark#(recip(X)) -> mark#(X)
p6: mark#(cons(X1,X2)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((0,0),(1,0)) x1 + (2,2)
          terms_A(x1) = ((1,0),(0,0)) x1 + (3,1)
          add_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (3,3)
          sqr_A(x1) = ((1,0),(0,0)) x1 + (3,3)
          recip_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1)
  
    precedence:
    
      terms = add > recip > mark# = sqr > cons
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(add) = []
      pi(sqr) = []
      pi(recip) = []
      pi(cons) = [2]

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(add(X1,X2)) -> mark#(X2)
p3: mark#(add(X1,X2)) -> mark#(X1)
p4: mark#(recip(X)) -> mark#(X)
p5: mark#(cons(X1,X2)) -> mark#(X1)

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, p2, p3, p4, p5}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> mark#(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          terms_A(x1) = x1 + (2,2)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,2)
          recip_A(x1) = x1 + (2,2)
          add_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (2,2)
  
    precedence:
    
      terms = cons > mark# = recip = add
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(cons) = [2]
      pi(recip) = [1]
      pi(add) = [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: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(recip(X)) -> mark#(X)
p3: mark#(add(X1,X2)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> mark#(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, p2, p3, p4}


-- Reduction pair.

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

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(add(X1,X2)) -> mark#(X2)
p3: mark#(add(X1,X2)) -> mark#(X1)
p4: mark#(recip(X)) -> 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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1
          cons_A(x1,x2) = x1 + x2 + (1,1)
          add_A(x1,x2) = x1 + x2 + (1,1)
          recip_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      cons > mark# = add = recip
    
    partial status:
  
      pi(mark#) = []
      pi(cons) = [1, 2]
      pi(add) = [1, 2]
      pi(recip) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(add(X1,X2)) -> mark#(X2)
p3: mark#(add(X1,X2)) -> mark#(X1)

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, p2, p3}


-- Reduction pair.

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

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(add(X1,X2)) -> mark#(X1)
p3: mark#(add(X1,X2)) -> mark#(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1
          cons_A(x1,x2) = x1 + x2 + (1,1)
          add_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (1,1)
  
    precedence:
    
      cons = add > mark#
    
    partial status:
  
      pi(mark#) = []
      pi(cons) = [2]
      pi(add) = []

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: mark#(add(X1,X2)) -> mark#(X1)
p2: mark#(add(X1,X2)) -> mark#(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, p2}


-- Reduction pair.

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

p1: mark#(add(X1,X2)) -> mark#(X1)
p2: mark#(add(X1,X2)) -> mark#(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1
          add_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (1,1)
  
    precedence:
    
      add > mark#
    
    partial status:
  
      pi(mark#) = []
      pi(add) = [2]

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: mark#(add(X1,X2)) -> mark#(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}


-- Reduction pair.

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

p1: mark#(add(X1,X2)) -> mark#(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,1)) x1
          add_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (1,1)
  
    precedence:
    
      mark# = add
    
    partial status:
  
      pi(mark#) = [1]
      pi(add) = [2]

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: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          cons#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (1,2)
          mark_A(x1) = ((1,0),(0,0)) x1 + (2,3)
          active_A(x1) = ((1,0),(0,0)) x1 + (2,1)
  
    precedence:
    
      cons# = mark > active
    
    partial status:
  
      pi(cons#) = []
      pi(mark) = []
      pi(active) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(active(X1),X2) -> cons#(X1,X2)
p3: 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(X1,mark(X2)) -> cons#(X1,X2)
p3: cons#(active(X1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          cons#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (2,2)
          mark_A(x1) = ((1,0),(0,0)) x1 + (3,1)
          active_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      cons# = mark = active
    
    partial status:
  
      pi(cons#) = []
      pi(mark) = []
      pi(active) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(active(X1),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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(active(X1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          cons#_A(x1,x2) = x1
          mark_A(x1) = ((1,0),(1,0)) x1 + (1,1)
          active_A(x1) = x1 + (1,1)
  
    precedence:
    
      mark > cons# > active
    
    partial status:
  
      pi(cons#) = [1]
      pi(mark) = []
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: cons#(mark(X1),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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: cons#(mark(X1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          cons#_A(x1,x2) = x1
          mark_A(x1) = x1 + (1,1)
  
    precedence:
    
      cons# = mark
    
    partial status:
  
      pi(cons#) = []
      pi(mark) = [1]

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: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          recip#_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          mark_A(x1) = x1 + (2,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (2,2)
  
    precedence:
    
      recip# = mark = active
    
    partial status:
  
      pi(recip#) = []
      pi(mark) = [1]
      pi(active) = []

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: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          recip#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      recip# = active
    
    partial status:
  
      pi(recip#) = []
      pi(active) = []

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: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          sqr#_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          mark_A(x1) = x1 + (2,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (2,2)
  
    precedence:
    
      sqr# = mark = active
    
    partial status:
  
      pi(sqr#) = []
      pi(mark) = [1]
      pi(active) = []

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: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          sqr#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      sqr# = active
    
    partial status:
  
      pi(sqr#) = []
      pi(active) = []

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: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          terms#_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          mark_A(x1) = x1 + (2,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (2,2)
  
    precedence:
    
      terms# = mark = active
    
    partial status:
  
      pi(terms#) = []
      pi(mark) = [1]
      pi(active) = []

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: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          terms#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      terms# = active
    
    partial status:
  
      pi(terms#) = []
      pi(active) = []

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: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          s#_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          mark_A(x1) = x1 + (2,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (2,2)
  
    precedence:
    
      s# = mark = active
    
    partial status:
  
      pi(s#) = []
      pi(mark) = [1]
      pi(active) = []

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: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          s#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      s# = active
    
    partial status:
  
      pi(s#) = []
      pi(active) = []

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: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          add#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (1,2)
          mark_A(x1) = ((1,0),(0,0)) x1 + (2,3)
          active_A(x1) = ((1,0),(0,0)) x1 + (2,1)
  
    precedence:
    
      add# = mark > active
    
    partial status:
  
      pi(add#) = []
      pi(mark) = []
      pi(active) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: add#(mark(X1),X2) -> add#(X1,X2)
p2: add#(active(X1),X2) -> add#(X1,X2)
p3: 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: add#(mark(X1),X2) -> add#(X1,X2)
p2: add#(X1,mark(X2)) -> add#(X1,X2)
p3: add#(active(X1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          add#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (1,2)
          mark_A(x1) = ((1,0),(0,0)) x1 + (2,1)
          active_A(x1) = ((1,0),(0,0)) x1 + (2,1)
  
    precedence:
    
      add# = mark = active
    
    partial status:
  
      pi(add#) = []
      pi(mark) = []
      pi(active) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: add#(mark(X1),X2) -> add#(X1,X2)
p2: add#(active(X1),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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: add#(mark(X1),X2) -> add#(X1,X2)
p2: add#(active(X1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          add#_A(x1,x2) = x1
          mark_A(x1) = ((1,0),(1,0)) x1 + (1,1)
          active_A(x1) = x1 + (1,1)
  
    precedence:
    
      mark > add# > active
    
    partial status:
  
      pi(add#) = [1]
      pi(mark) = []
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: add#(mark(X1),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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: add#(mark(X1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          add#_A(x1,x2) = x1
          mark_A(x1) = x1 + (1,1)
  
    precedence:
    
      add# = mark
    
    partial status:
  
      pi(add#) = []
      pi(mark) = [1]

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: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          dbl#_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          mark_A(x1) = x1 + (2,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (2,2)
  
    precedence:
    
      dbl# = mark = active
    
    partial status:
  
      pi(dbl#) = []
      pi(mark) = [1]
      pi(active) = []

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: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          dbl#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      dbl# = active
    
    partial status:
  
      pi(dbl#) = []
      pi(active) = []

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: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          first#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (1,2)
          mark_A(x1) = ((1,0),(0,0)) x1 + (2,3)
          active_A(x1) = ((1,0),(0,0)) x1 + (2,1)
  
    precedence:
    
      first# = mark > active
    
    partial status:
  
      pi(first#) = []
      pi(mark) = []
      pi(active) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: first#(mark(X1),X2) -> first#(X1,X2)
p2: first#(active(X1),X2) -> first#(X1,X2)
p3: 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: first#(mark(X1),X2) -> first#(X1,X2)
p2: first#(X1,mark(X2)) -> first#(X1,X2)
p3: first#(active(X1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          first#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (1,2)
          mark_A(x1) = ((1,0),(0,0)) x1 + (2,1)
          active_A(x1) = ((1,0),(0,0)) x1 + (2,1)
  
    precedence:
    
      first# = mark = active
    
    partial status:
  
      pi(first#) = []
      pi(mark) = []
      pi(active) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: first#(mark(X1),X2) -> first#(X1,X2)
p2: first#(active(X1),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, p2}


-- Reduction pair.

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

p1: first#(mark(X1),X2) -> first#(X1,X2)
p2: first#(active(X1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          first#_A(x1,x2) = x1
          mark_A(x1) = ((1,0),(1,0)) x1 + (1,1)
          active_A(x1) = x1 + (1,1)
  
    precedence:
    
      mark > first# > active
    
    partial status:
  
      pi(first#) = [1]
      pi(mark) = []
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: first#(mark(X1),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}


-- Reduction pair.

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

p1: first#(mark(X1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          first#_A(x1,x2) = x1
          mark_A(x1) = x1 + (1,1)
  
    precedence:
    
      first# = mark
    
    partial status:
  
      pi(first#) = []
      pi(mark) = [1]

The next rules are strictly ordered:

  p1

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