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: standard order
        interpretations:
          active#_A(x1) = ((1,0),(1,0)) x1 + (7,0)
          terms_A(x1) = ((1,1),(1,1)) x1 + (35,5)
          mark#_A(x1) = ((1,0),(1,0)) x1 + (8,2)
          cons_A(x1,x2) = ((1,1),(0,0)) x1 + (9,9)
          recip_A(x1) = ((1,1),(0,0)) x1 + (1,1)
          sqr_A(x1) = ((1,1),(0,0)) x1 + (9,5)
          s_A(x1) = (3,0)
          first_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (8,30)
          mark_A(x1) = ((1,1),(0,0)) x1
          dbl_A(x1) = ((1,1),(1,1)) x1 + (18,18)
          add_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(0,0)) x2 + (9,12)
          |0|_A() = (0,0)
          active_A(x1) = ((1,1),(0,0)) x1
          nil_A() = (6,1)
  
    precedence:
    
      recip = s = first = dbl = add > terms > active# = mark# = cons = sqr = mark = active = nil > |0|
    
    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:

  p14

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: 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#(recip(X)) -> active#(recip(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#(add(|0|(),X)) -> mark#(X)
p7: mark#(cons(X1,X2)) -> mark#(X1)
p8: mark#(recip(X)) -> active#(recip(mark(X)))
p9: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p10: mark#(recip(X)) -> mark#(X)
p11: mark#(sqr(X)) -> active#(sqr(mark(X)))
p12: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p13: mark#(sqr(X)) -> mark#(X)
p14: mark#(s(X)) -> active#(s(X))
p15: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
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: standard order
        interpretations:
          active#_A(x1) = ((0,1),(0,1)) x1 + (5,2)
          terms_A(x1) = ((1,0),(1,0)) x1 + (37,31)
          mark#_A(x1) = ((1,0),(1,0)) x1 + (4,0)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + (8,2)
          recip_A(x1) = ((1,0),(0,0)) x1 + (3,0)
          sqr_A(x1) = ((1,0),(1,0)) x1 + (20,0)
          s_A(x1) = (14,8)
          mark_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          add_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (5,0)
          dbl_A(x1) = ((1,0),(1,0)) x1 + (3,1)
          |0|_A() = (5,0)
          first_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (4,2)
          active_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          nil_A() = (4,1)
  
    precedence:
    
      |0| > active# = mark# = recip = sqr = dbl = first > s > mark = active > cons > terms > add = 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:

  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#(terms(X)) -> active#(terms(mark(X)))
p3: mark#(terms(X)) -> mark#(X)
p4: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p5: active#(add(|0|(),X)) -> mark#(X)
p6: mark#(cons(X1,X2)) -> mark#(X1)
p7: mark#(recip(X)) -> active#(recip(mark(X)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(recip(X)) -> mark#(X)
p10: mark#(sqr(X)) -> active#(sqr(mark(X)))
p11: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p12: mark#(sqr(X)) -> mark#(X)
p13: mark#(s(X)) -> active#(s(X))
p14: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
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: mark#(sqr(X)) -> mark#(X)
p15: mark#(sqr(X)) -> active#(sqr(mark(X)))
p16: mark#(recip(X)) -> mark#(X)
p17: mark#(recip(X)) -> active#(recip(mark(X)))
p18: mark#(cons(X1,X2)) -> mark#(X1)
p19: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
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: standard order
        interpretations:
          active#_A(x1) = ((1,0),(0,0)) x1
          terms_A(x1) = ((1,1),(1,1)) x1 + (15,12)
          mark#_A(x1) = ((0,1),(0,0)) x1 + (4,0)
          cons_A(x1,x2) = ((0,0),(1,1)) x1
          recip_A(x1) = ((0,0),(1,1)) x1 + (5,2)
          sqr_A(x1) = ((0,0),(1,1)) x1 + (3,0)
          s_A(x1) = (4,0)
          first_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (5,2)
          mark_A(x1) = ((0,0),(1,1)) x1
          dbl_A(x1) = ((0,0),(1,1)) x1 + (5,8)
          add_A(x1,x2) = ((0,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (4,1)
          |0|_A() = (2,0)
          active_A(x1) = ((0,0),(1,1)) x1
          nil_A() = (1,10)
  
    precedence:
    
      terms = nil > active# = mark# = cons = recip = sqr = s = first = mark = dbl = add = |0| = active
    
    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:

  p17

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: mark#(sqr(X)) -> mark#(X)
p15: mark#(sqr(X)) -> active#(sqr(mark(X)))
p16: mark#(recip(X)) -> mark#(X)
p17: mark#(cons(X1,X2)) -> mark#(X1)
p18: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
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#(add(|0|(),X)) -> mark#(X)
p4: mark#(terms(X)) -> mark#(X)
p5: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(cons(X1,X2)) -> mark#(X1)
p8: mark#(recip(X)) -> mark#(X)
p9: mark#(sqr(X)) -> active#(sqr(mark(X)))
p10: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p11: mark#(sqr(X)) -> mark#(X)
p12: mark#(s(X)) -> active#(s(X))
p13: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p14: mark#(add(X1,X2)) -> mark#(X1)
p15: mark#(add(X1,X2)) -> mark#(X2)
p16: mark#(dbl(X)) -> active#(dbl(mark(X)))
p17: mark#(dbl(X)) -> mark#(X)
p18: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
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: standard order
        interpretations:
          active#_A(x1) = ((0,1),(1,0)) x1
          terms_A(x1) = ((1,1),(1,1)) x1 + (34,28)
          mark#_A(x1) = ((1,1),(1,1)) x1 + (5,2)
          cons_A(x1,x2) = x1 + (7,2)
          recip_A(x1) = ((0,1),(1,0)) x1 + (5,2)
          sqr_A(x1) = x1 + (4,2)
          s_A(x1) = (3,3)
          mark_A(x1) = ((1,1),(1,1)) x1
          add_A(x1,x2) = ((0,1),(1,0)) x1 + ((1,1),(1,1)) x2 + (10,9)
          |0|_A() = (2,2)
          first_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (6,8)
          dbl_A(x1) = ((0,1),(1,0)) x1 + (6,2)
          active_A(x1) = x1 + (2,1)
          nil_A() = (1,11)
  
    precedence:
    
      recip = mark = active = nil > |0| > cons = sqr = first > add > active# = terms = mark# = s = dbl
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = [1]
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = [1]
      pi(s) = []
      pi(mark) = []
      pi(add) = []
      pi(|0|) = []
      pi(first) = [1]
      pi(dbl) = []
      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#(terms(X)) -> active#(terms(mark(X)))
p3: mark#(terms(X)) -> mark#(X)
p4: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p5: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p6: mark#(cons(X1,X2)) -> mark#(X1)
p7: mark#(recip(X)) -> mark#(X)
p8: mark#(sqr(X)) -> active#(sqr(mark(X)))
p9: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p10: mark#(sqr(X)) -> mark#(X)
p11: mark#(s(X)) -> active#(s(X))
p12: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p13: mark#(add(X1,X2)) -> mark#(X1)
p14: mark#(add(X1,X2)) -> mark#(X2)
p15: mark#(dbl(X)) -> active#(dbl(mark(X)))
p16: mark#(dbl(X)) -> mark#(X)
p17: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
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: mark#(s(X)) -> active#(s(X))
p13: mark#(sqr(X)) -> mark#(X)
p14: mark#(sqr(X)) -> active#(sqr(mark(X)))
p15: mark#(recip(X)) -> mark#(X)
p16: mark#(cons(X1,X2)) -> mark#(X1)
p17: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p18: mark#(terms(X)) -> mark#(X)
p19: 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: standard order
        interpretations:
          active#_A(x1) = ((1,0),(1,1)) x1 + (11,1)
          terms_A(x1) = ((1,1),(0,0)) x1 + (10,11)
          mark#_A(x1) = ((1,1),(1,1)) x1 + (11,2)
          cons_A(x1,x2) = ((0,0),(1,1)) x1 + (1,2)
          recip_A(x1) = ((1,1),(0,0)) x1 + (1,2)
          sqr_A(x1) = ((1,1),(0,0)) x1 + (1,2)
          s_A(x1) = (0,0)
          first_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(0,0)) x2 + (12,2)
          mark_A(x1) = ((0,0),(1,1)) x1
          dbl_A(x1) = ((1,1),(0,0)) x1 + (12,0)
          add_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + (0,1)
          active_A(x1) = ((0,0),(1,1)) x1
          |0|_A() = (3,0)
          nil_A() = (0,0)
  
    precedence:
    
      terms > sqr = s > cons = add > mark = active = |0| > dbl > active# = mark# > recip = first = 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(active) = []
      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: 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#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: mark#(s(X)) -> active#(s(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)
p16: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p17: mark#(terms(X)) -> mark#(X)
p18: 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}


-- 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#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p4: mark#(terms(X)) -> mark#(X)
p5: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p6: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p7: mark#(cons(X1,X2)) -> mark#(X1)
p8: mark#(recip(X)) -> mark#(X)
p9: mark#(sqr(X)) -> active#(sqr(mark(X)))
p10: mark#(sqr(X)) -> mark#(X)
p11: mark#(s(X)) -> active#(s(X))
p12: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p13: mark#(add(X1,X2)) -> mark#(X1)
p14: mark#(dbl(X)) -> active#(dbl(mark(X)))
p15: mark#(dbl(X)) -> mark#(X)
p16: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p17: mark#(first(X1,X2)) -> mark#(X1)
p18: 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: standard order
        interpretations:
          active#_A(x1) = x1
          terms_A(x1) = ((1,1),(0,1)) x1 + (67,20)
          mark#_A(x1) = ((1,1),(0,1)) x1 + (5,1)
          cons_A(x1,x2) = x1 + (37,6)
          recip_A(x1) = x1 + (1,5)
          sqr_A(x1) = x1 + (4,8)
          s_A(x1) = (36,7)
          mark_A(x1) = ((1,1),(0,1)) x1
          add_A(x1,x2) = x1 + ((1,1),(0,1)) x2 + (12,8)
          first_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,1)) x2 + (0,8)
          dbl_A(x1) = x1 + (3,16)
          active_A(x1) = x1 + (5,0)
          |0|_A() = (2,8)
          nil_A() = (1,14)
  
    precedence:
    
      add > cons = mark = active > s > |0| > first > terms > mark# > sqr > nil > active# = recip = dbl
    
    partial status:
  
      pi(active#) = [1]
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(mark) = []
      pi(add) = []
      pi(first) = []
      pi(dbl) = []
      pi(active) = []
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p14

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


-- 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#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p9: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p10: mark#(s(X)) -> active#(s(X))
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)
p15: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p16: mark#(terms(X)) -> mark#(X)
p17: 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: standard order
        interpretations:
          active#_A(x1) = ((1,1),(0,0)) x1 + (1,0)
          terms_A(x1) = (1,2)
          mark#_A(x1) = (4,0)
          cons_A(x1,x2) = (0,0)
          recip_A(x1) = (4,3)
          sqr_A(x1) = (2,0)
          s_A(x1) = (1,1)
          first_A(x1,x2) = (1,2)
          mark_A(x1) = (1,2)
          dbl_A(x1) = (1,0)
          add_A(x1,x2) = (1,2)
          active_A(x1) = (1,2)
          |0|_A() = (0,0)
          nil_A() = (1,0)
  
    precedence:
    
      active# = terms = mark# = cons = recip = s = first = mark = dbl = add = active = |0| > sqr > 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(active) = []
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p10

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#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p9: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
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)
p14: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p15: mark#(terms(X)) -> mark#(X)
p16: 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}


-- 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#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p4: mark#(terms(X)) -> mark#(X)
p5: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p6: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p7: mark#(cons(X1,X2)) -> mark#(X1)
p8: mark#(recip(X)) -> mark#(X)
p9: mark#(sqr(X)) -> active#(sqr(mark(X)))
p10: mark#(sqr(X)) -> mark#(X)
p11: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p12: mark#(add(X1,X2)) -> mark#(X1)
p13: mark#(dbl(X)) -> mark#(X)
p14: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p15: mark#(first(X1,X2)) -> mark#(X1)
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: standard order
        interpretations:
          active#_A(x1) = ((0,1),(0,0)) x1 + (2,6)
          terms_A(x1) = (3,7)
          mark#_A(x1) = (9,6)
          cons_A(x1,x2) = (0,0)
          recip_A(x1) = ((1,0),(0,0)) x1 + (1,4)
          sqr_A(x1) = (6,5)
          s_A(x1) = ((0,0),(1,0)) x1
          mark_A(x1) = ((1,0),(0,0)) x1 + (2,5)
          add_A(x1,x2) = ((1,0),(0,0)) x2 + (9,7)
          first_A(x1,x2) = ((1,0),(0,0)) x1 + (10,7)
          dbl_A(x1) = ((1,0),(0,0)) x1 + (5,5)
          active_A(x1) = ((1,0),(0,0)) x1 + (0,5)
          |0|_A() = (4,0)
          nil_A() = (1,8)
  
    precedence:
    
      recip > first = dbl > add > sqr = s = mark = active = |0| > active# = mark# > cons > terms = nil
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(mark) = []
      pi(add) = []
      pi(first) = []
      pi(dbl) = []
      pi(active) = []
      pi(|0|) = []
      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#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p4: mark#(terms(X)) -> mark#(X)
p5: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p6: mark#(cons(X1,X2)) -> mark#(X1)
p7: mark#(recip(X)) -> mark#(X)
p8: mark#(sqr(X)) -> active#(sqr(mark(X)))
p9: mark#(sqr(X)) -> mark#(X)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: mark#(add(X1,X2)) -> mark#(X1)
p12: mark#(dbl(X)) -> mark#(X)
p13: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p14: mark#(first(X1,X2)) -> mark#(X1)
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: 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#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p9: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
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)
p14: mark#(terms(X)) -> mark#(X)
p15: 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: standard order
        interpretations:
          active#_A(x1) = ((0,1),(0,1)) x1 + (22,0)
          terms_A(x1) = ((0,0),(0,1)) x1 + (49,24)
          mark#_A(x1) = ((0,1),(0,1)) x1 + (28,1)
          cons_A(x1,x2) = ((0,0),(0,1)) x1 + (43,8)
          recip_A(x1) = ((0,0),(0,1)) x1 + (29,1)
          sqr_A(x1) = ((0,0),(0,1)) x1 + (23,8)
          s_A(x1) = ((0,1),(0,0)) x1 + (1,7)
          first_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (27,0)
          mark_A(x1) = ((0,1),(0,1)) x1 + (30,0)
          dbl_A(x1) = ((0,0),(0,1)) x1 + (9,7)
          add_A(x1,x2) = ((0,0),(0,1)) x1 + ((0,1),(0,1)) x2 + (33,6)
          active_A(x1) = ((0,1),(0,1)) x1 + (24,0)
          |0|_A() = (0,8)
          nil_A() = (30,1)
  
    precedence:
    
      recip > add = |0| > active# = mark# = nil > first > dbl > terms = cons = sqr = s = mark = active
    
    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(active) = []
      pi(|0|) = []
      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#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> mark#(X1)
p4: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p5: mark#(dbl(X)) -> mark#(X)
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
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)
p13: mark#(terms(X)) -> mark#(X)
p14: 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}


-- 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#(add(s(X),Y)) -> mark#(s(add(X,Y)))
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: mark#(sqr(X)) -> mark#(X)
p9: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(dbl(X)) -> mark#(X)
p12: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p13: mark#(first(X1,X2)) -> mark#(X1)
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: standard order
        interpretations:
          active#_A(x1) = ((1,1),(1,0)) x1 + (8,2)
          terms_A(x1) = ((1,1),(1,1)) x1
          mark#_A(x1) = ((1,1),(1,1)) x1 + (8,2)
          cons_A(x1,x2) = ((0,0),(1,1)) x1
          recip_A(x1) = ((0,0),(1,1)) x1
          sqr_A(x1) = ((1,1),(0,0)) x1
          s_A(x1) = ((0,1),(0,1)) x1 + (7,2)
          mark_A(x1) = ((0,0),(1,1)) x1
          add_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(0,0)) x2 + (9,0)
          dbl_A(x1) = ((1,1),(0,0)) x1 + (8,4)
          first_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(0,0)) x2 + (9,11)
          active_A(x1) = ((0,0),(1,1)) x1
          |0|_A() = (2,2)
          nil_A() = (1,1)
  
    precedence:
    
      cons = first > dbl > active# = mark# > recip > add > nil > s = mark = active = |0| > terms = sqr
    
    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:

  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#(terms(X)) -> active#(terms(mark(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: mark#(sqr(X)) -> mark#(X)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(dbl(X)) -> mark#(X)
p11: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p12: mark#(first(X1,X2)) -> mark#(X1)
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: 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: mark#(dbl(X)) -> mark#(X)
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#(sqr(X)) -> active#(sqr(mark(X)))
p10: mark#(recip(X)) -> mark#(X)
p11: mark#(cons(X1,X2)) -> mark#(X1)
p12: mark#(terms(X)) -> mark#(X)
p13: 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: standard order
        interpretations:
          active#_A(x1) = ((0,1),(0,1)) x1 + (3,1)
          terms_A(x1) = ((1,1),(0,0)) x1 + (13,16)
          mark#_A(x1) = (19,17)
          cons_A(x1,x2) = ((0,0),(1,1)) x1 + (2,0)
          recip_A(x1) = ((1,1),(0,0)) x1 + (1,18)
          sqr_A(x1) = ((1,1),(0,0)) x1 + (1,6)
          s_A(x1) = (3,2)
          first_A(x1,x2) = ((1,1),(0,0)) x2 + (4,15)
          mark_A(x1) = ((0,0),(1,1)) x1 + (0,1)
          dbl_A(x1) = ((0,0),(1,1)) x1 + (20,0)
          add_A(x1,x2) = ((1,1),(0,0)) x2 + (17,13)
          active_A(x1) = ((0,0),(1,1)) x1
          |0|_A() = (1,7)
          nil_A() = (1,6)
  
    precedence:
    
      dbl = |0| > recip = nil > active# = mark# > mark = active > add > terms = s > sqr > cons = first
    
    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(active) = []
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p7

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: mark#(dbl(X)) -> mark#(X)
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(sqr(X)) -> mark#(X)
p8: mark#(sqr(X)) -> active#(sqr(mark(X)))
p9: mark#(recip(X)) -> mark#(X)
p10: mark#(cons(X1,X2)) -> mark#(X1)
p11: mark#(terms(X)) -> mark#(X)
p12: 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}


-- 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: 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: mark#(sqr(X)) -> mark#(X)
p8: mark#(add(X1,X2)) -> mark#(X1)
p9: mark#(dbl(X)) -> mark#(X)
p10: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p11: mark#(first(X1,X2)) -> mark#(X1)
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: standard order
        interpretations:
          active#_A(x1) = ((0,1),(0,0)) x1 + (1,0)
          terms_A(x1) = (35,30)
          mark#_A(x1) = (31,0)
          cons_A(x1,x2) = (1,26)
          recip_A(x1) = (36,32)
          sqr_A(x1) = (10,29)
          s_A(x1) = (0,3)
          mark_A(x1) = ((0,1),(1,1)) x1 + (8,1)
          add_A(x1,x2) = ((0,1),(1,1)) x2 + (9,4)
          dbl_A(x1) = (11,15)
          first_A(x1,x2) = (32,28)
          active_A(x1) = x1 + (2,1)
          |0|_A() = (10,4)
          nil_A() = (0,23)
  
    precedence:
    
      terms > mark = dbl = first = active > add > recip > active# = mark# = cons = s = |0| > sqr > 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:

  p6

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: mark#(terms(X)) -> mark#(X)
p4: mark#(cons(X1,X2)) -> mark#(X1)
p5: mark#(recip(X)) -> mark#(X)
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(dbl(X)) -> mark#(X)
p9: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p10: mark#(first(X1,X2)) -> mark#(X1)
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: 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: mark#(dbl(X)) -> mark#(X)
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(sqr(X)) -> mark#(X)
p8: mark#(recip(X)) -> mark#(X)
p9: mark#(cons(X1,X2)) -> mark#(X1)
p10: mark#(terms(X)) -> mark#(X)
p11: 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: standard order
        interpretations:
          active#_A(x1) = ((0,1),(0,0)) x1 + (1,0)
          terms_A(x1) = (0,2)
          mark#_A(x1) = (3,0)
          cons_A(x1,x2) = (3,0)
          recip_A(x1) = (7,1)
          sqr_A(x1) = (7,4)
          s_A(x1) = (4,3)
          first_A(x1,x2) = (0,1)
          mark_A(x1) = (6,0)
          dbl_A(x1) = (5,5)
          add_A(x1,x2) = (5,5)
          active_A(x1) = (6,0)
          |0|_A() = (1,1)
          nil_A() = (1,0)
  
    precedence:
    
      active# = mark# = first > terms > sqr = s = mark = dbl = add = active > recip > cons > |0| > 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(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)) -> mark#(X1)
p4: mark#(dbl(X)) -> mark#(X)
p5: mark#(add(X1,X2)) -> mark#(X1)
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(recip(X)) -> mark#(X)
p8: mark#(cons(X1,X2)) -> mark#(X1)
p9: mark#(terms(X)) -> mark#(X)
p10: 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}


-- 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: mark#(terms(X)) -> mark#(X)
p4: mark#(cons(X1,X2)) -> mark#(X1)
p5: mark#(recip(X)) -> mark#(X)
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(dbl(X)) -> mark#(X)
p9: mark#(first(X1,X2)) -> mark#(X1)
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: standard order
        interpretations:
          active#_A(x1) = ((0,1),(0,1)) x1 + (1,4)
          terms_A(x1) = x1 + (17,15)
          mark#_A(x1) = ((0,1),(0,1)) x1 + (12,16)
          cons_A(x1,x2) = ((0,0),(0,1)) x1 + (8,1)
          recip_A(x1) = ((0,1),(0,1)) x1 + (0,1)
          sqr_A(x1) = ((0,0),(0,1)) x1 + (15,1)
          s_A(x1) = (0,5)
          mark_A(x1) = x1 + (9,0)
          add_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (9,16)
          dbl_A(x1) = ((1,1),(0,1)) x1 + (13,16)
          first_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (13,16)
          active_A(x1) = x1
          |0|_A() = (5,1)
          nil_A() = (0,17)
  
    precedence:
    
      mark = active > |0| > recip > nil > active# = cons > sqr > add > first > dbl > s > terms = mark#
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(mark) = []
      pi(add) = []
      pi(dbl) = []
      pi(first) = [2]
      pi(active) = []
      pi(|0|) = []
      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: mark#(terms(X)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(recip(X)) -> mark#(X)
p5: mark#(sqr(X)) -> mark#(X)
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(dbl(X)) -> mark#(X)
p8: mark#(first(X1,X2)) -> mark#(X1)
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:

  {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#(first(X1,X2)) -> mark#(X1)
p4: mark#(dbl(X)) -> mark#(X)
p5: mark#(add(X1,X2)) -> mark#(X1)
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 set of usable rules consists of

  (no rules)

Take the reduction pair:

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

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: mark#(dbl(X)) -> mark#(X)
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: mark#(sqr(X)) -> mark#(X)
p6: mark#(recip(X)) -> mark#(X)
p7: 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}


-- 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)) -> mark#(X1)
p6: mark#(dbl(X)) -> mark#(X)
p7: 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

  (no rules)

Take the reduction pair:

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

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#(cons(X1,X2)) -> mark#(X1)
p3: mark#(sqr(X)) -> mark#(X)
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: mark#(dbl(X)) -> mark#(X)
p6: 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}


-- 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)) -> mark#(X)
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: mark#(sqr(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: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          terms_A(x1) = x1 + (3,1)
          first_A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (1,1)
          dbl_A(x1) = x1 + (1,1)
          add_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (1,1)
          sqr_A(x1) = x1 + (1,1)
          cons_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (1,1)
  
    precedence:
    
      dbl > mark# = terms = first = add = sqr = cons
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = [1]
      pi(first) = []
      pi(dbl) = [1]
      pi(add) = []
      pi(sqr) = []
      pi(cons) = [2]

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: mark#(add(X1,X2)) -> mark#(X1)
p4: mark#(sqr(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#(sqr(X)) -> mark#(X)
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: 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

  (no rules)

Take the reduction pair:

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

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#(sqr(X)) -> mark#(X)
p4: 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}


-- 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#(sqr(X)) -> mark#(X)
p4: 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: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,1)) x1 + (2,2)
          terms_A(x1) = ((1,1),(1,1)) x1 + (3,2)
          first_A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (1,1)
          sqr_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          cons_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (3,1)
  
    precedence:
    
      terms > mark# = sqr = cons > first
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = [1]
      pi(first) = [2]
      pi(sqr) = []
      pi(cons) = []

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#(sqr(X)) -> mark#(X)
p3: 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}


-- 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#(sqr(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: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          terms_A(x1) = x1 + (1,1)
          cons_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (0,1)
          sqr_A(x1) = ((0,0),(0,1)) x1 + (3,1)
  
    precedence:
    
      mark# > terms = cons = sqr
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = [1]
      pi(cons) = [2]
      pi(sqr) = []

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#(sqr(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}


-- Reduction pair.

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

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(sqr(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: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          terms_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          sqr_A(x1) = ((0,1),(0,1)) x1 + (1,1)
  
    precedence:
    
      mark# = terms = sqr
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(sqr) = []

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#(sqr(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}


-- Reduction pair.

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

p1: mark#(sqr(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: standard order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          sqr_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      mark# = sqr
    
    partial status:
  
      pi(mark#) = []
      pi(sqr) = []

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: standard order
        interpretations:
          cons#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (2,2)
          mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
          active_A(x1) = ((1,1),(1,1)) x1 + (1,1)
  
    precedence:
    
      cons# = mark = active
    
    partial status:
  
      pi(cons#) = [1]
      pi(mark) = [1]
      pi(active) = [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: cons#(X1,active(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#(X1,active(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: standard order
        interpretations:
          cons#_A(x1,x2) = ((1,1),(1,1)) x1 + (2,2)
          active_A(x1) = ((1,1),(1,1)) x1 + (3,2)
          mark_A(x1) = (1,1)
  
    precedence:
    
      cons# > active = mark
    
    partial status:
  
      pi(cons#) = []
      pi(active) = []
      pi(mark) = []

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


-- Reduction pair.

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

p1: cons#(X1,active(X2)) -> cons#(X1,X2)
p2: 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: standard order
        interpretations:
          cons#_A(x1,x2) = x1 + x2 + (2,2)
          active_A(x1) = x1 + (1,0)
          mark_A(x1) = x1 + (1,1)
  
    precedence:
    
      cons# = active = mark
    
    partial status:
  
      pi(cons#) = [2]
      pi(active) = [1]
      pi(mark) = [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: 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}


-- Reduction pair.

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

p1: 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: standard order
        interpretations:
          cons#_A(x1,x2) = x1 + ((0,1),(0,0)) x2 + (2,2)
          mark_A(x1) = ((0,0),(0,1)) x1 + (1,1)
  
    precedence:
    
      cons# = mark
    
    partial status:
  
      pi(cons#) = []
      pi(mark) = []

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: 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: standard order
        interpretations:
          recip#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          mark_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          active_A(x1) = ((0,1),(0,1)) x1 + (1,1)
  
    precedence:
    
      recip# = mark = active
    
    partial status:
  
      pi(recip#) = []
      pi(mark) = []
      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: standard 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: standard order
        interpretations:
          sqr#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          mark_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          active_A(x1) = ((0,1),(0,1)) x1 + (1,1)
  
    precedence:
    
      sqr# = mark = active
    
    partial status:
  
      pi(sqr#) = []
      pi(mark) = []
      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: standard 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: standard order
        interpretations:
          terms#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          mark_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          active_A(x1) = ((0,1),(0,1)) x1 + (1,1)
  
    precedence:
    
      terms# = mark = active
    
    partial status:
  
      pi(terms#) = []
      pi(mark) = []
      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: standard 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: standard order
        interpretations:
          s#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          mark_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          active_A(x1) = ((0,1),(0,1)) x1 + (1,1)
  
    precedence:
    
      s# = mark = active
    
    partial status:
  
      pi(s#) = []
      pi(mark) = []
      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: standard 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: standard order
        interpretations:
          add#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (2,2)
          mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
          active_A(x1) = ((1,1),(1,1)) x1 + (1,1)
  
    precedence:
    
      add# = mark = active
    
    partial status:
  
      pi(add#) = [1]
      pi(mark) = [1]
      pi(active) = [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: add#(X1,active(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#(X1,active(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: standard order
        interpretations:
          add#_A(x1,x2) = ((1,1),(1,1)) x1 + (2,2)
          active_A(x1) = ((1,1),(1,1)) x1 + (3,2)
          mark_A(x1) = (1,1)
  
    precedence:
    
      add# > active = mark
    
    partial status:
  
      pi(add#) = []
      pi(active) = []
      pi(mark) = []

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


-- Reduction pair.

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

p1: add#(X1,active(X2)) -> add#(X1,X2)
p2: 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: standard order
        interpretations:
          add#_A(x1,x2) = x1 + x2 + (2,2)
          active_A(x1) = x1 + (1,0)
          mark_A(x1) = x1 + (1,1)
  
    precedence:
    
      add# = active = mark
    
    partial status:
  
      pi(add#) = [2]
      pi(active) = [1]
      pi(mark) = [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: 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}


-- Reduction pair.

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

p1: 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: standard order
        interpretations:
          add#_A(x1,x2) = x1 + ((0,1),(0,0)) x2 + (2,2)
          mark_A(x1) = ((0,0),(0,1)) x1 + (1,1)
  
    precedence:
    
      add# = mark
    
    partial status:
  
      pi(add#) = []
      pi(mark) = []

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: 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: standard order
        interpretations:
          dbl#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          mark_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          active_A(x1) = ((0,1),(0,1)) x1 + (1,1)
  
    precedence:
    
      dbl# = mark = active
    
    partial status:
  
      pi(dbl#) = []
      pi(mark) = []
      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: standard 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: standard order
        interpretations:
          first#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (2,2)
          mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
          active_A(x1) = ((1,1),(1,1)) x1 + (1,1)
  
    precedence:
    
      first# = mark = active
    
    partial status:
  
      pi(first#) = [1]
      pi(mark) = [1]
      pi(active) = [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: first#(X1,active(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#(X1,active(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: standard order
        interpretations:
          first#_A(x1,x2) = ((1,1),(1,1)) x1 + (2,2)
          active_A(x1) = ((1,1),(1,1)) x1 + (3,2)
          mark_A(x1) = (1,1)
  
    precedence:
    
      first# > active = mark
    
    partial status:
  
      pi(first#) = []
      pi(active) = []
      pi(mark) = []

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


-- Reduction pair.

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

p1: first#(X1,active(X2)) -> first#(X1,X2)
p2: 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: standard order
        interpretations:
          first#_A(x1,x2) = x1 + x2 + (2,2)
          active_A(x1) = x1 + (1,0)
          mark_A(x1) = x1 + (1,1)
  
    precedence:
    
      first# = active = mark
    
    partial status:
  
      pi(first#) = [2]
      pi(active) = [1]
      pi(mark) = [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: 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}


-- Reduction pair.

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

p1: 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: standard order
        interpretations:
          first#_A(x1,x2) = x1 + ((0,1),(0,0)) x2 + (2,2)
          mark_A(x1) = ((0,0),(0,1)) x1 + (1,1)
  
    precedence:
    
      first# = mark
    
    partial status:
  
      pi(first#) = []
      pi(mark) = []

The next rules are strictly ordered:

  p1

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