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)))
  active(terms(X)) -> terms(active(X))
  active(cons(X1,X2)) -> cons(active(X1),X2)
  active(recip(X)) -> recip(active(X))
  active(sqr(X)) -> sqr(active(X))
  active(add(X1,X2)) -> add(active(X1),X2)
  active(add(X1,X2)) -> add(X1,active(X2))
  active(dbl(X)) -> dbl(active(X))
  active(first(X1,X2)) -> first(active(X1),X2)
  active(first(X1,X2)) -> first(X1,active(X2))
  terms(mark(X)) -> mark(terms(X))
  cons(mark(X1),X2) -> mark(cons(X1,X2))
  recip(mark(X)) -> mark(recip(X))
  sqr(mark(X)) -> mark(sqr(X))
  add(mark(X1),X2) -> mark(add(X1,X2))
  add(X1,mark(X2)) -> mark(add(X1,X2))
  dbl(mark(X)) -> mark(dbl(X))
  first(mark(X1),X2) -> mark(first(X1,X2))
  first(X1,mark(X2)) -> mark(first(X1,X2))
  proper(terms(X)) -> terms(proper(X))
  proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
  proper(recip(X)) -> recip(proper(X))
  proper(sqr(X)) -> sqr(proper(X))
  proper(s(X)) -> s(proper(X))
  proper(|0|()) -> ok(|0|())
  proper(add(X1,X2)) -> add(proper(X1),proper(X2))
  proper(dbl(X)) -> dbl(proper(X))
  proper(first(X1,X2)) -> first(proper(X1),proper(X2))
  proper(nil()) -> ok(nil())
  terms(ok(X)) -> ok(terms(X))
  cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
  recip(ok(X)) -> ok(recip(X))
  sqr(ok(X)) -> ok(sqr(X))
  s(ok(X)) -> ok(s(X))
  add(ok(X1),ok(X2)) -> ok(add(X1,X2))
  dbl(ok(X)) -> ok(dbl(X))
  first(ok(X1),ok(X2)) -> ok(first(X1,X2))
  top(mark(X)) -> top(proper(X))
  top(ok(X)) -> top(active(X))

-- SCC decomposition.

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

p1: active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
p2: active#(terms(N)) -> recip#(sqr(N))
p3: active#(terms(N)) -> sqr#(N)
p4: active#(terms(N)) -> terms#(s(N))
p5: active#(terms(N)) -> s#(N)
p6: active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
p7: active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
p8: active#(sqr(s(X))) -> sqr#(X)
p9: active#(sqr(s(X))) -> dbl#(X)
p10: active#(dbl(s(X))) -> s#(s(dbl(X)))
p11: active#(dbl(s(X))) -> s#(dbl(X))
p12: active#(dbl(s(X))) -> dbl#(X)
p13: active#(add(s(X),Y)) -> s#(add(X,Y))
p14: active#(add(s(X),Y)) -> add#(X,Y)
p15: active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
p16: active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
p17: active#(terms(X)) -> terms#(active(X))
p18: active#(terms(X)) -> active#(X)
p19: active#(cons(X1,X2)) -> cons#(active(X1),X2)
p20: active#(cons(X1,X2)) -> active#(X1)
p21: active#(recip(X)) -> recip#(active(X))
p22: active#(recip(X)) -> active#(X)
p23: active#(sqr(X)) -> sqr#(active(X))
p24: active#(sqr(X)) -> active#(X)
p25: active#(add(X1,X2)) -> add#(active(X1),X2)
p26: active#(add(X1,X2)) -> active#(X1)
p27: active#(add(X1,X2)) -> add#(X1,active(X2))
p28: active#(add(X1,X2)) -> active#(X2)
p29: active#(dbl(X)) -> dbl#(active(X))
p30: active#(dbl(X)) -> active#(X)
p31: active#(first(X1,X2)) -> first#(active(X1),X2)
p32: active#(first(X1,X2)) -> active#(X1)
p33: active#(first(X1,X2)) -> first#(X1,active(X2))
p34: active#(first(X1,X2)) -> active#(X2)
p35: terms#(mark(X)) -> terms#(X)
p36: cons#(mark(X1),X2) -> cons#(X1,X2)
p37: recip#(mark(X)) -> recip#(X)
p38: sqr#(mark(X)) -> sqr#(X)
p39: add#(mark(X1),X2) -> add#(X1,X2)
p40: add#(X1,mark(X2)) -> add#(X1,X2)
p41: dbl#(mark(X)) -> dbl#(X)
p42: first#(mark(X1),X2) -> first#(X1,X2)
p43: first#(X1,mark(X2)) -> first#(X1,X2)
p44: proper#(terms(X)) -> terms#(proper(X))
p45: proper#(terms(X)) -> proper#(X)
p46: proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
p47: proper#(cons(X1,X2)) -> proper#(X1)
p48: proper#(cons(X1,X2)) -> proper#(X2)
p49: proper#(recip(X)) -> recip#(proper(X))
p50: proper#(recip(X)) -> proper#(X)
p51: proper#(sqr(X)) -> sqr#(proper(X))
p52: proper#(sqr(X)) -> proper#(X)
p53: proper#(s(X)) -> s#(proper(X))
p54: proper#(s(X)) -> proper#(X)
p55: proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
p56: proper#(add(X1,X2)) -> proper#(X1)
p57: proper#(add(X1,X2)) -> proper#(X2)
p58: proper#(dbl(X)) -> dbl#(proper(X))
p59: proper#(dbl(X)) -> proper#(X)
p60: proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
p61: proper#(first(X1,X2)) -> proper#(X1)
p62: proper#(first(X1,X2)) -> proper#(X2)
p63: terms#(ok(X)) -> terms#(X)
p64: cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
p65: recip#(ok(X)) -> recip#(X)
p66: sqr#(ok(X)) -> sqr#(X)
p67: s#(ok(X)) -> s#(X)
p68: add#(ok(X1),ok(X2)) -> add#(X1,X2)
p69: dbl#(ok(X)) -> dbl#(X)
p70: first#(ok(X1),ok(X2)) -> first#(X1,X2)
p71: top#(mark(X)) -> top#(proper(X))
p72: top#(mark(X)) -> proper#(X)
p73: top#(ok(X)) -> top#(active(X))
p74: top#(ok(X)) -> active#(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The estimated dependency graph contains the following SCCs:

  {p71, p73}
  {p18, p20, p22, p24, p26, p28, p30, p32, p34}
  {p45, p47, p48, p50, p52, p54, p56, p57, p59, p61, p62}
  {p36, p64}
  {p37, p65}
  {p38, p66}
  {p35, p63}
  {p67}
  {p39, p40, p68}
  {p41, p69}
  {p42, p43, p70}


-- Reduction pair.

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

p1: top#(ok(X)) -> top#(active(X))
p2: top#(mark(X)) -> top#(proper(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

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, r42, r43, r44, r45

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            top#_A(x1) = ((0,0),(1,0)) x1 + (0,1)
            ok_A(x1) = ((1,0),(0,0)) x1 + (0,38)
            active_A(x1) = ((1,0),(1,1)) x1 + (0,2)
            mark_A(x1) = ((1,0),(0,0)) x1 + (4,79)
            proper_A(x1) = ((1,0),(1,1)) x1 + (0,12)
            terms_A(x1) = ((1,0),(0,0)) x1 + (11,82)
            cons_A(x1,x2) = ((1,0),(1,0)) x1 + (1,78)
            recip_A(x1) = ((1,0),(1,0)) x1 + (0,76)
            sqr_A(x1) = ((1,0),(1,1)) x1 + (5,0)
            add_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (13,12)
            dbl_A(x1) = x1 + (14,11)
            first_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (15,80)
            s_A(x1) = (14,38)
            |0|_A() = (1,69)
            nil_A() = (2,69)
    
      precedence:
      
        sqr > add > proper > active > first > dbl > recip > terms > top# = mark = s = |0| > cons > nil > ok
      
      partial status:
    
        pi(top#) = []
        pi(ok) = []
        pi(active) = [1]
        pi(mark) = []
        pi(proper) = [1]
        pi(terms) = []
        pi(cons) = []
        pi(recip) = []
        pi(sqr) = []
        pi(add) = [1, 2]
        pi(dbl) = [1]
        pi(first) = []
        pi(s) = []
        pi(|0|) = []
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            top#_A(x1) = (0,10)
            ok_A(x1) = (5,6)
            active_A(x1) = ((0,0),(1,0)) x1 + (4,9)
            mark_A(x1) = (2,3)
            proper_A(x1) = (7,2)
            terms_A(x1) = (1,4)
            cons_A(x1,x2) = (0,2)
            recip_A(x1) = (3,5)
            sqr_A(x1) = (8,8)
            add_A(x1,x2) = (3,1)
            dbl_A(x1) = (7,1)
            first_A(x1,x2) = (3,2)
            s_A(x1) = (6,7)
            |0|_A() = (6,0)
            nil_A() = (0,0)
    
      precedence:
      
        |0| > proper = nil > sqr = dbl = s > ok = recip > active > top# > first > mark = add > terms > cons
      
      partial status:
    
        pi(top#) = []
        pi(ok) = []
        pi(active) = []
        pi(mark) = []
        pi(proper) = []
        pi(terms) = []
        pi(cons) = []
        pi(recip) = []
        pi(sqr) = []
        pi(add) = []
        pi(dbl) = []
        pi(first) = []
        pi(s) = []
        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: top#(ok(X)) -> top#(active(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: top#(ok(X)) -> top#(active(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

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, r38, r39, r40, r41, r42, r43, r44, r45

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          top#_A(x1) = x1 + 138
          ok_A(x1) = max{136, x1 + 83}
          active_A(x1) = x1
          terms_A(x1) = x1 + 51
          mark_A(x1) = 51
          cons_A(x1,x2) = max{x1 - 169, x2 + 67}
          recip_A(x1) = x1 + 82
          sqr_A(x1) = x1 + 137
          add_A(x1,x2) = x1 + 51
          dbl_A(x1) = x1 + 51
          first_A(x1,x2) = max{x1 + 54, x2 + 54}
          s_A(x1) = x1 + 39
          |0|_A = 0
          nil_A = 0
    
      precedence:
      
        active = terms = mark = cons = recip = sqr = add = dbl = first = s > top# = ok = |0| > nil
      
      partial status:
    
        pi(top#) = [1]
        pi(ok) = [1]
        pi(active) = [1]
        pi(terms) = [1]
        pi(mark) = []
        pi(cons) = [2]
        pi(recip) = [1]
        pi(sqr) = [1]
        pi(add) = [1]
        pi(dbl) = [1]
        pi(first) = [1, 2]
        pi(s) = []
        pi(|0|) = []
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          top#_A(x1) = max{405, x1 + 152}
          ok_A(x1) = max{404, x1 + 255}
          active_A(x1) = max{406, x1 + 195}
          terms_A(x1) = x1 + 38
          mark_A(x1) = 686
          cons_A(x1,x2) = x2 + 164
          recip_A(x1) = max{232, x1 + 128}
          sqr_A(x1) = max{297, x1 + 106}
          add_A(x1,x2) = max{109, x1 + 3}
          dbl_A(x1) = max{403, x1 + 194}
          first_A(x1,x2) = max{38, x1 - 256, x2 + 19}
          s_A(x1) = 296
          |0|_A = 17
          nil_A = 18
    
      precedence:
      
        top# = ok = active = terms = mark = cons = recip = sqr = add = dbl = first = s = |0| = nil
      
      partial status:
    
        pi(top#) = []
        pi(ok) = [1]
        pi(active) = []
        pi(terms) = [1]
        pi(mark) = []
        pi(cons) = [2]
        pi(recip) = [1]
        pi(sqr) = [1]
        pi(add) = [1]
        pi(dbl) = [1]
        pi(first) = [2]
        pi(s) = []
        pi(|0|) = []
        pi(nil) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: active#(first(X1,X2)) -> active#(X2)
p2: active#(first(X1,X2)) -> active#(X1)
p3: active#(dbl(X)) -> active#(X)
p4: active#(add(X1,X2)) -> active#(X2)
p5: active#(add(X1,X2)) -> active#(X1)
p6: active#(sqr(X)) -> active#(X)
p7: active#(recip(X)) -> active#(X)
p8: active#(cons(X1,X2)) -> active#(X1)
p9: active#(terms(X)) -> active#(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          active#_A(x1) = x1 + 4
          first_A(x1,x2) = max{x1 + 2, x2 + 1}
          dbl_A(x1) = x1 + 2
          add_A(x1,x2) = max{x1 + 4, x2 + 3}
          sqr_A(x1) = x1 + 2
          recip_A(x1) = x1 + 4
          cons_A(x1,x2) = max{x1 + 1, x2 - 1}
          terms_A(x1) = x1 + 5
    
      precedence:
      
        active# = first = dbl = add = sqr = recip = cons = terms
      
      partial status:
    
        pi(active#) = []
        pi(first) = []
        pi(dbl) = []
        pi(add) = []
        pi(sqr) = []
        pi(recip) = []
        pi(cons) = []
        pi(terms) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          active#_A(x1) = 1
          first_A(x1,x2) = 0
          dbl_A(x1) = x1
          add_A(x1,x2) = 0
          sqr_A(x1) = x1
          recip_A(x1) = x1
          cons_A(x1,x2) = 0
          terms_A(x1) = x1
    
      precedence:
      
        active# = first = dbl = add = sqr = recip = cons = terms
      
      partial status:
    
        pi(active#) = []
        pi(first) = []
        pi(dbl) = [1]
        pi(add) = []
        pi(sqr) = [1]
        pi(recip) = [1]
        pi(cons) = []
        pi(terms) = [1]
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8, p9

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

-- Reduction pair.

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

p1: proper#(first(X1,X2)) -> proper#(X2)
p2: proper#(first(X1,X2)) -> proper#(X1)
p3: proper#(dbl(X)) -> proper#(X)
p4: proper#(add(X1,X2)) -> proper#(X2)
p5: proper#(add(X1,X2)) -> proper#(X1)
p6: proper#(s(X)) -> proper#(X)
p7: proper#(sqr(X)) -> proper#(X)
p8: proper#(recip(X)) -> proper#(X)
p9: proper#(cons(X1,X2)) -> proper#(X2)
p10: proper#(cons(X1,X2)) -> proper#(X1)
p11: proper#(terms(X)) -> proper#(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          proper#_A(x1) = max{2, x1 + 1}
          first_A(x1,x2) = max{x1, x2}
          dbl_A(x1) = max{1, x1}
          add_A(x1,x2) = max{x1, x2}
          s_A(x1) = max{1, x1}
          sqr_A(x1) = max{1, x1}
          recip_A(x1) = max{1, x1}
          cons_A(x1,x2) = max{x1, x2}
          terms_A(x1) = max{1, x1}
    
      precedence:
      
        proper# = first = dbl = add = s = sqr = recip = cons = terms
      
      partial status:
    
        pi(proper#) = [1]
        pi(first) = [1, 2]
        pi(dbl) = [1]
        pi(add) = [1, 2]
        pi(s) = [1]
        pi(sqr) = [1]
        pi(recip) = [1]
        pi(cons) = [1, 2]
        pi(terms) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          proper#_A(x1) = x1 + 1
          first_A(x1,x2) = max{x1, x2}
          dbl_A(x1) = x1
          add_A(x1,x2) = max{x1, x2}
          s_A(x1) = x1
          sqr_A(x1) = x1
          recip_A(x1) = x1
          cons_A(x1,x2) = max{x1, x2}
          terms_A(x1) = x1
    
      precedence:
      
        proper# = first = dbl = add = s = sqr = recip = cons = terms
      
      partial status:
    
        pi(proper#) = [1]
        pi(first) = [1, 2]
        pi(dbl) = [1]
        pi(add) = [1, 2]
        pi(s) = [1]
        pi(sqr) = [1]
        pi(recip) = [1]
        pi(cons) = [1, 2]
        pi(terms) = [1]
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11

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#(ok(X1),ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          cons#_A(x1,x2) = max{2, x1 + 1, x2 + 1}
          mark_A(x1) = max{1, x1}
          ok_A(x1) = max{1, x1}
    
      precedence:
      
        cons# = mark = ok
      
      partial status:
    
        pi(cons#) = [1, 2]
        pi(mark) = [1]
        pi(ok) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          cons#_A(x1,x2) = max{x1 + 1, x2 - 2}
          mark_A(x1) = x1
          ok_A(x1) = max{4, x1 + 1}
    
      precedence:
      
        cons# = mark = ok
      
      partial status:
    
        pi(cons#) = [1]
        pi(mark) = [1]
        pi(ok) = [1]
    

The next rules are strictly ordered:

  p1, p2

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

-- Reduction pair.

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

p1: recip#(mark(X)) -> recip#(X)
p2: recip#(ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

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

The next rules are strictly ordered:

  p1, p2

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

-- Reduction pair.

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

p1: sqr#(mark(X)) -> sqr#(X)
p2: sqr#(ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

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

The next rules are strictly ordered:

  p1, p2

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

-- Reduction pair.

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

p1: terms#(mark(X)) -> terms#(X)
p2: terms#(ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

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

The next rules are strictly ordered:

  p1, p2

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

-- Reduction pair.

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

p1: s#(ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

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

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: add#(mark(X1),X2) -> add#(X1,X2)
p2: add#(ok(X1),ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          add#_A(x1,x2) = max{2, x1}
          mark_A(x1) = max{2, x1 + 1}
          ok_A(x1) = max{7, x1 + 3}
    
      precedence:
      
        add# = ok > mark
      
      partial status:
    
        pi(add#) = [1]
        pi(mark) = [1]
        pi(ok) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          add#_A(x1,x2) = 0
          mark_A(x1) = x1 + 1
          ok_A(x1) = max{3, x1 + 1}
    
      precedence:
      
        add# = ok > mark
      
      partial status:
    
        pi(add#) = []
        pi(mark) = [1]
        pi(ok) = []
    

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#(ok(X1),ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: add#(ok(X1),ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          add#_A(x1,x2) = x2 + 1
          ok_A(x1) = x1 + 2
          mark_A(x1) = x1 + 1
    
      precedence:
      
        add# = ok = mark
      
      partial status:
    
        pi(add#) = []
        pi(ok) = []
        pi(mark) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          add#_A(x1,x2) = 0
          ok_A(x1) = max{2, x1}
          mark_A(x1) = x1
    
      precedence:
      
        add# = ok = mark
      
      partial status:
    
        pi(add#) = []
        pi(ok) = [1]
        pi(mark) = [1]
    

The next rules are strictly ordered:

  p1, p2

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

-- Reduction pair.

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

p1: dbl#(mark(X)) -> dbl#(X)
p2: dbl#(ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

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

The next rules are strictly ordered:

  p1, p2

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

-- Reduction pair.

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

p1: first#(mark(X1),X2) -> first#(X1,X2)
p2: first#(ok(X1),ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          first#_A(x1,x2) = max{2, x1}
          mark_A(x1) = max{2, x1 + 1}
          ok_A(x1) = max{7, x1 + 3}
    
      precedence:
      
        first# = ok > mark
      
      partial status:
    
        pi(first#) = [1]
        pi(mark) = [1]
        pi(ok) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          first#_A(x1,x2) = 0
          mark_A(x1) = x1 + 1
          ok_A(x1) = max{3, x1 + 1}
    
      precedence:
      
        first# = ok > mark
      
      partial status:
    
        pi(first#) = []
        pi(mark) = [1]
        pi(ok) = []
    

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#(ok(X1),ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: first#(ok(X1),ok(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: active(terms(X)) -> terms(active(X))
r11: active(cons(X1,X2)) -> cons(active(X1),X2)
r12: active(recip(X)) -> recip(active(X))
r13: active(sqr(X)) -> sqr(active(X))
r14: active(add(X1,X2)) -> add(active(X1),X2)
r15: active(add(X1,X2)) -> add(X1,active(X2))
r16: active(dbl(X)) -> dbl(active(X))
r17: active(first(X1,X2)) -> first(active(X1),X2)
r18: active(first(X1,X2)) -> first(X1,active(X2))
r19: terms(mark(X)) -> mark(terms(X))
r20: cons(mark(X1),X2) -> mark(cons(X1,X2))
r21: recip(mark(X)) -> mark(recip(X))
r22: sqr(mark(X)) -> mark(sqr(X))
r23: add(mark(X1),X2) -> mark(add(X1,X2))
r24: add(X1,mark(X2)) -> mark(add(X1,X2))
r25: dbl(mark(X)) -> mark(dbl(X))
r26: first(mark(X1),X2) -> mark(first(X1,X2))
r27: first(X1,mark(X2)) -> mark(first(X1,X2))
r28: proper(terms(X)) -> terms(proper(X))
r29: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r30: proper(recip(X)) -> recip(proper(X))
r31: proper(sqr(X)) -> sqr(proper(X))
r32: proper(s(X)) -> s(proper(X))
r33: proper(|0|()) -> ok(|0|())
r34: proper(add(X1,X2)) -> add(proper(X1),proper(X2))
r35: proper(dbl(X)) -> dbl(proper(X))
r36: proper(first(X1,X2)) -> first(proper(X1),proper(X2))
r37: proper(nil()) -> ok(nil())
r38: terms(ok(X)) -> ok(terms(X))
r39: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r40: recip(ok(X)) -> ok(recip(X))
r41: sqr(ok(X)) -> ok(sqr(X))
r42: s(ok(X)) -> ok(s(X))
r43: add(ok(X1),ok(X2)) -> ok(add(X1,X2))
r44: dbl(ok(X)) -> ok(dbl(X))
r45: first(ok(X1),ok(X2)) -> ok(first(X1,X2))
r46: top(mark(X)) -> top(proper(X))
r47: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          first#_A(x1,x2) = x2 + 1
          ok_A(x1) = x1 + 2
          mark_A(x1) = x1 + 1
    
      precedence:
      
        first# = ok = mark
      
      partial status:
    
        pi(first#) = []
        pi(ok) = []
        pi(mark) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          first#_A(x1,x2) = 0
          ok_A(x1) = max{2, x1}
          mark_A(x1) = x1
    
      precedence:
      
        first# = ok = mark
      
      partial status:
    
        pi(first#) = []
        pi(ok) = [1]
        pi(mark) = [1]
    

The next rules are strictly ordered:

  p1, p2

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