YES

We show the termination of the TRS R:

  active(and(true(),X)) -> mark(X)
  active(and(false(),Y)) -> mark(false())
  active(if(true(),X,Y)) -> mark(X)
  active(if(false(),X,Y)) -> mark(Y)
  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(from(X)) -> mark(cons(X,from(s(X))))
  mark(and(X1,X2)) -> active(and(mark(X1),X2))
  mark(true()) -> active(true())
  mark(false()) -> active(false())
  mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
  mark(add(X1,X2)) -> active(add(mark(X1),X2))
  mark(|0|()) -> active(|0|())
  mark(s(X)) -> active(s(X))
  mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
  mark(nil()) -> active(nil())
  mark(cons(X1,X2)) -> active(cons(X1,X2))
  mark(from(X)) -> active(from(X))
  and(mark(X1),X2) -> and(X1,X2)
  and(X1,mark(X2)) -> and(X1,X2)
  and(active(X1),X2) -> and(X1,X2)
  and(X1,active(X2)) -> and(X1,X2)
  if(mark(X1),X2,X3) -> if(X1,X2,X3)
  if(X1,mark(X2),X3) -> if(X1,X2,X3)
  if(X1,X2,mark(X3)) -> if(X1,X2,X3)
  if(active(X1),X2,X3) -> if(X1,X2,X3)
  if(X1,active(X2),X3) -> if(X1,X2,X3)
  if(X1,X2,active(X3)) -> if(X1,X2,X3)
  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)
  s(mark(X)) -> s(X)
  s(active(X)) -> s(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)
  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)
  from(mark(X)) -> from(X)
  from(active(X)) -> from(X)

-- SCC decomposition.

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

p1: active#(and(true(),X)) -> mark#(X)
p2: active#(and(false(),Y)) -> mark#(false())
p3: active#(if(true(),X,Y)) -> mark#(X)
p4: active#(if(false(),X,Y)) -> mark#(Y)
p5: active#(add(|0|(),X)) -> mark#(X)
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: active#(add(s(X),Y)) -> s#(add(X,Y))
p8: active#(add(s(X),Y)) -> add#(X,Y)
p9: active#(first(|0|(),X)) -> mark#(nil())
p10: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p11: active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
p12: active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
p13: active#(from(X)) -> mark#(cons(X,from(s(X))))
p14: active#(from(X)) -> cons#(X,from(s(X)))
p15: active#(from(X)) -> from#(s(X))
p16: active#(from(X)) -> s#(X)
p17: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p18: mark#(and(X1,X2)) -> and#(mark(X1),X2)
p19: mark#(and(X1,X2)) -> mark#(X1)
p20: mark#(true()) -> active#(true())
p21: mark#(false()) -> active#(false())
p22: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p23: mark#(if(X1,X2,X3)) -> if#(mark(X1),X2,X3)
p24: mark#(if(X1,X2,X3)) -> mark#(X1)
p25: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p26: mark#(add(X1,X2)) -> add#(mark(X1),X2)
p27: mark#(add(X1,X2)) -> mark#(X1)
p28: mark#(|0|()) -> active#(|0|())
p29: mark#(s(X)) -> active#(s(X))
p30: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p31: mark#(first(X1,X2)) -> first#(mark(X1),mark(X2))
p32: mark#(first(X1,X2)) -> mark#(X1)
p33: mark#(first(X1,X2)) -> mark#(X2)
p34: mark#(nil()) -> active#(nil())
p35: mark#(cons(X1,X2)) -> active#(cons(X1,X2))
p36: mark#(from(X)) -> active#(from(X))
p37: and#(mark(X1),X2) -> and#(X1,X2)
p38: and#(X1,mark(X2)) -> and#(X1,X2)
p39: and#(active(X1),X2) -> and#(X1,X2)
p40: and#(X1,active(X2)) -> and#(X1,X2)
p41: if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
p42: if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
p43: if#(X1,X2,mark(X3)) -> if#(X1,X2,X3)
p44: if#(active(X1),X2,X3) -> if#(X1,X2,X3)
p45: if#(X1,active(X2),X3) -> if#(X1,X2,X3)
p46: if#(X1,X2,active(X3)) -> if#(X1,X2,X3)
p47: add#(mark(X1),X2) -> add#(X1,X2)
p48: add#(X1,mark(X2)) -> add#(X1,X2)
p49: add#(active(X1),X2) -> add#(X1,X2)
p50: add#(X1,active(X2)) -> add#(X1,X2)
p51: s#(mark(X)) -> s#(X)
p52: s#(active(X)) -> s#(X)
p53: first#(mark(X1),X2) -> first#(X1,X2)
p54: first#(X1,mark(X2)) -> first#(X1,X2)
p55: first#(active(X1),X2) -> first#(X1,X2)
p56: first#(X1,active(X2)) -> first#(X1,X2)
p57: cons#(mark(X1),X2) -> cons#(X1,X2)
p58: cons#(X1,mark(X2)) -> cons#(X1,X2)
p59: cons#(active(X1),X2) -> cons#(X1,X2)
p60: cons#(X1,active(X2)) -> cons#(X1,X2)
p61: from#(mark(X)) -> from#(X)
p62: from#(active(X)) -> from#(X)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1, p3, p4, p5, p6, p10, p13, p17, p19, p22, p24, p25, p27, p29, p30, p32, p33, p35, p36}
  {p51, p52}
  {p47, p48, p49, p50}
  {p57, p58, p59, p60}
  {p53, p54, p55, p56}
  {p61, p62}
  {p37, p38, p39, p40}
  {p41, p42, p43, p44, p45, p46}


-- Reduction pair.

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

p1: active#(and(true(),X)) -> mark#(X)
p2: mark#(from(X)) -> active#(from(X))
p3: active#(from(X)) -> mark#(cons(X,from(s(X))))
p4: mark#(cons(X1,X2)) -> active#(cons(X1,X2))
p5: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p6: mark#(first(X1,X2)) -> mark#(X2)
p7: mark#(first(X1,X2)) -> mark#(X1)
p8: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p9: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p10: mark#(s(X)) -> active#(s(X))
p11: active#(add(|0|(),X)) -> mark#(X)
p12: mark#(add(X1,X2)) -> mark#(X1)
p13: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p14: active#(if(false(),X,Y)) -> mark#(Y)
p15: mark#(if(X1,X2,X3)) -> mark#(X1)
p16: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p17: active#(if(true(),X,Y)) -> mark#(X)
p18: mark#(and(X1,X2)) -> mark#(X1)
p19: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        active#_A(x1) = x1 + 14
        and_A(x1,x2) = max{x1 + 6, x2 + 6}
        true_A = 4
        mark#_A(x1) = x1 + 14
        from_A(x1) = max{12, x1 - 1}
        cons_A(x1,x2) = max{5, x2}
        s_A(x1) = max{5, x1}
        first_A(x1,x2) = max{x1 + 6, x2 + 10}
        mark_A(x1) = x1
        add_A(x1,x2) = max{x1 + 6, x2}
        |0|_A = 4
        if_A(x1,x2,x3) = max{x1 + 12, x2 + 15, x3 + 13}
        false_A = 5
        active_A(x1) = max{3, x1}
        nil_A = 9
  
    precedence:
    
      active# = and = true = mark# = from = cons = s = first = mark = add = |0| = if = false = active = nil
    
    partial status:
  
      pi(active#) = [1]
      pi(and) = []
      pi(true) = []
      pi(mark#) = [1]
      pi(from) = []
      pi(cons) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = [1]
      pi(add) = [2]
      pi(|0|) = []
      pi(if) = []
      pi(false) = []
      pi(active) = [1]
      pi(nil) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(from(X)) -> active#(from(X))
p2: active#(from(X)) -> mark#(cons(X,from(s(X))))
p3: mark#(cons(X1,X2)) -> active#(cons(X1,X2))
p4: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p5: mark#(first(X1,X2)) -> mark#(X2)
p6: mark#(first(X1,X2)) -> mark#(X1)
p7: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(s(X)) -> active#(s(X))
p10: active#(add(|0|(),X)) -> mark#(X)
p11: mark#(add(X1,X2)) -> mark#(X1)
p12: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p13: active#(if(false(),X,Y)) -> mark#(Y)
p14: mark#(if(X1,X2,X3)) -> mark#(X1)
p15: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p16: active#(if(true(),X,Y)) -> mark#(X)
p17: mark#(and(X1,X2)) -> mark#(X1)
p18: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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: mark#(from(X)) -> active#(from(X))
p2: active#(if(true(),X,Y)) -> mark#(X)
p3: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p4: active#(if(false(),X,Y)) -> mark#(Y)
p5: mark#(and(X1,X2)) -> mark#(X1)
p6: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p7: active#(add(|0|(),X)) -> mark#(X)
p8: mark#(if(X1,X2,X3)) -> mark#(X1)
p9: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p10: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p11: mark#(add(X1,X2)) -> mark#(X1)
p12: mark#(s(X)) -> active#(s(X))
p13: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p14: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p15: active#(from(X)) -> mark#(cons(X,from(s(X))))
p16: mark#(first(X1,X2)) -> mark#(X1)
p17: mark#(first(X1,X2)) -> mark#(X2)
p18: mark#(cons(X1,X2)) -> active#(cons(X1,X2))

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{18, x1}
        from_A(x1) = max{38, x1 + 30}
        active#_A(x1) = max{5, x1 - 9}
        if_A(x1,x2,x3) = max{x1 + 17, x2 + 17, x3 + 17}
        true_A = 10
        and_A(x1,x2) = max{x1 + 7, x2 + 12}
        mark_A(x1) = x1
        false_A = 11
        add_A(x1,x2) = max{x1 + 80, x2 + 80}
        |0|_A = 9
        s_A(x1) = 8
        first_A(x1,x2) = max{x1 + 36, x2 + 29}
        cons_A(x1,x2) = max{29, x1 + 21}
        active_A(x1) = max{8, x1}
        nil_A = 8
  
    precedence:
    
      add > mark# = active# = true = and = false > from = if = mark = |0| = s = first = cons = active = nil
    
    partial status:
  
      pi(mark#) = []
      pi(from) = []
      pi(active#) = []
      pi(if) = []
      pi(true) = []
      pi(and) = []
      pi(mark) = []
      pi(false) = []
      pi(add) = []
      pi(|0|) = []
      pi(s) = []
      pi(first) = []
      pi(cons) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p12

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(from(X)) -> active#(from(X))
p2: active#(if(true(),X,Y)) -> mark#(X)
p3: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p4: active#(if(false(),X,Y)) -> mark#(Y)
p5: mark#(and(X1,X2)) -> mark#(X1)
p6: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p7: active#(add(|0|(),X)) -> mark#(X)
p8: mark#(if(X1,X2,X3)) -> mark#(X1)
p9: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p10: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p11: mark#(add(X1,X2)) -> mark#(X1)
p12: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p13: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p14: active#(from(X)) -> mark#(cons(X,from(s(X))))
p15: mark#(first(X1,X2)) -> mark#(X1)
p16: mark#(first(X1,X2)) -> mark#(X2)
p17: mark#(cons(X1,X2)) -> active#(cons(X1,X2))

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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: mark#(from(X)) -> active#(from(X))
p2: active#(from(X)) -> mark#(cons(X,from(s(X))))
p3: mark#(cons(X1,X2)) -> active#(cons(X1,X2))
p4: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p5: mark#(first(X1,X2)) -> mark#(X2)
p6: mark#(first(X1,X2)) -> mark#(X1)
p7: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p11: active#(add(|0|(),X)) -> mark#(X)
p12: mark#(if(X1,X2,X3)) -> mark#(X1)
p13: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p14: active#(if(false(),X,Y)) -> mark#(Y)
p15: mark#(and(X1,X2)) -> mark#(X1)
p16: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p17: active#(if(true(),X,Y)) -> mark#(X)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 10
        from_A(x1) = x1 + 94
        active#_A(x1) = max{45, x1 - 7}
        cons_A(x1,x2) = max{76, x1 - 11}
        s_A(x1) = 95
        first_A(x1,x2) = max{x1 + 35, x2 + 35}
        mark_A(x1) = x1
        add_A(x1,x2) = max{x1 + 113, x2 + 113}
        |0|_A = 6
        if_A(x1,x2,x3) = max{x1 + 4, x2 + 36, x3 + 36}
        false_A = 6
        and_A(x1,x2) = max{x1 + 1, x2 + 36}
        true_A = 6
        active_A(x1) = max{5, x1}
        nil_A = 34
  
    precedence:
    
      mark# > from > active# = cons = s = first = mark = add = |0| = if = false = and = true = active = nil
    
    partial status:
  
      pi(mark#) = [1]
      pi(from) = []
      pi(active#) = []
      pi(cons) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = [1]
      pi(add) = [2]
      pi(|0|) = []
      pi(if) = []
      pi(false) = []
      pi(and) = [2]
      pi(true) = []
      pi(active) = [1]
      pi(nil) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: active#(from(X)) -> mark#(cons(X,from(s(X))))
p2: mark#(cons(X1,X2)) -> active#(cons(X1,X2))
p3: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p4: mark#(first(X1,X2)) -> mark#(X2)
p5: mark#(first(X1,X2)) -> mark#(X1)
p6: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p7: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p8: mark#(add(X1,X2)) -> mark#(X1)
p9: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p10: active#(add(|0|(),X)) -> mark#(X)
p11: mark#(if(X1,X2,X3)) -> mark#(X1)
p12: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p13: active#(if(false(),X,Y)) -> mark#(Y)
p14: mark#(and(X1,X2)) -> mark#(X1)
p15: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p16: active#(if(true(),X,Y)) -> mark#(X)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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#(from(X)) -> mark#(cons(X,from(s(X))))
p2: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p3: active#(if(true(),X,Y)) -> mark#(X)
p4: mark#(and(X1,X2)) -> mark#(X1)
p5: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p6: active#(if(false(),X,Y)) -> mark#(Y)
p7: mark#(if(X1,X2,X3)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p9: active#(add(|0|(),X)) -> mark#(X)
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p12: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p13: mark#(first(X1,X2)) -> mark#(X1)
p14: mark#(first(X1,X2)) -> mark#(X2)
p15: mark#(cons(X1,X2)) -> active#(cons(X1,X2))
p16: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        active#_A(x1) = x1 + 41
        from_A(x1) = 169
        mark#_A(x1) = 96
        cons_A(x1,x2) = 55
        s_A(x1) = 169
        and_A(x1,x2) = 0
        mark_A(x1) = 167
        if_A(x1,x2,x3) = 55
        true_A = 124
        false_A = 124
        add_A(x1,x2) = 55
        |0|_A = 147
        first_A(x1,x2) = 55
        active_A(x1) = max{167, x1 - 3}
        nil_A = 170
  
    precedence:
    
      active# = from = mark# = cons = s = and = mark = if = true = false = add = |0| = first = active = nil
    
    partial status:
  
      pi(active#) = []
      pi(from) = []
      pi(mark#) = []
      pi(cons) = []
      pi(s) = []
      pi(and) = []
      pi(mark) = []
      pi(if) = []
      pi(true) = []
      pi(false) = []
      pi(add) = []
      pi(|0|) = []
      pi(first) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p2: active#(if(true(),X,Y)) -> mark#(X)
p3: mark#(and(X1,X2)) -> mark#(X1)
p4: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p5: active#(if(false(),X,Y)) -> mark#(Y)
p6: mark#(if(X1,X2,X3)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p8: active#(add(|0|(),X)) -> mark#(X)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p11: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p12: mark#(first(X1,X2)) -> mark#(X1)
p13: mark#(first(X1,X2)) -> mark#(X2)
p14: mark#(cons(X1,X2)) -> active#(cons(X1,X2))
p15: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p2: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p3: mark#(cons(X1,X2)) -> active#(cons(X1,X2))
p4: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p5: mark#(first(X1,X2)) -> mark#(X2)
p6: mark#(first(X1,X2)) -> mark#(X1)
p7: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p8: active#(add(|0|(),X)) -> mark#(X)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p11: active#(if(false(),X,Y)) -> mark#(Y)
p12: mark#(if(X1,X2,X3)) -> mark#(X1)
p13: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p14: active#(if(true(),X,Y)) -> mark#(X)
p15: mark#(and(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 22
        and_A(x1,x2) = max{x1 + 25, x2 + 24}
        active#_A(x1) = max{27, x1}
        mark_A(x1) = x1 + 3
        first_A(x1,x2) = max{x1 + 75, x2 + 75}
        s_A(x1) = 14
        cons_A(x1,x2) = 6
        add_A(x1,x2) = max{26, x1 + 23, x2 + 22}
        |0|_A = 7
        if_A(x1,x2,x3) = max{x1 + 27, x2 + 27, x3 + 27}
        false_A = 6
        true_A = 7
        active_A(x1) = max{9, x1}
        nil_A = 6
        from_A(x1) = x1 + 13
  
    precedence:
    
      mark = s = cons = add = nil > mark# = and = |0| = if > active# = first > false = true = active > from
    
    partial status:
  
      pi(mark#) = [1]
      pi(and) = [1]
      pi(active#) = [1]
      pi(mark) = [1]
      pi(first) = []
      pi(s) = []
      pi(cons) = []
      pi(add) = [1]
      pi(|0|) = []
      pi(if) = []
      pi(false) = []
      pi(true) = []
      pi(active) = [1]
      pi(nil) = []
      pi(from) = []

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: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p2: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p3: mark#(cons(X1,X2)) -> active#(cons(X1,X2))
p4: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p5: mark#(first(X1,X2)) -> mark#(X2)
p6: mark#(first(X1,X2)) -> mark#(X1)
p7: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p8: active#(add(|0|(),X)) -> mark#(X)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p11: active#(if(false(),X,Y)) -> mark#(Y)
p12: mark#(if(X1,X2,X3)) -> mark#(X1)
p13: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p14: mark#(and(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p2: active#(if(false(),X,Y)) -> mark#(Y)
p3: mark#(and(X1,X2)) -> mark#(X1)
p4: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p5: active#(add(|0|(),X)) -> mark#(X)
p6: mark#(if(X1,X2,X3)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p11: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p12: mark#(first(X1,X2)) -> mark#(X1)
p13: mark#(first(X1,X2)) -> mark#(X2)
p14: mark#(cons(X1,X2)) -> active#(cons(X1,X2))

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = 28
        and_A(x1,x2) = 28
        active#_A(x1) = max{6, x1}
        mark_A(x1) = 47
        if_A(x1,x2,x3) = 28
        false_A = 37
        add_A(x1,x2) = 28
        |0|_A = 36
        s_A(x1) = 7
        first_A(x1,x2) = 28
        cons_A(x1,x2) = 6
        active_A(x1) = max{47, x1 + 10}
        true_A = 37
        nil_A = 37
        from_A(x1) = 4
  
    precedence:
    
      mark# = and = active# = mark = if = false = add = |0| = s = first = cons = active = true = nil = from
    
    partial status:
  
      pi(mark#) = []
      pi(and) = []
      pi(active#) = []
      pi(mark) = []
      pi(if) = []
      pi(false) = []
      pi(add) = []
      pi(|0|) = []
      pi(s) = []
      pi(first) = []
      pi(cons) = []
      pi(active) = []
      pi(true) = []
      pi(nil) = []
      pi(from) = []

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: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p2: active#(if(false(),X,Y)) -> mark#(Y)
p3: mark#(and(X1,X2)) -> mark#(X1)
p4: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p5: active#(add(|0|(),X)) -> mark#(X)
p6: mark#(if(X1,X2,X3)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p11: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p12: mark#(first(X1,X2)) -> mark#(X1)
p13: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(and(X1,X2)) -> active#(and(mark(X1),X2))
p2: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p3: mark#(first(X1,X2)) -> mark#(X2)
p4: mark#(first(X1,X2)) -> mark#(X1)
p5: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p9: active#(add(|0|(),X)) -> mark#(X)
p10: mark#(if(X1,X2,X3)) -> mark#(X1)
p11: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p12: active#(if(false(),X,Y)) -> mark#(Y)
p13: mark#(and(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = 60
        and_A(x1,x2) = 56
        active#_A(x1) = max{8, x1}
        mark_A(x1) = 22
        first_A(x1,x2) = 60
        s_A(x1) = 60
        cons_A(x1,x2) = 21
        add_A(x1,x2) = 60
        |0|_A = 21
        if_A(x1,x2,x3) = 60
        false_A = 24
        active_A(x1) = max{22, x1 - 63}
        true_A = 54
        nil_A = 1
        from_A(x1) = 85
  
    precedence:
    
      mark# = active# > and = mark = first = s = cons = add = |0| = if = false = active = true = nil = from
    
    partial status:
  
      pi(mark#) = []
      pi(and) = []
      pi(active#) = []
      pi(mark) = []
      pi(first) = []
      pi(s) = []
      pi(cons) = []
      pi(add) = []
      pi(|0|) = []
      pi(if) = []
      pi(false) = []
      pi(active) = []
      pi(true) = []
      pi(nil) = []
      pi(from) = []

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: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
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#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p8: active#(add(|0|(),X)) -> mark#(X)
p9: mark#(if(X1,X2,X3)) -> mark#(X1)
p10: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p11: active#(if(false(),X,Y)) -> mark#(Y)
p12: mark#(and(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p2: mark#(and(X1,X2)) -> mark#(X1)
p3: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p4: active#(if(false(),X,Y)) -> mark#(Y)
p5: mark#(if(X1,X2,X3)) -> mark#(X1)
p6: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p7: active#(add(|0|(),X)) -> mark#(X)
p8: mark#(add(X1,X2)) -> mark#(X1)
p9: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p10: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p11: mark#(first(X1,X2)) -> mark#(X1)
p12: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        active#_A(x1) = max{1, x1 - 11}
        first_A(x1,x2) = max{x1, x2 + 27}
        s_A(x1) = 15
        cons_A(x1,x2) = max{29, x1 + 1}
        mark#_A(x1) = x1 + 8
        and_A(x1,x2) = max{x1 + 4, x2 + 5}
        if_A(x1,x2,x3) = max{x1 + 7, x2 + 14, x3 + 19}
        mark_A(x1) = x1
        false_A = 5
        add_A(x1,x2) = max{x1 + 32, x2 + 33}
        |0|_A = 7
        active_A(x1) = max{5, x1}
        true_A = 5
        nil_A = 6
        from_A(x1) = max{52, x1 + 36}
  
    precedence:
    
      active# = first = cons > s = and = if = false = add > mark# = mark = |0| = active = true = nil = from
    
    partial status:
  
      pi(active#) = []
      pi(first) = []
      pi(s) = []
      pi(cons) = []
      pi(mark#) = []
      pi(and) = []
      pi(if) = [2]
      pi(mark) = [1]
      pi(false) = []
      pi(add) = [2]
      pi(|0|) = []
      pi(active) = [1]
      pi(true) = []
      pi(nil) = []
      pi(from) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
p2: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p3: active#(if(false(),X,Y)) -> mark#(Y)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(add(|0|(),X)) -> mark#(X)
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p9: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p10: mark#(first(X1,X2)) -> mark#(X1)
p11: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
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#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p8: active#(add(|0|(),X)) -> mark#(X)
p9: mark#(if(X1,X2,X3)) -> mark#(X1)
p10: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p11: active#(if(false(),X,Y)) -> mark#(Y)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1
        and_A(x1,x2) = max{x1 + 3, x2 + 3}
        first_A(x1,x2) = max{x1 + 3, x2 + 4}
        active#_A(x1) = x1
        mark_A(x1) = x1
        add_A(x1,x2) = max{x1, x2}
        s_A(x1) = x1
        |0|_A = 2
        if_A(x1,x2,x3) = max{x1 + 1, x2 + 2, x3 + 2}
        false_A = 1
        active_A(x1) = x1
        true_A = 1
        nil_A = 1
        cons_A(x1,x2) = x2
        from_A(x1) = max{3, x1 + 2}
  
    precedence:
    
      mark# = and = first = active# = mark = add = s = |0| = if = false = active = true = nil = cons = from
    
    partial status:
  
      pi(mark#) = []
      pi(and) = []
      pi(first) = []
      pi(active#) = []
      pi(mark) = [1]
      pi(add) = [2]
      pi(s) = []
      pi(|0|) = []
      pi(if) = []
      pi(false) = []
      pi(active) = [1]
      pi(true) = []
      pi(nil) = []
      pi(cons) = []
      pi(from) = [1]

The next rules are strictly ordered:

  p11

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
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#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p8: active#(add(|0|(),X)) -> mark#(X)
p9: mark#(if(X1,X2,X3)) -> mark#(X1)
p10: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}


-- Reduction pair.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
p2: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p3: active#(add(|0|(),X)) -> mark#(X)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p9: mark#(first(X1,X2)) -> mark#(X1)
p10: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = 18
        and_A(x1,x2) = 21
        if_A(x1,x2,x3) = 21
        active#_A(x1) = max{17, x1 - 3}
        mark_A(x1) = 25
        add_A(x1,x2) = 21
        |0|_A = 21
        s_A(x1) = 20
        first_A(x1,x2) = 20
        active_A(x1) = max{25, x1 + 4}
        true_A = 21
        false_A = 1
        nil_A = 1
        cons_A(x1,x2) = 21
        from_A(x1) = 1
  
    precedence:
    
      cons > from > and = if > mark# = active# > mark = add = |0| = s = first = active = true = false = nil
    
    partial status:
  
      pi(mark#) = []
      pi(and) = []
      pi(if) = []
      pi(active#) = []
      pi(mark) = []
      pi(add) = []
      pi(|0|) = []
      pi(s) = []
      pi(first) = []
      pi(active) = []
      pi(true) = []
      pi(false) = []
      pi(nil) = []
      pi(cons) = []
      pi(from) = []

The next rules are strictly ordered:

  p8

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
p2: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p3: active#(add(|0|(),X)) -> mark#(X)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(first(X1,X2)) -> mark#(X1)
p9: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(if(X1,X2,X3)) -> mark#(X1)
p8: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p9: active#(add(|0|(),X)) -> mark#(X)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{0, x1 - 9}
        and_A(x1,x2) = max{x1, x2 + 42}
        first_A(x1,x2) = max{123, x1 + 2, x2 + 43}
        add_A(x1,x2) = max{x1 + 28, x2 + 70}
        active#_A(x1) = max{18, x1 - 35}
        mark_A(x1) = max{42, x1}
        s_A(x1) = 39
        if_A(x1,x2,x3) = max{x1, x2 + 52, x3}
        |0|_A = 45
        active_A(x1) = max{37, x1}
        true_A = 43
        false_A = 43
        nil_A = 44
        cons_A(x1,x2) = 36
        from_A(x1) = x1 + 42
  
    precedence:
    
      mark# > and = first = add = active# = mark = s = if = |0| = active = true = false = nil = cons = from
    
    partial status:
  
      pi(mark#) = []
      pi(and) = []
      pi(first) = []
      pi(add) = []
      pi(active#) = []
      pi(mark) = [1]
      pi(s) = []
      pi(if) = [3]
      pi(|0|) = []
      pi(active) = [1]
      pi(true) = []
      pi(false) = []
      pi(nil) = []
      pi(cons) = []
      pi(from) = [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#(and(X1,X2)) -> mark#(X1)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p5: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p6: mark#(if(X1,X2,X3)) -> mark#(X1)
p7: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p8: active#(add(|0|(),X)) -> mark#(X)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8}


-- Reduction pair.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
p2: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p3: active#(add(|0|(),X)) -> mark#(X)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p7: mark#(first(X1,X2)) -> mark#(X1)
p8: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1
        and_A(x1,x2) = max{x1 + 5, x2 + 21}
        if_A(x1,x2,x3) = max{x1 + 14, x2 + 18, x3 + 18}
        active#_A(x1) = max{10, x1 - 8}
        mark_A(x1) = max{12, x1 + 8}
        add_A(x1,x2) = max{x1 + 8, x2 + 12}
        |0|_A = 4
        s_A(x1) = 9
        first_A(x1,x2) = max{25, x1 + 21, x2 + 8}
        active_A(x1) = max{12, x1}
        true_A = 11
        false_A = 13
        nil_A = 13
        cons_A(x1,x2) = max{4, x1 - 8}
        from_A(x1) = x1
  
    precedence:
    
      if > mark# = active# > and = mark = add = |0| = s = first = active = true = false = nil = cons = from
    
    partial status:
  
      pi(mark#) = [1]
      pi(and) = []
      pi(if) = []
      pi(active#) = []
      pi(mark) = []
      pi(add) = []
      pi(|0|) = []
      pi(s) = []
      pi(first) = []
      pi(active) = []
      pi(true) = []
      pi(false) = []
      pi(nil) = []
      pi(cons) = []
      pi(from) = []

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#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p2: active#(add(|0|(),X)) -> mark#(X)
p3: mark#(if(X1,X2,X3)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p5: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p6: mark#(first(X1,X2)) -> mark#(X1)
p7: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p2: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p3: mark#(first(X1,X2)) -> mark#(X2)
p4: mark#(first(X1,X2)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(add(|0|(),X)) -> mark#(X)
p7: mark#(if(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{13, x1}
        if_A(x1,x2,x3) = max{x1 + 7, x2 + 15, x3 + 15}
        active#_A(x1) = max{6, x1}
        mark_A(x1) = x1
        add_A(x1,x2) = max{x1 + 14, x2 + 17}
        s_A(x1) = max{5, x1}
        first_A(x1,x2) = max{15, x1 + 7, x2 + 12}
        |0|_A = 4
        active_A(x1) = max{2, x1}
        and_A(x1,x2) = x2 + 3
        true_A = 3
        false_A = 3
        nil_A = 3
        cons_A(x1,x2) = 2
        from_A(x1) = x1 + 5
  
    precedence:
    
      add = nil = cons = from > mark# = if = active# > and = true > false > mark > s = first = |0| = active
    
    partial status:
  
      pi(mark#) = [1]
      pi(if) = [3]
      pi(active#) = [1]
      pi(mark) = [1]
      pi(add) = []
      pi(s) = []
      pi(first) = [1, 2]
      pi(|0|) = []
      pi(active) = [1]
      pi(and) = []
      pi(true) = []
      pi(false) = []
      pi(nil) = []
      pi(cons) = []
      pi(from) = [1]

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p2: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p3: mark#(first(X1,X2)) -> mark#(X2)
p4: mark#(first(X1,X2)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: mark#(if(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p2: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p3: mark#(if(X1,X2,X3)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p5: mark#(first(X1,X2)) -> mark#(X1)
p6: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = 5
        if_A(x1,x2,x3) = 71
        active#_A(x1) = max{4, x1 - 76}
        mark_A(x1) = 79
        add_A(x1,x2) = 81
        s_A(x1) = 8
        first_A(x1,x2) = 80
        active_A(x1) = max{79, x1 - 2}
        and_A(x1,x2) = 71
        true_A = 21
        false_A = 81
        |0|_A = 33
        nil_A = 79
        cons_A(x1,x2) = 13
        from_A(x1) = 81
  
    precedence:
    
      mark# = if = active# = mark = add = s = first = active = and = true = false = |0| = nil = cons = from
    
    partial status:
  
      pi(mark#) = []
      pi(if) = []
      pi(active#) = []
      pi(mark) = []
      pi(add) = []
      pi(s) = []
      pi(first) = []
      pi(active) = []
      pi(and) = []
      pi(true) = []
      pi(false) = []
      pi(|0|) = []
      pi(nil) = []
      pi(cons) = []
      pi(from) = []

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: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p2: mark#(if(X1,X2,X3)) -> mark#(X1)
p3: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p4: mark#(first(X1,X2)) -> mark#(X1)
p5: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p5: mark#(if(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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, r46

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        active#_A(x1) = max{1, x1}
        add_A(x1,x2) = max{x1 + 10, x2 + 10}
        s_A(x1) = max{6, x1 - 3}
        mark#_A(x1) = x1
        first_A(x1,x2) = max{x1 + 5, x2 + 4}
        mark_A(x1) = x1
        if_A(x1,x2,x3) = max{x1 + 1, x2 + 3, x3 + 3}
        active_A(x1) = x1
        and_A(x1,x2) = x2 + 2
        true_A = 1
        false_A = 1
        |0|_A = 0
        nil_A = 1
        cons_A(x1,x2) = x1 + 4
        from_A(x1) = max{6, x1 + 4}
  
    precedence:
    
      true = nil > from > and = false = |0| = cons > active# = add > s = mark = if > mark# = active > first
    
    partial status:
  
      pi(active#) = []
      pi(add) = []
      pi(s) = []
      pi(mark#) = [1]
      pi(first) = [2]
      pi(mark) = [1]
      pi(if) = [2, 3]
      pi(active) = [1]
      pi(and) = [2]
      pi(true) = []
      pi(false) = []
      pi(|0|) = []
      pi(nil) = []
      pi(cons) = []
      pi(from) = []

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#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> mark#(X1)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4}


-- Reduction pair.

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

p1: mark#(first(X1,X2)) -> mark#(X2)
p2: mark#(if(X1,X2,X3)) -> mark#(X1)
p3: mark#(first(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{5, x1 + 2}
        first_A(x1,x2) = max{x1 + 1, x2 + 3}
        if_A(x1,x2,x3) = max{x1 + 5, x2 + 4, x3 + 5}
  
    precedence:
    
      mark# = first = if
    
    partial status:
  
      pi(mark#) = []
      pi(first) = [2]
      pi(if) = [3]

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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: mark#(first(X1,X2)) -> mark#(X2)
p2: mark#(first(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 3
        first_A(x1,x2) = max{x1, x2 + 1}
  
    precedence:
    
      mark# = first
    
    partial status:
  
      pi(mark#) = []
      pi(first) = [2]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 2
        first_A(x1,x2) = max{x1 + 1, x2 + 1}
  
    precedence:
    
      mark# = first
    
    partial status:
  
      pi(mark#) = []
      pi(first) = [2]

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: s#(mark(X)) -> s#(X)
p2: s#(active(X)) -> s#(X)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        s#_A(x1) = max{3, x1 + 2}
        mark_A(x1) = x1
        active_A(x1) = x1 + 1
  
    precedence:
    
      s# = mark = active
    
    partial status:
  
      pi(s#) = [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: s#(active(X)) -> s#(X)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        s#_A(x1) = x1 + 1
        active_A(x1) = x1
  
    precedence:
    
      s# = active
    
    partial status:
  
      pi(s#) = [1]
      pi(active) = [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#(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(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        add#_A(x1,x2) = max{0, x2 - 2}
        mark_A(x1) = x1
        active_A(x1) = max{3, x1 + 1}
  
    precedence:
    
      add# = mark = active
    
    partial status:
  
      pi(add#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        add#_A(x1,x2) = max{0, x1 - 2}
        mark_A(x1) = max{3, x1 + 1}
        active_A(x1) = x1
  
    precedence:
    
      add# = mark = active
    
    partial status:
  
      pi(add#) = []
      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,mark(X2)) -> add#(X1,X2)
p2: add#(active(X1),X2) -> add#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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#(X1,mark(X2)) -> add#(X1,X2)
p2: add#(active(X1),X2) -> add#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        add#_A(x1,x2) = x2 + 1
        mark_A(x1) = x1 + 1
        active_A(x1) = x1 + 1
  
    precedence:
    
      add# = mark = active
    
    partial status:
  
      pi(add#) = []
      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#(active(X1),X2) -> add#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        add#_A(x1,x2) = max{1, x1, x2}
        active_A(x1) = x1 + 1
  
    precedence:
    
      add# = active
    
    partial status:
  
      pi(add#) = [1, 2]
      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: 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(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        cons#_A(x1,x2) = max{0, x2 - 2}
        mark_A(x1) = x1
        active_A(x1) = max{3, x1 + 1}
  
    precedence:
    
      cons# = mark = active
    
    partial status:
  
      pi(cons#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        cons#_A(x1,x2) = max{0, x1 - 2}
        mark_A(x1) = max{3, x1 + 1}
        active_A(x1) = x1
  
    precedence:
    
      cons# = mark = active
    
    partial status:
  
      pi(cons#) = []
      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,mark(X2)) -> cons#(X1,X2)
p2: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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: cons#(X1,mark(X2)) -> cons#(X1,X2)
p2: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        cons#_A(x1,x2) = x2 + 1
        mark_A(x1) = x1 + 1
        active_A(x1) = x1 + 1
  
    precedence:
    
      cons# = mark = active
    
    partial status:
  
      pi(cons#) = []
      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#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        cons#_A(x1,x2) = max{1, x1, x2}
        active_A(x1) = x1 + 1
  
    precedence:
    
      cons# = active
    
    partial status:
  
      pi(cons#) = [1, 2]
      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(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        first#_A(x1,x2) = max{0, x2 - 2}
        mark_A(x1) = x1
        active_A(x1) = max{3, x1 + 1}
  
    precedence:
    
      first# = mark = active
    
    partial status:
  
      pi(first#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        first#_A(x1,x2) = max{0, x1 - 2}
        mark_A(x1) = max{3, x1 + 1}
        active_A(x1) = x1
  
    precedence:
    
      first# = mark = active
    
    partial status:
  
      pi(first#) = []
      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,mark(X2)) -> first#(X1,X2)
p2: first#(active(X1),X2) -> first#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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#(X1,mark(X2)) -> first#(X1,X2)
p2: first#(active(X1),X2) -> first#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        first#_A(x1,x2) = x2 + 1
        mark_A(x1) = x1 + 1
        active_A(x1) = x1 + 1
  
    precedence:
    
      first# = mark = active
    
    partial status:
  
      pi(first#) = []
      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#(active(X1),X2) -> first#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        first#_A(x1,x2) = max{1, x1, x2}
        active_A(x1) = x1 + 1
  
    precedence:
    
      first# = active
    
    partial status:
  
      pi(first#) = [1, 2]
      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: from#(mark(X)) -> from#(X)
p2: from#(active(X)) -> from#(X)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        from#_A(x1) = x1 + 1
        mark_A(x1) = x1
        active_A(x1) = x1
  
    precedence:
    
      from# = mark = active
    
    partial status:
  
      pi(from#) = [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: from#(active(X)) -> from#(X)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: from#(active(X)) -> from#(X)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        from#_A(x1) = x1 + 1
        active_A(x1) = x1
  
    precedence:
    
      from# = active
    
    partial status:
  
      pi(from#) = [1]
      pi(active) = [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: and#(mark(X1),X2) -> and#(X1,X2)
p2: and#(X1,active(X2)) -> and#(X1,X2)
p3: and#(active(X1),X2) -> and#(X1,X2)
p4: and#(X1,mark(X2)) -> and#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        and#_A(x1,x2) = max{0, x2 - 2}
        mark_A(x1) = x1
        active_A(x1) = max{3, x1 + 1}
  
    precedence:
    
      and# = mark = active
    
    partial status:
  
      pi(and#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        and#_A(x1,x2) = max{0, x1 - 2}
        mark_A(x1) = max{3, x1 + 1}
        active_A(x1) = x1
  
    precedence:
    
      and# = mark = active
    
    partial status:
  
      pi(and#) = []
      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: and#(X1,mark(X2)) -> and#(X1,X2)
p2: and#(active(X1),X2) -> and#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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: and#(X1,mark(X2)) -> and#(X1,X2)
p2: and#(active(X1),X2) -> and#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        and#_A(x1,x2) = x2 + 1
        mark_A(x1) = x1 + 1
        active_A(x1) = x1 + 1
  
    precedence:
    
      and# = mark = active
    
    partial status:
  
      pi(and#) = []
      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: and#(active(X1),X2) -> and#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: and#(active(X1),X2) -> and#(X1,X2)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        and#_A(x1,x2) = max{1, x1, x2}
        active_A(x1) = x1 + 1
  
    precedence:
    
      and# = active
    
    partial status:
  
      pi(and#) = [1, 2]
      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: if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
p2: if#(X1,X2,active(X3)) -> if#(X1,X2,X3)
p3: if#(X1,active(X2),X3) -> if#(X1,X2,X3)
p4: if#(active(X1),X2,X3) -> if#(X1,X2,X3)
p5: if#(X1,X2,mark(X3)) -> if#(X1,X2,X3)
p6: if#(X1,mark(X2),X3) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        if#_A(x1,x2,x3) = max{x1 - 1, x2 - 1, x3 + 1}
        mark_A(x1) = x1
        active_A(x1) = x1
  
    precedence:
    
      if# = mark = active
    
    partial status:
  
      pi(if#) = [3]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
p2: if#(X1,active(X2),X3) -> if#(X1,X2,X3)
p3: if#(active(X1),X2,X3) -> if#(X1,X2,X3)
p4: if#(X1,X2,mark(X3)) -> if#(X1,X2,X3)
p5: if#(X1,mark(X2),X3) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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: if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
p2: if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
p3: if#(X1,X2,mark(X3)) -> if#(X1,X2,X3)
p4: if#(active(X1),X2,X3) -> if#(X1,X2,X3)
p5: if#(X1,active(X2),X3) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        if#_A(x1,x2,x3) = max{0, x2 - 2}
        mark_A(x1) = max{3, x1 + 1}
        active_A(x1) = x1
  
    precedence:
    
      if# = mark = active
    
    partial status:
  
      pi(if#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
p2: if#(X1,X2,mark(X3)) -> if#(X1,X2,X3)
p3: if#(active(X1),X2,X3) -> if#(X1,X2,X3)
p4: if#(X1,active(X2),X3) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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: if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
p2: if#(X1,active(X2),X3) -> if#(X1,X2,X3)
p3: if#(active(X1),X2,X3) -> if#(X1,X2,X3)
p4: if#(X1,X2,mark(X3)) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        if#_A(x1,x2,x3) = max{x1 - 1, x2 - 1, x3 + 1}
        mark_A(x1) = x1
        active_A(x1) = x1
  
    precedence:
    
      if# = mark = active
    
    partial status:
  
      pi(if#) = [3]
      pi(mark) = [1]
      pi(active) = [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: if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
p2: if#(X1,active(X2),X3) -> if#(X1,X2,X3)
p3: if#(active(X1),X2,X3) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

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: if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
p2: if#(active(X1),X2,X3) -> if#(X1,X2,X3)
p3: if#(X1,active(X2),X3) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        if#_A(x1,x2,x3) = max{x1 - 1, x2 + 1, x3 + 1}
        mark_A(x1) = x1
        active_A(x1) = x1
  
    precedence:
    
      if# = mark = active
    
    partial status:
  
      pi(if#) = [2]
      pi(mark) = [1]
      pi(active) = [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: if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
p2: if#(active(X1),X2,X3) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(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: if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
p2: if#(active(X1),X2,X3) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        if#_A(x1,x2,x3) = max{x1, x3 + 1}
        mark_A(x1) = x1 + 2
        active_A(x1) = x1 + 1
  
    precedence:
    
      if# = mark = active
    
    partial status:
  
      pi(if#) = [1]
      pi(mark) = []
      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: if#(active(X1),X2,X3) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: if#(active(X1),X2,X3) -> if#(X1,X2,X3)

and R consists of:

r1: active(and(true(),X)) -> mark(X)
r2: active(and(false(),Y)) -> mark(false())
r3: active(if(true(),X,Y)) -> mark(X)
r4: active(if(false(),X,Y)) -> mark(Y)
r5: active(add(|0|(),X)) -> mark(X)
r6: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r7: active(first(|0|(),X)) -> mark(nil())
r8: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r9: active(from(X)) -> mark(cons(X,from(s(X))))
r10: mark(and(X1,X2)) -> active(and(mark(X1),X2))
r11: mark(true()) -> active(true())
r12: mark(false()) -> active(false())
r13: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3))
r14: mark(add(X1,X2)) -> active(add(mark(X1),X2))
r15: mark(|0|()) -> active(|0|())
r16: mark(s(X)) -> active(s(X))
r17: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r18: mark(nil()) -> active(nil())
r19: mark(cons(X1,X2)) -> active(cons(X1,X2))
r20: mark(from(X)) -> active(from(X))
r21: and(mark(X1),X2) -> and(X1,X2)
r22: and(X1,mark(X2)) -> and(X1,X2)
r23: and(active(X1),X2) -> and(X1,X2)
r24: and(X1,active(X2)) -> and(X1,X2)
r25: if(mark(X1),X2,X3) -> if(X1,X2,X3)
r26: if(X1,mark(X2),X3) -> if(X1,X2,X3)
r27: if(X1,X2,mark(X3)) -> if(X1,X2,X3)
r28: if(active(X1),X2,X3) -> if(X1,X2,X3)
r29: if(X1,active(X2),X3) -> if(X1,X2,X3)
r30: if(X1,X2,active(X3)) -> if(X1,X2,X3)
r31: add(mark(X1),X2) -> add(X1,X2)
r32: add(X1,mark(X2)) -> add(X1,X2)
r33: add(active(X1),X2) -> add(X1,X2)
r34: add(X1,active(X2)) -> add(X1,X2)
r35: s(mark(X)) -> s(X)
r36: s(active(X)) -> s(X)
r37: first(mark(X1),X2) -> first(X1,X2)
r38: first(X1,mark(X2)) -> first(X1,X2)
r39: first(active(X1),X2) -> first(X1,X2)
r40: first(X1,active(X2)) -> first(X1,X2)
r41: cons(mark(X1),X2) -> cons(X1,X2)
r42: cons(X1,mark(X2)) -> cons(X1,X2)
r43: cons(active(X1),X2) -> cons(X1,X2)
r44: cons(X1,active(X2)) -> cons(X1,X2)
r45: from(mark(X)) -> from(X)
r46: from(active(X)) -> from(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        if#_A(x1,x2,x3) = max{x1 + 1, x2 + 1, x3 + 1}
        active_A(x1) = x1
  
    precedence:
    
      if# = active
    
    partial status:
  
      pi(if#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

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