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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1
          and_A(x1,x2) = 3
          true_A() = 4
          mark#_A(x1) = 3
          from_A(x1) = 3
          cons_A(x1,x2) = 1
          s_A(x1) = 2
          first_A(x1,x2) = 3
          mark_A(x1) = 5
          add_A(x1,x2) = 3
          |0|_A() = 1
          if_A(x1,x2,x3) = 3
          false_A() = 2
          active_A(x1) = 5
          nil_A() = 2
  
    precedence:
    
      true > from = cons > active# = and = mark# = s = first = mark = add = |0| = if = false = active = nil
    
    partial status:
  
      pi(active#) = []
      pi(and) = []
      pi(true) = []
      pi(mark#) = []
      pi(from) = []
      pi(cons) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = []
      pi(add) = []
      pi(|0|) = []
      pi(if) = []
      pi(false) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: active#(and(true(),X)) -> mark#(X)
p2: mark#(from(X)) -> active#(from(X))
p3: active#(from(X)) -> mark#(cons(X,from(s(X))))
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: active#(and(true(),X)) -> mark#(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#(s(X)) -> active#(s(X))
p12: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p13: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p14: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p15: mark#(first(X1,X2)) -> mark#(X1)
p16: mark#(first(X1,X2)) -> mark#(X2)
p17: mark#(from(X)) -> active#(from(X))
p18: active#(from(X)) -> mark#(cons(X,from(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

  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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1
          and_A(x1,x2) = x1 + x2 + 5
          true_A() = 1
          mark#_A(x1) = x1 + 4
          mark_A(x1) = x1
          if_A(x1,x2,x3) = x1 + x2 + x3 + 5
          false_A() = 1
          add_A(x1,x2) = x1 + x2 + 5
          |0|_A() = 2
          s_A(x1) = 1
          first_A(x1,x2) = x1 + x2 + 3
          cons_A(x1,x2) = x1 + 2
          from_A(x1) = x1 + 6
          active_A(x1) = x1
          nil_A() = 1
  
    precedence:
    
      mark = if > false > and = true = mark# = add = |0| = s = first = from = active > active# > cons > nil
    
    partial status:
  
      pi(active#) = [1]
      pi(and) = [1, 2]
      pi(true) = []
      pi(mark#) = []
      pi(mark) = [1]
      pi(if) = [2]
      pi(false) = []
      pi(add) = []
      pi(|0|) = []
      pi(s) = []
      pi(first) = [1]
      pi(cons) = [1]
      pi(from) = []
      pi(active) = [1]
      pi(nil) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

p1: active#(and(true(),X)) -> mark#(X)
p2: mark#(from(X)) -> active#(from(X))
p3: active#(from(X)) -> mark#(cons(X,from(s(X))))
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#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p8: mark#(s(X)) -> active#(s(X))
p9: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p12: active#(add(|0|(),X)) -> mark#(X)
p13: mark#(if(X1,X2,X3)) -> mark#(X1)
p14: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p15: active#(if(false(),X,Y)) -> mark#(Y)
p16: mark#(and(X1,X2)) -> mark#(X1)
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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1 + 2
          and_A(x1,x2) = 7
          true_A() = 1
          mark#_A(x1) = 6
          from_A(x1) = 4
          cons_A(x1,x2) = 5
          s_A(x1) = 2
          first_A(x1,x2) = 4
          mark_A(x1) = 3
          add_A(x1,x2) = 4
          |0|_A() = 1
          if_A(x1,x2,x3) = 4
          false_A() = 2
          active_A(x1) = 3
          nil_A() = 2
  
    precedence:
    
      true > from = cons = s > first = mark = active > and > if > active# = mark# > add = |0| = false = nil
    
    partial status:
  
      pi(active#) = []
      pi(and) = []
      pi(true) = []
      pi(mark#) = []
      pi(from) = []
      pi(cons) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = []
      pi(add) = []
      pi(|0|) = []
      pi(if) = []
      pi(false) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p8

We remove them from the problem.

-- SCC decomposition.

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

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

  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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1
          and_A(x1,x2) = x1 + x2
          true_A() = 3
          mark#_A(x1) = x1 + 2
          if_A(x1,x2,x3) = x1 + x2 + x3
          mark_A(x1) = x1
          add_A(x1,x2) = x1 + x2 + 3
          false_A() = 3
          first_A(x1,x2) = x1 + x2 + 3
          |0|_A() = 1
          from_A(x1) = 4
          s_A(x1) = 1
          cons_A(x1,x2) = 2
          active_A(x1) = x1
          nil_A() = 3
  
    precedence:
    
      and = true = if = mark = add = false = first = |0| = s = cons = active = nil > mark# = from > active#
    
    partial status:
  
      pi(active#) = [1]
      pi(and) = []
      pi(true) = []
      pi(mark#) = []
      pi(if) = []
      pi(mark) = []
      pi(add) = []
      pi(false) = []
      pi(first) = []
      pi(|0|) = []
      pi(from) = []
      pi(s) = []
      pi(cons) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p15

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

p1: active#(and(true(),X)) -> mark#(X)
p2: mark#(from(X)) -> active#(from(X))
p3: active#(from(X)) -> mark#(cons(X,from(s(X))))
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: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1 + 1
          and_A(x1,x2) = x1 + x2 + 2
          true_A() = 2
          mark#_A(x1) = x1 + 1
          from_A(x1) = 4
          cons_A(x1,x2) = 1
          s_A(x1) = 3
          first_A(x1,x2) = x1 + x2 + 2
          mark_A(x1) = x1
          add_A(x1,x2) = x1 + x2 + 4
          |0|_A() = 2
          if_A(x1,x2,x3) = x1 + x2 + x3
          false_A() = 2
          active_A(x1) = x1
          nil_A() = 1
  
    precedence:
    
      first > and > true > s > if > mark# = from > active# = cons = mark = add = |0| = false = active = nil
    
    partial status:
  
      pi(active#) = [1]
      pi(and) = []
      pi(true) = []
      pi(mark#) = []
      pi(from) = []
      pi(cons) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = []
      pi(add) = []
      pi(|0|) = []
      pi(if) = []
      pi(false) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

p1: active#(and(true(),X)) -> mark#(X)
p2: mark#(from(X)) -> active#(from(X))
p3: active#(from(X)) -> mark#(cons(X,from(s(X))))
p4: mark#(first(X1,X2)) -> mark#(X2)
p5: mark#(first(X1,X2)) -> mark#(X1)
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)
p14: 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}


-- Reduction pair.

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

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

  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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          active#_A(x1) = x1 + 2
          and_A(x1,x2) = 3
          true_A() = 0
          mark#_A(x1) = 2
          if_A(x1,x2,x3) = 0
          mark_A(x1) = 0
          add_A(x1,x2) = 0
          false_A() = 0
          first_A(x1,x2) = 3
          from_A(x1) = 0
          |0|_A() = 0
          s_A(x1) = 4
          cons_A(x1,x2) = 1
          active_A(x1) = 0
          nil_A() = 0
  
    precedence:
    
      |0| > and > true = false = first > mark = active > cons = nil > s > active# = mark# = if = add = from
    
    partial status:
  
      pi(active#) = []
      pi(and) = []
      pi(true) = []
      pi(mark#) = []
      pi(if) = []
      pi(mark) = []
      pi(add) = []
      pi(false) = []
      pi(first) = []
      pi(from) = []
      pi(|0|) = []
      pi(s) = []
      pi(cons) = []
      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)) -> mark#(X1)
p2: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p3: active#(if(true(),X,Y)) -> mark#(X)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(if(false(),X,Y)) -> mark#(Y)
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(first(X1,X2)) -> mark#(X1)
p9: mark#(first(X1,X2)) -> mark#(X2)
p10: mark#(from(X)) -> active#(from(X))
p11: active#(add(|0|(),X)) -> mark#(X)
p12: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p13: active#(from(X)) -> mark#(cons(X,from(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, 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)) -> mark#(X1)
p2: mark#(from(X)) -> active#(from(X))
p3: active#(from(X)) -> mark#(cons(X,from(s(X))))
p4: mark#(first(X1,X2)) -> mark#(X2)
p5: mark#(first(X1,X2)) -> mark#(X1)
p6: mark#(add(X1,X2)) -> 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#(if(X1,X2,X3)) -> mark#(X1)
p10: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p11: active#(add(|0|(),X)) -> mark#(X)
p12: active#(if(false(),X,Y)) -> mark#(Y)
p13: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + 2
          and_A(x1,x2) = x1 + x2 + 1
          from_A(x1) = 9
          active#_A(x1) = x1 + 1
          cons_A(x1,x2) = 6
          s_A(x1) = 4
          first_A(x1,x2) = x1 + x2 + 5
          add_A(x1,x2) = x1 + x2 + 3
          mark_A(x1) = x1
          if_A(x1,x2,x3) = x1 + x2 + x3
          |0|_A() = 1
          false_A() = 3
          true_A() = 3
          active_A(x1) = x1
          nil_A() = 1
  
    precedence:
    
      mark# = s = add = mark = if = |0| = false = true = active > from > active# > cons > first > and = nil
    
    partial status:
  
      pi(mark#) = []
      pi(and) = [2]
      pi(from) = []
      pi(active#) = []
      pi(cons) = []
      pi(s) = []
      pi(first) = [1]
      pi(add) = []
      pi(mark) = []
      pi(if) = []
      pi(|0|) = []
      pi(false) = []
      pi(true) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p11

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
p2: mark#(from(X)) -> active#(from(X))
p3: active#(from(X)) -> mark#(cons(X,from(s(X))))
p4: mark#(first(X1,X2)) -> mark#(X2)
p5: mark#(first(X1,X2)) -> mark#(X1)
p6: mark#(add(X1,X2)) -> 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#(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: 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}


-- 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#(if(true(),X,Y)) -> mark#(X)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(if(false(),X,Y)) -> mark#(Y)
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(first(X1,X2)) -> mark#(X1)
p9: mark#(first(X1,X2)) -> mark#(X2)
p10: mark#(from(X)) -> active#(from(X))
p11: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p12: active#(from(X)) -> mark#(cons(X,from(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

  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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          and_A(x1,x2) = x1 + x2 + 1
          if_A(x1,x2,x3) = x1 + x2 + x3
          active#_A(x1) = x1
          mark_A(x1) = x1
          true_A() = 1
          add_A(x1,x2) = x1 + x2 + 2
          false_A() = 1
          first_A(x1,x2) = x1 + x2
          from_A(x1) = x1
          s_A(x1) = 1
          cons_A(x1,x2) = x1
          active_A(x1) = x1
          |0|_A() = 1
          nil_A() = 1
  
    precedence:
    
      mark# = and = true = add = from > active# = mark = first = s = |0| = nil > if > false = cons = active
    
    partial status:
  
      pi(mark#) = [1]
      pi(and) = [2]
      pi(if) = [1, 2]
      pi(active#) = [1]
      pi(mark) = [1]
      pi(true) = []
      pi(add) = []
      pi(false) = []
      pi(first) = [1, 2]
      pi(from) = [1]
      pi(s) = []
      pi(cons) = []
      pi(active) = [1]
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p12

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
p2: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p3: active#(if(true(),X,Y)) -> mark#(X)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(if(false(),X,Y)) -> mark#(Y)
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(first(X1,X2)) -> mark#(X1)
p9: mark#(first(X1,X2)) -> mark#(X2)
p10: mark#(from(X)) -> active#(from(X))
p11: active#(add(s(X),Y)) -> mark#(s(add(X,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 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#(from(X)) -> active#(from(X))
p3: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p4: mark#(first(X1,X2)) -> mark#(X2)
p5: mark#(first(X1,X2)) -> mark#(X1)
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p8: active#(if(false(),X,Y)) -> mark#(Y)
p9: mark#(if(X1,X2,X3)) -> mark#(X1)
p10: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p11: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = 3
          and_A(x1,x2) = 2
          from_A(x1) = 1
          active#_A(x1) = x1 + 1
          add_A(x1,x2) = 2
          s_A(x1) = 5
          first_A(x1,x2) = 4
          mark_A(x1) = 2
          if_A(x1,x2,x3) = 2
          false_A() = 1
          true_A() = 0
          active_A(x1) = 2
          |0|_A() = 3
          nil_A() = 1
          cons_A(x1,x2) = 3
  
    precedence:
    
      add > mark# = and = active# = mark = false = true = active > first > |0| = nil = cons > from = s = if
    
    partial status:
  
      pi(mark#) = []
      pi(and) = []
      pi(from) = []
      pi(active#) = []
      pi(add) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = []
      pi(if) = []
      pi(false) = []
      pi(true) = []
      pi(active) = []
      pi(|0|) = []
      pi(nil) = []
      pi(cons) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

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


-- 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#(if(true(),X,Y)) -> mark#(X)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(if(false(),X,Y)) -> mark#(Y)
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(first(X1,X2)) -> mark#(X1)
p9: mark#(first(X1,X2)) -> mark#(X2)
p10: active#(add(s(X),Y)) -> mark#(s(add(X,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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + 1
          and_A(x1,x2) = x1 + x2 + 2
          if_A(x1,x2,x3) = x1 + x2 + x3
          active#_A(x1) = x1
          mark_A(x1) = x1
          true_A() = 2
          add_A(x1,x2) = x1 + x2 + 4
          false_A() = 2
          first_A(x1,x2) = x1 + x2 + 2
          s_A(x1) = 3
          active_A(x1) = x1
          |0|_A() = 1
          nil_A() = 2
          cons_A(x1,x2) = x1
          from_A(x1) = x1
  
    precedence:
    
      mark# = false = first > and = if = active# = mark = true = add = s = active = |0| = nil = cons = from
    
    partial status:
  
      pi(mark#) = []
      pi(and) = []
      pi(if) = []
      pi(active#) = [1]
      pi(mark) = [1]
      pi(true) = []
      pi(add) = []
      pi(false) = []
      pi(first) = []
      pi(s) = []
      pi(active) = [1]
      pi(|0|) = []
      pi(nil) = []
      pi(cons) = []
      pi(from) = []

The next rules are strictly ordered:

  p10

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
p2: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p3: active#(if(true(),X,Y)) -> mark#(X)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),X2))
p6: active#(if(false(),X,Y)) -> mark#(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#(if(false(),X,Y)) -> mark#(Y)
p7: mark#(if(X1,X2,X3)) -> mark#(X1)
p8: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p9: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = 6
          and_A(x1,x2) = 7
          first_A(x1,x2) = 2
          add_A(x1,x2) = 4
          active#_A(x1) = x1
          mark_A(x1) = 5
          if_A(x1,x2,x3) = 6
          false_A() = 6
          true_A() = 7
          active_A(x1) = 5
          |0|_A() = 1
          s_A(x1) = 3
          nil_A() = 1
          cons_A(x1,x2) = 2
          from_A(x1) = 4
  
    precedence:
    
      mark# = add = active# = mark = if = true = active > first = nil > |0| = cons > and = false = from > s
    
    partial status:
  
      pi(mark#) = []
      pi(and) = []
      pi(first) = []
      pi(add) = []
      pi(active#) = []
      pi(mark) = []
      pi(if) = []
      pi(false) = []
      pi(true) = []
      pi(active) = []
      pi(|0|) = []
      pi(s) = []
      pi(nil) = []
      pi(cons) = []
      pi(from) = []

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

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

p1: 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: active#(if(false(),X,Y)) -> mark#(Y)
p6: mark#(if(X1,X2,X3)) -> mark#(X1)
p7: mark#(if(X1,X2,X3)) -> active#(if(mark(X1),X2,X3))
p8: 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}


-- 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#(if(true(),X,Y)) -> mark#(X)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> mark#(X1)
p6: mark#(first(X1,X2)) -> mark#(X1)
p7: mark#(first(X1,X2)) -> mark#(X2)
p8: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + 2
          and_A(x1,x2) = x1 + x2 + 3
          if_A(x1,x2,x3) = x1 + x2 + x3 + 3
          active#_A(x1) = x1
          mark_A(x1) = x1
          true_A() = 1
          add_A(x1,x2) = x1 + x2 + 3
          first_A(x1,x2) = x1 + x2 + 3
          false_A() = 1
          active_A(x1) = x1
          |0|_A() = 1
          s_A(x1) = 5
          nil_A() = 2
          cons_A(x1,x2) = x1 + 4
          from_A(x1) = x1 + 6
  
    precedence:
    
      if = add = first = false = |0| > active# = nil > cons > mark# = and = mark > true = active = s > from
    
    partial status:
  
      pi(mark#) = [1]
      pi(and) = [1, 2]
      pi(if) = [2]
      pi(active#) = []
      pi(mark) = [1]
      pi(true) = []
      pi(add) = []
      pi(first) = []
      pi(false) = []
      pi(active) = [1]
      pi(|0|) = []
      pi(s) = []
      pi(nil) = []
      pi(cons) = []
      pi(from) = []

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#(and(X1,X2)) -> mark#(X1)
p2: active#(if(true(),X,Y)) -> mark#(X)
p3: mark#(if(X1,X2,X3)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: mark#(first(X1,X2)) -> mark#(X1)
p6: mark#(first(X1,X2)) -> mark#(X2)
p7: 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 estimated dependency graph contains the following SCCs:

  {p1, p3, p4, p5, p6}


-- 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#(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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          and_A(x1,x2) = x1 + x2 + 1
          first_A(x1,x2) = x1 + x2 + 1
          add_A(x1,x2) = x1 + x2 + 1
          if_A(x1,x2,x3) = x1 + x3 + 1
  
    precedence:
    
      mark# = and = first = add = if
    
    partial status:
  
      pi(mark#) = [1]
      pi(and) = [1, 2]
      pi(first) = [1, 2]
      pi(add) = [1, 2]
      pi(if) = [1, 3]

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

  {p1, 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#(add(X1,X2)) -> mark#(X1)
p4: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          first_A(x1,x2) = x1 + x2 + 1
          if_A(x1,x2,x3) = x1 + x3 + 1
          add_A(x1,x2) = x1 + x2 + 1
  
    precedence:
    
      mark# = first = if = add
    
    partial status:
  
      pi(mark#) = [1]
      pi(first) = [1, 2]
      pi(if) = [1, 3]
      pi(add) = [1, 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#(if(X1,X2,X3)) -> mark#(X1)
p2: mark#(add(X1,X2)) -> 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: mark#(if(X1,X2,X3)) -> mark#(X1)
p2: mark#(first(X1,X2)) -> mark#(X1)
p3: mark#(add(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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          if_A(x1,x2,x3) = x1 + x2 + x3 + 1
          first_A(x1,x2) = x1 + x2 + 1
          add_A(x1,x2) = x1 + x2 + 1
  
    precedence:
    
      mark# = if = first = add
    
    partial status:
  
      pi(mark#) = []
      pi(if) = [3]
      pi(first) = [2]
      pi(add) = [2]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(first(X1,X2)) -> mark#(X1)
p2: mark#(add(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#(X1)
p2: mark#(add(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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          first_A(x1,x2) = x1 + x2 + 1
          add_A(x1,x2) = x1 + x2 + 1
  
    precedence:
    
      mark# = first = add
    
    partial status:
  
      pi(mark#) = []
      pi(first) = [2]
      pi(add) = [2]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(add(X1,X2)) -> mark#(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#(add(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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          add_A(x1,x2) = x1 + x2 + 1
  
    precedence:
    
      mark# = add
    
    partial status:
  
      pi(mark#) = []
      pi(add) = [2]

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          s#_A(x1) = x1
          mark_A(x1) = x1 + 1
          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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          s#_A(x1) = x1
          active_A(x1) = x1 + 1
  
    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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          add#_A(x1,x2) = x1 + x2 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      add# = mark = active
    
    partial status:
  
      pi(add#) = []
      pi(mark) = [1]
      pi(active) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(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#(X1,active(X2)) -> add#(X1,X2)
p2: add#(X1,mark(X2)) -> add#(X1,X2)
p3: add#(active(X1),X2) -> add#(X1,X2)

and R consists of:

r1: active(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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          add#_A(x1,x2) = x1 + x2 + 2
          active_A(x1) = x1 + 1
          mark_A(x1) = x1 + 1
  
    precedence:
    
      add# = active = mark
    
    partial status:
  
      pi(add#) = [1]
      pi(active) = [1]
      pi(mark) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: add#(X1,mark(X2)) -> add#(X1,X2)
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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          add#_A(x1,x2) = x1 + x2 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      mark = active > add#
    
    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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          add#_A(x1,x2) = x1
          active_A(x1) = x1 + 1
  
    precedence:
    
      add# = active
    
    partial status:
  
      pi(add#) = [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: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          cons#_A(x1,x2) = x1 + x2 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      cons# = mark = active
    
    partial status:
  
      pi(cons#) = []
      pi(mark) = [1]
      pi(active) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(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#(X1,active(X2)) -> cons#(X1,X2)
p2: cons#(X1,mark(X2)) -> cons#(X1,X2)
p3: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          cons#_A(x1,x2) = x1 + x2 + 2
          active_A(x1) = x1 + 1
          mark_A(x1) = x1 + 1
  
    precedence:
    
      cons# = active = mark
    
    partial status:
  
      pi(cons#) = [1]
      pi(active) = [1]
      pi(mark) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: cons#(X1,mark(X2)) -> cons#(X1,X2)
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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          cons#_A(x1,x2) = x1 + x2 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      mark = active > cons#
    
    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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          cons#_A(x1,x2) = x1
          active_A(x1) = x1 + 1
  
    precedence:
    
      cons# = active
    
    partial status:
  
      pi(cons#) = [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: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          first#_A(x1,x2) = x1 + x2 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      first# = mark = active
    
    partial status:
  
      pi(first#) = []
      pi(mark) = [1]
      pi(active) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: active(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#(X1,active(X2)) -> first#(X1,X2)
p2: first#(X1,mark(X2)) -> first#(X1,X2)
p3: first#(active(X1),X2) -> first#(X1,X2)

and R consists of:

r1: active(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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          first#_A(x1,x2) = x1 + x2 + 2
          active_A(x1) = x1 + 1
          mark_A(x1) = x1 + 1
  
    precedence:
    
      first# = active = mark
    
    partial status:
  
      pi(first#) = [1]
      pi(active) = [1]
      pi(mark) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: first#(X1,mark(X2)) -> first#(X1,X2)
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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          first#_A(x1,x2) = x1 + x2 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      mark = active > first#
    
    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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          first#_A(x1,x2) = x1
          active_A(x1) = x1 + 1
  
    precedence:
    
      first# = active
    
    partial status:
  
      pi(first#) = [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: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          from#_A(x1) = x1
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          from#_A(x1) = x1
          active_A(x1) = x1 + 1
  
    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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          and#_A(x1,x2) = x1 + x2 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      and# = mark = active
    
    partial status:
  
      pi(and#) = []
      pi(mark) = [1]
      pi(active) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: and#(X1,active(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#(X1,active(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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          and#_A(x1,x2) = x1 + x2 + 2
          active_A(x1) = x1 + 1
          mark_A(x1) = x1 + 1
  
    precedence:
    
      and# = active = mark
    
    partial status:
  
      pi(and#) = [1]
      pi(active) = [1]
      pi(mark) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          and#_A(x1,x2) = x1 + x2 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      mark = active > and#
    
    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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          and#_A(x1,x2) = x1
          active_A(x1) = x1 + 1
  
    precedence:
    
      and# = active
    
    partial status:
  
      pi(and#) = [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: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          if#_A(x1,x2,x3) = x1 + x2 + x3 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      if# = mark = active
    
    partial status:
  
      pi(if#) = [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: if#(X1,X2,active(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#(X1,X2,active(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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          if#_A(x1,x2,x3) = x1 + x2 + x3 + 2
          active_A(x1) = x1 + 1
          mark_A(x1) = x1 + 1
  
    precedence:
    
      if# = active = mark
    
    partial status:
  
      pi(if#) = [1]
      pi(active) = [1]
      pi(mark) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: if#(X1,mark(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#(X1,mark(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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          if#_A(x1,x2,x3) = x1 + x2 + x3 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      if# = mark = active
    
    partial status:
  
      pi(if#) = [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: if#(X1,active(X2),X3) -> if#(X1,X2,X3)
p2: if#(active(X1),X2,X3) -> if#(X1,X2,X3)
p3: 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 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#(X1,active(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)

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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          if#_A(x1,x2,x3) = x1 + x2 + x3 + 2
          active_A(x1) = x1 + 1
          mark_A(x1) = x1 + 1
  
    precedence:
    
      if# = active = mark
    
    partial status:
  
      pi(if#) = [1]
      pi(active) = [1]
      pi(mark) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: if#(X1,X2,mark(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#(X1,X2,mark(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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          if#_A(x1,x2,x3) = x1 + x2 + x3 + 2
          mark_A(x1) = x1 + 1
          active_A(x1) = x1 + 1
  
    precedence:
    
      if# = mark = active
    
    partial status:
  
      pi(if#) = [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: 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:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          if#_A(x1,x2,x3) = x1
          active_A(x1) = x1 + 1
  
    precedence:
    
      if# = active
    
    partial status:
  
      pi(if#) = [1]
      pi(active) = []

The next rules are strictly ordered:

  p1

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