YES

We show the termination of the TRS R:

  a__and(tt(),T) -> mark(T)
  a__isNatIList(IL) -> a__isNatList(IL)
  a__isNat(|0|()) -> tt()
  a__isNat(s(N)) -> a__isNat(N)
  a__isNat(length(L)) -> a__isNatList(L)
  a__isNatIList(zeros()) -> tt()
  a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
  a__isNatList(nil()) -> tt()
  a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
  a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
  a__zeros() -> cons(|0|(),zeros())
  a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
  a__uTake1(tt()) -> nil()
  a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
  a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
  a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
  a__uLength(tt(),L) -> s(a__length(mark(L)))
  mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
  mark(isNatIList(X)) -> a__isNatIList(X)
  mark(isNatList(X)) -> a__isNatList(X)
  mark(isNat(X)) -> a__isNat(X)
  mark(length(X)) -> a__length(mark(X))
  mark(zeros()) -> a__zeros()
  mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
  mark(uTake1(X)) -> a__uTake1(mark(X))
  mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
  mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
  mark(tt()) -> tt()
  mark(|0|()) -> |0|()
  mark(s(X)) -> s(mark(X))
  mark(cons(X1,X2)) -> cons(mark(X1),X2)
  mark(nil()) -> nil()
  a__and(X1,X2) -> and(X1,X2)
  a__isNatIList(X) -> isNatIList(X)
  a__isNatList(X) -> isNatList(X)
  a__isNat(X) -> isNat(X)
  a__length(X) -> length(X)
  a__zeros() -> zeros()
  a__take(X1,X2) -> take(X1,X2)
  a__uTake1(X) -> uTake1(X)
  a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
  a__uLength(X1,X2) -> uLength(X1,X2)

-- SCC decomposition.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: a__isNatIList#(IL) -> a__isNatList#(IL)
p3: a__isNat#(s(N)) -> a__isNat#(N)
p4: a__isNat#(length(L)) -> a__isNatList#(L)
p5: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p6: a__isNatIList#(cons(N,IL)) -> a__isNat#(N)
p7: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p8: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p9: a__isNatList#(cons(N,L)) -> a__isNat#(N)
p10: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p11: a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p12: a__isNatList#(take(N,IL)) -> a__isNat#(N)
p13: a__isNatList#(take(N,IL)) -> a__isNatIList#(IL)
p14: a__take#(|0|(),IL) -> a__uTake1#(a__isNatIList(IL))
p15: a__take#(|0|(),IL) -> a__isNatIList#(IL)
p16: a__take#(s(M),cons(N,IL)) -> a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
p17: a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL)))
p18: a__take#(s(M),cons(N,IL)) -> a__isNat#(M)
p19: a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p20: a__take#(s(M),cons(N,IL)) -> a__isNat#(N)
p21: a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL)
p22: a__uTake2#(tt(),M,N,IL) -> mark#(N)
p23: a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L)
p24: a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p25: a__length#(cons(N,L)) -> a__isNat#(N)
p26: a__length#(cons(N,L)) -> a__isNatList#(L)
p27: a__uLength#(tt(),L) -> a__length#(mark(L))
p28: a__uLength#(tt(),L) -> mark#(L)
p29: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))
p30: mark#(and(X1,X2)) -> mark#(X1)
p31: mark#(and(X1,X2)) -> mark#(X2)
p32: mark#(isNatIList(X)) -> a__isNatIList#(X)
p33: mark#(isNatList(X)) -> a__isNatList#(X)
p34: mark#(isNat(X)) -> a__isNat#(X)
p35: mark#(length(X)) -> a__length#(mark(X))
p36: mark#(length(X)) -> mark#(X)
p37: mark#(zeros()) -> a__zeros#()
p38: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p39: mark#(take(X1,X2)) -> mark#(X1)
p40: mark#(take(X1,X2)) -> mark#(X2)
p41: mark#(uTake1(X)) -> a__uTake1#(mark(X))
p42: mark#(uTake1(X)) -> mark#(X)
p43: mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4)
p44: mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1)
p45: mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2)
p46: mark#(uLength(X1,X2)) -> mark#(X1)
p47: mark#(s(X)) -> mark#(X)
p48: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p38, p39, p40, p42, p43, p44, p45, p46, p47, p48}


-- Reduction pair.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(s(X)) -> mark#(X)
p4: mark#(uLength(X1,X2)) -> mark#(X1)
p5: mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2)
p6: a__uLength#(tt(),L) -> mark#(L)
p7: mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4)
p9: a__uTake2#(tt(),M,N,IL) -> mark#(N)
p10: mark#(uTake1(X)) -> mark#(X)
p11: mark#(take(X1,X2)) -> mark#(X2)
p12: mark#(take(X1,X2)) -> mark#(X1)
p13: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p14: a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL)
p15: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p16: a__isNatIList#(cons(N,IL)) -> a__isNat#(N)
p17: a__isNat#(length(L)) -> a__isNatList#(L)
p18: a__isNatList#(take(N,IL)) -> a__isNatIList#(IL)
p19: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p20: a__isNatIList#(IL) -> a__isNatList#(IL)
p21: a__isNatList#(take(N,IL)) -> a__isNat#(N)
p22: a__isNat#(s(N)) -> a__isNat#(N)
p23: a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p24: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p25: a__isNatList#(cons(N,L)) -> a__isNat#(N)
p26: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p27: a__take#(s(M),cons(N,IL)) -> a__isNat#(N)
p28: a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p29: a__take#(s(M),cons(N,IL)) -> a__isNat#(M)
p30: a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL)))
p31: a__take#(s(M),cons(N,IL)) -> a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
p32: a__take#(|0|(),IL) -> a__isNatIList#(IL)
p33: mark#(length(X)) -> mark#(X)
p34: mark#(length(X)) -> a__length#(mark(X))
p35: a__length#(cons(N,L)) -> a__isNatList#(L)
p36: a__length#(cons(N,L)) -> a__isNat#(N)
p37: a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p38: a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L)
p39: a__uLength#(tt(),L) -> a__length#(mark(L))
p40: mark#(isNat(X)) -> a__isNat#(X)
p41: mark#(isNatList(X)) -> a__isNatList#(X)
p42: mark#(isNatIList(X)) -> a__isNatIList#(X)
p43: mark#(and(X1,X2)) -> mark#(X2)
p44: mark#(and(X1,X2)) -> mark#(X1)
p45: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__and#_A(x1,x2) = max{6, x1 - 8, x2 - 8}
    tt_A = 12
    mark#_A(x1) = max{6, x1 - 8}
    cons_A(x1,x2) = max{10, x1, x2}
    s_A(x1) = max{29, x1}
    uLength_A(x1,x2) = max{29, x1 + 1, x2 + 15}
    a__uLength#_A(x1,x2) = max{10, x1 - 19, x2 - 3}
    mark_A(x1) = max{13, x1}
    uTake2_A(x1,x2,x3,x4) = max{38, x1 + 22, x2 + 9, x3 + 21, x4 + 21}
    a__uTake2#_A(x1,x2,x3,x4) = max{x1 - 5, x3 + 12, x4 + 12}
    uTake1_A(x1) = max{22, x1 + 5}
    take_A(x1,x2) = max{34, x1 + 9, x2 + 21}
    a__take#_A(x1,x2) = x2 + 12
    a__isNatIList#_A(x1) = 8
    a__isNat#_A(x1) = 8
    length_A(x1) = max{29, x1 + 15}
    a__isNatList#_A(x1) = 8
    a__isNat_A(x1) = 16
    a__isNatIList_A(x1) = 16
    a__isNatList_A(x1) = 16
    a__and_A(x1,x2) = max{13, x1, x2}
    |0|_A = 0
    a__length#_A(x1) = max{10, x1 - 3}
    isNat_A(x1) = 16
    isNatList_A(x1) = 16
    isNatIList_A(x1) = 16
    and_A(x1,x2) = max{12, x1, x2}
    a__zeros_A = 13
    zeros_A = 12
    a__take_A(x1,x2) = max{34, x1 + 9, x2 + 21}
    a__uTake1_A(x1) = max{22, x1 + 5}
    nil_A = 18
    a__uTake2_A(x1,x2,x3,x4) = max{38, x1 + 22, x2 + 9, x3 + 21, x4 + 21}
    a__length_A(x1) = max{29, x1 + 15}
    a__uLength_A(x1,x2) = max{29, x1 + 1, x2 + 15}

The next rules are strictly ordered:

  p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p27, p28, p29, p30, p32, p33, p34, p35, p36, p37

We remove them from the problem.

-- SCC decomposition.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(s(X)) -> mark#(X)
p4: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p5: a__isNatIList#(cons(N,IL)) -> a__isNat#(N)
p6: a__isNat#(length(L)) -> a__isNatList#(L)
p7: a__isNatList#(take(N,IL)) -> a__isNatIList#(IL)
p8: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p9: a__isNatIList#(IL) -> a__isNatList#(IL)
p10: a__isNatList#(take(N,IL)) -> a__isNat#(N)
p11: a__isNat#(s(N)) -> a__isNat#(N)
p12: a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p13: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p14: a__isNatList#(cons(N,L)) -> a__isNat#(N)
p15: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p16: a__take#(s(M),cons(N,IL)) -> a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
p17: a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L)
p18: a__uLength#(tt(),L) -> a__length#(mark(L))
p19: mark#(isNat(X)) -> a__isNat#(X)
p20: mark#(isNatList(X)) -> a__isNatList#(X)
p21: mark#(isNatIList(X)) -> a__isNatIList#(X)
p22: mark#(and(X1,X2)) -> mark#(X2)
p23: mark#(and(X1,X2)) -> mark#(X1)
p24: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p19, p20, p21, p22, p23, p24}
  {p17, p18}


-- Reduction pair.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))
p3: mark#(and(X1,X2)) -> mark#(X1)
p4: mark#(and(X1,X2)) -> mark#(X2)
p5: mark#(isNatIList(X)) -> a__isNatIList#(X)
p6: a__isNatIList#(IL) -> a__isNatList#(IL)
p7: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p8: a__isNatList#(cons(N,L)) -> a__isNat#(N)
p9: a__isNat#(s(N)) -> a__isNat#(N)
p10: a__isNat#(length(L)) -> a__isNatList#(L)
p11: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p12: a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p13: a__isNatList#(take(N,IL)) -> a__isNat#(N)
p14: a__isNatList#(take(N,IL)) -> a__isNatIList#(IL)
p15: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p16: a__isNatIList#(cons(N,IL)) -> a__isNat#(N)
p17: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p18: mark#(isNatList(X)) -> a__isNatList#(X)
p19: mark#(isNat(X)) -> a__isNat#(X)
p20: mark#(s(X)) -> mark#(X)
p21: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__and#_A(x1,x2) = max{3, x1 - 16, x2 - 7}
    tt_A = 8
    mark#_A(x1) = max{3, x1 - 7}
    and_A(x1,x2) = max{4, x1, x2}
    mark_A(x1) = max{8, x1}
    isNatIList_A(x1) = x1 + 5
    a__isNatIList#_A(x1) = max{2, x1 - 2}
    a__isNatList#_A(x1) = max{0, x1 - 3}
    cons_A(x1,x2) = max{8, x1, x2}
    a__isNat_A(x1) = max{11, x1 + 4}
    a__isNatList_A(x1) = max{6, x1 + 4}
    a__isNat#_A(x1) = max{4, x1 - 3}
    s_A(x1) = x1
    length_A(x1) = x1
    take_A(x1,x2) = max{17, x1, x2 + 9}
    a__isNatIList_A(x1) = max{8, x1 + 5}
    isNatList_A(x1) = max{6, x1 + 4}
    isNat_A(x1) = max{11, x1 + 4}
    a__and_A(x1,x2) = max{4, x1, x2}
    a__zeros_A = 8
    |0|_A = 3
    zeros_A = 2
    a__take_A(x1,x2) = max{17, x1, x2 + 9}
    a__uTake1_A(x1) = max{9, x1 - 5}
    nil_A = 9
    a__uTake2_A(x1,x2,x3,x4) = max{17, x2, x3 + 9, x4 + 9}
    a__length_A(x1) = x1
    a__uLength_A(x1,x2) = max{8, x1 - 11, x2}
    uTake1_A(x1) = max{9, x1 - 5}
    uTake2_A(x1,x2,x3,x4) = max{17, x2, x3 + 9, x4 + 9}
    uLength_A(x1,x2) = max{x1 - 11, x2}

The next rules are strictly ordered:

  p6, p12, p14, p16

We remove them from the problem.

-- SCC decomposition.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))
p3: mark#(and(X1,X2)) -> mark#(X1)
p4: mark#(and(X1,X2)) -> mark#(X2)
p5: mark#(isNatIList(X)) -> a__isNatIList#(X)
p6: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p7: a__isNatList#(cons(N,L)) -> a__isNat#(N)
p8: a__isNat#(s(N)) -> a__isNat#(N)
p9: a__isNat#(length(L)) -> a__isNatList#(L)
p10: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p11: a__isNatList#(take(N,IL)) -> a__isNat#(N)
p12: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p13: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p14: mark#(isNatList(X)) -> a__isNatList#(X)
p15: mark#(isNat(X)) -> a__isNat#(X)
p16: mark#(s(X)) -> mark#(X)
p17: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> a__isNat#(X)
p5: a__isNat#(length(L)) -> a__isNatList#(L)
p6: a__isNatList#(take(N,IL)) -> a__isNat#(N)
p7: a__isNat#(s(N)) -> a__isNat#(N)
p8: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p9: a__isNatList#(cons(N,L)) -> a__isNat#(N)
p10: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p11: mark#(isNatList(X)) -> a__isNatList#(X)
p12: mark#(isNatIList(X)) -> a__isNatIList#(X)
p13: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p14: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p15: mark#(and(X1,X2)) -> mark#(X2)
p16: mark#(and(X1,X2)) -> mark#(X1)
p17: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__and#_A(x1,x2) = max{x1 - 6, x2 + 2}
    tt_A = 25
    mark#_A(x1) = max{9, x1 + 2}
    cons_A(x1,x2) = max{16, x1 + 8, x2}
    s_A(x1) = max{4, x1}
    isNat_A(x1) = x1 + 19
    a__isNat#_A(x1) = x1 + 21
    length_A(x1) = max{7, x1 - 8}
    a__isNatList#_A(x1) = x1 + 13
    take_A(x1,x2) = max{19, x1 + 8, x2 + 13}
    a__isNat_A(x1) = x1 + 19
    a__isNatList_A(x1) = x1 + 11
    isNatList_A(x1) = x1 + 11
    isNatIList_A(x1) = x1 + 23
    a__isNatIList#_A(x1) = x1 + 25
    a__isNatIList_A(x1) = x1 + 23
    and_A(x1,x2) = max{21, x1, x2}
    mark_A(x1) = max{6, x1}
    a__and_A(x1,x2) = max{21, x1, x2}
    a__zeros_A = 16
    |0|_A = 8
    zeros_A = 16
    a__take_A(x1,x2) = max{19, x1 + 8, x2 + 13}
    a__uTake1_A(x1) = max{14, x1 - 10}
    nil_A = 15
    a__uTake2_A(x1,x2,x3,x4) = max{x2 + 8, x3 + 19, x4 + 13}
    a__length_A(x1) = max{7, x1 - 8}
    a__uLength_A(x1,x2) = max{8, x1 - 28, x2 - 8}
    uTake1_A(x1) = max{14, x1 - 10}
    uTake2_A(x1,x2,x3,x4) = max{x2 + 8, x3 + 19, x4 + 13}
    uLength_A(x1,x2) = max{8, x1 - 28, x2 - 8}

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: a__and#(tt(),T) -> mark#(T)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(isNat(X)) -> a__isNat#(X)
p4: a__isNat#(length(L)) -> a__isNatList#(L)
p5: a__isNatList#(take(N,IL)) -> a__isNat#(N)
p6: a__isNat#(s(N)) -> a__isNat#(N)
p7: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p8: a__isNatList#(cons(N,L)) -> a__isNat#(N)
p9: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p10: mark#(isNatList(X)) -> a__isNatList#(X)
p11: mark#(isNatIList(X)) -> a__isNatIList#(X)
p12: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p13: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p14: mark#(and(X1,X2)) -> mark#(X2)
p15: mark#(and(X1,X2)) -> mark#(X1)
p16: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))
p3: mark#(and(X1,X2)) -> mark#(X1)
p4: mark#(and(X1,X2)) -> mark#(X2)
p5: mark#(isNatIList(X)) -> a__isNatIList#(X)
p6: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p7: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p8: mark#(isNatList(X)) -> a__isNatList#(X)
p9: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p10: a__isNatList#(cons(N,L)) -> a__isNat#(N)
p11: a__isNat#(s(N)) -> a__isNat#(N)
p12: a__isNat#(length(L)) -> a__isNatList#(L)
p13: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p14: a__isNatList#(take(N,IL)) -> a__isNat#(N)
p15: mark#(isNat(X)) -> a__isNat#(X)
p16: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__and#_A(x1,x2) = max{17, x1 + 1, x2 + 2}
    tt_A = 18
    mark#_A(x1) = x1 + 2
    and_A(x1,x2) = max{15, x1 + 14, x2}
    mark_A(x1) = x1
    isNatIList_A(x1) = x1 + 32
    a__isNatIList#_A(x1) = x1 + 34
    cons_A(x1,x2) = max{x1 - 5, x2}
    a__isNat_A(x1) = x1 + 13
    a__isNatIList_A(x1) = x1 + 32
    isNatList_A(x1) = x1 + 32
    a__isNatList#_A(x1) = x1 + 34
    a__isNatList_A(x1) = x1 + 32
    a__isNat#_A(x1) = x1 + 15
    s_A(x1) = max{6, x1}
    length_A(x1) = x1 + 19
    take_A(x1,x2) = max{x1 + 18, x2 + 14}
    isNat_A(x1) = x1 + 13
    a__and_A(x1,x2) = max{15, x1 + 14, x2}
    a__zeros_A = 0
    |0|_A = 5
    zeros_A = 0
    a__take_A(x1,x2) = max{x1 + 18, x2 + 14}
    a__uTake1_A(x1) = 19
    nil_A = 1
    a__uTake2_A(x1,x2,x3,x4) = max{x2 + 18, x3 - 1, x4 + 14}
    a__length_A(x1) = x1 + 19
    a__uLength_A(x1,x2) = max{x1 - 13, x2 + 19}
    uTake1_A(x1) = 19
    uTake2_A(x1,x2,x3,x4) = max{x2 + 18, x3 - 1, x4 + 14}
    uLength_A(x1,x2) = max{x1 - 13, x2 + 19}

The next rules are strictly ordered:

  p3, p10, p14

We remove them from the problem.

-- SCC decomposition.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))
p3: mark#(and(X1,X2)) -> mark#(X2)
p4: mark#(isNatIList(X)) -> a__isNatIList#(X)
p5: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p6: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p7: mark#(isNatList(X)) -> a__isNatList#(X)
p8: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p9: a__isNat#(s(N)) -> a__isNat#(N)
p10: a__isNat#(length(L)) -> a__isNatList#(L)
p11: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p12: mark#(isNat(X)) -> a__isNat#(X)
p13: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(isNat(X)) -> a__isNat#(X)
p4: a__isNat#(length(L)) -> a__isNatList#(L)
p5: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p6: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p7: a__isNat#(s(N)) -> a__isNat#(N)
p8: mark#(isNatList(X)) -> a__isNatList#(X)
p9: mark#(isNatIList(X)) -> a__isNatIList#(X)
p10: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p11: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p12: mark#(and(X1,X2)) -> mark#(X2)
p13: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__and#_A(x1,x2) = x2 + 5
    tt_A = 1
    mark#_A(x1) = x1 + 5
    s_A(x1) = max{2, x1}
    isNat_A(x1) = 3
    a__isNat#_A(x1) = 6
    length_A(x1) = x1 + 5
    a__isNatList#_A(x1) = 6
    cons_A(x1,x2) = max{11, x1, x2}
    a__isNat_A(x1) = 3
    a__isNatList_A(x1) = 1
    isNatList_A(x1) = 1
    isNatIList_A(x1) = 1
    a__isNatIList#_A(x1) = 6
    a__isNatIList_A(x1) = 1
    and_A(x1,x2) = max{1, x1 - 4, x2}
    mark_A(x1) = x1
    a__and_A(x1,x2) = max{1, x1 - 4, x2}
    a__zeros_A = 11
    |0|_A = 1
    zeros_A = 11
    a__take_A(x1,x2) = x2 + 12
    a__uTake1_A(x1) = x1 + 11
    nil_A = 11
    a__uTake2_A(x1,x2,x3,x4) = max{x1 + 11, x3 + 12, x4 + 12}
    take_A(x1,x2) = x2 + 12
    a__length_A(x1) = x1 + 5
    a__uLength_A(x1,x2) = max{x1 + 4, x2 + 5}
    uTake1_A(x1) = x1 + 11
    uTake2_A(x1,x2,x3,x4) = max{x1 + 11, x3 + 12, x4 + 12}
    uLength_A(x1,x2) = max{x1 + 4, x2 + 5}

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(s(X)) -> mark#(X)
p3: a__isNat#(length(L)) -> a__isNatList#(L)
p4: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p5: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p6: a__isNat#(s(N)) -> a__isNat#(N)
p7: mark#(isNatList(X)) -> a__isNatList#(X)
p8: mark#(isNatIList(X)) -> a__isNatIList#(X)
p9: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p10: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p11: mark#(and(X1,X2)) -> mark#(X2)
p12: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__isNat#(s(N)) -> a__isNat#(N)

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    a__isNat#_A(x1) = x1
    s_A(x1) = x1 + 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: a__and#(tt(),T) -> mark#(T)
p2: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))
p3: mark#(and(X1,X2)) -> mark#(X2)
p4: mark#(isNatIList(X)) -> a__isNatIList#(X)
p5: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p6: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p7: mark#(isNatList(X)) -> a__isNatList#(X)
p8: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p9: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p10: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__and#_A(x1,x2) = x2 + 2
    tt_A = 10
    mark#_A(x1) = x1 + 2
    and_A(x1,x2) = max{15, x2}
    mark_A(x1) = max{12, x1}
    isNatIList_A(x1) = x1 + 26
    a__isNatIList#_A(x1) = max{28, x1 + 16}
    cons_A(x1,x2) = max{x1, x2 + 12}
    a__isNat_A(x1) = x1 + 39
    a__isNatIList_A(x1) = x1 + 26
    isNatList_A(x1) = x1 + 26
    a__isNatList#_A(x1) = x1 + 17
    a__isNatList_A(x1) = x1 + 26
    s_A(x1) = max{1, x1}
    a__and_A(x1,x2) = max{15, x2}
    a__zeros_A = 12
    |0|_A = 0
    zeros_A = 0
    a__take_A(x1,x2) = x2
    a__uTake1_A(x1) = 0
    nil_A = 0
    a__uTake2_A(x1,x2,x3,x4) = max{x3, x4 + 12}
    take_A(x1,x2) = x2
    a__length_A(x1) = max{40, x1 + 28}
    a__uLength_A(x1,x2) = max{x1 + 1, x2 + 40}
    length_A(x1) = max{40, x1 + 28}
    uTake1_A(x1) = 0
    uTake2_A(x1,x2,x3,x4) = max{x3, x4 + 12}
    uLength_A(x1,x2) = max{x1 + 1, x2 + 40}
    isNat_A(x1) = x1 + 39

The next rules are strictly ordered:

  p7, p8, p9

We remove them from the problem.

-- SCC decomposition.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))
p3: mark#(and(X1,X2)) -> mark#(X2)
p4: mark#(isNatIList(X)) -> a__isNatIList#(X)
p5: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p6: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p7: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__and#(tt(),T) -> mark#(T)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(isNatIList(X)) -> a__isNatIList#(X)
p4: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p5: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p6: mark#(and(X1,X2)) -> mark#(X2)
p7: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__and#_A(x1,x2) = max{2, x1 - 16, x2 - 7}
    tt_A = 23
    mark#_A(x1) = max{0, x1 - 8}
    s_A(x1) = max{8, x1 + 4}
    isNatIList_A(x1) = x1 + 23
    a__isNatIList#_A(x1) = x1 + 15
    cons_A(x1,x2) = max{x1, x2 + 4}
    a__isNat_A(x1) = max{23, x1 + 22}
    a__isNatIList_A(x1) = x1 + 23
    and_A(x1,x2) = max{19, x1 - 8, x2 + 4}
    mark_A(x1) = max{4, x1}
    a__and_A(x1,x2) = max{19, x1 - 8, x2 + 4}
    a__isNatList_A(x1) = x1 + 15
    nil_A = 8
    take_A(x1,x2) = max{24, x1 + 20, x2 + 18}
    a__zeros_A = 4
    |0|_A = 0
    zeros_A = 0
    a__take_A(x1,x2) = max{24, x1 + 20, x2 + 18}
    a__uTake1_A(x1) = 24
    a__uTake2_A(x1,x2,x3,x4) = max{28, x2 + 24, x3, x4 + 22}
    a__length_A(x1) = x1
    a__uLength_A(x1,x2) = max{x1 - 15, x2 + 4}
    isNatList_A(x1) = x1 + 15
    length_A(x1) = x1
    uTake1_A(x1) = 24
    uTake2_A(x1,x2,x3,x4) = max{28, x2 + 24, x3, x4 + 22}
    uLength_A(x1,x2) = max{x1 - 15, x2 + 4}
    isNat_A(x1) = max{23, x1 + 22}

The next rules are strictly ordered:

  p1, p4, p5, p6

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(isNatIList(X)) -> a__isNatIList#(X)
p3: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2))

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1
    s_A(x1) = x1 + 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: a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L)
p2: a__uLength#(tt(),L) -> a__length#(mark(L))

and R consists of:

r1: a__and(tt(),T) -> mark(T)
r2: a__isNatIList(IL) -> a__isNatList(IL)
r3: a__isNat(|0|()) -> tt()
r4: a__isNat(s(N)) -> a__isNat(N)
r5: a__isNat(length(L)) -> a__isNatList(L)
r6: a__isNatIList(zeros()) -> tt()
r7: a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r8: a__isNatList(nil()) -> tt()
r9: a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L))
r10: a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL))
r11: a__zeros() -> cons(|0|(),zeros())
r12: a__take(|0|(),IL) -> a__uTake1(a__isNatIList(IL))
r13: a__uTake1(tt()) -> nil()
r14: a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL)
r15: a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL))
r16: a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L)
r17: a__uLength(tt(),L) -> s(a__length(mark(L)))
r18: mark(and(X1,X2)) -> a__and(mark(X1),mark(X2))
r19: mark(isNatIList(X)) -> a__isNatIList(X)
r20: mark(isNatList(X)) -> a__isNatList(X)
r21: mark(isNat(X)) -> a__isNat(X)
r22: mark(length(X)) -> a__length(mark(X))
r23: mark(zeros()) -> a__zeros()
r24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r25: mark(uTake1(X)) -> a__uTake1(mark(X))
r26: mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4)
r27: mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2)
r28: mark(tt()) -> tt()
r29: mark(|0|()) -> |0|()
r30: mark(s(X)) -> s(mark(X))
r31: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r32: mark(nil()) -> nil()
r33: a__and(X1,X2) -> and(X1,X2)
r34: a__isNatIList(X) -> isNatIList(X)
r35: a__isNatList(X) -> isNatList(X)
r36: a__isNat(X) -> isNat(X)
r37: a__length(X) -> length(X)
r38: a__zeros() -> zeros()
r39: a__take(X1,X2) -> take(X1,X2)
r40: a__uTake1(X) -> uTake1(X)
r41: a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4)
r42: a__uLength(X1,X2) -> uLength(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__length#_A(x1) = x1 + 4
    cons_A(x1,x2) = max{14, x2 + 11}
    a__uLength#_A(x1,x2) = max{x1 - 17, x2 + 5}
    a__and_A(x1,x2) = max{17, x2 + 3}
    a__isNat_A(x1) = x1 + 40
    a__isNatList_A(x1) = max{16, x1 + 5}
    tt_A = 36
    mark_A(x1) = max{14, x1}
    a__isNatIList_A(x1) = x1 + 33
    zeros_A = 3
    a__zeros_A = 14
    |0|_A = 18
    a__take_A(x1,x2) = max{47, x1 + 22, x2 + 33}
    a__uTake1_A(x1) = x1
    nil_A = 31
    s_A(x1) = max{36, x1 + 11}
    a__uTake2_A(x1,x2,x3,x4) = max{58, x1 - 26, x2 + 33, x4 + 44}
    take_A(x1,x2) = max{47, x1 + 22, x2 + 33}
    a__length_A(x1) = max{19, x1 - 3}
    a__uLength_A(x1,x2) = max{9, x1, x2 + 8}
    isNatIList_A(x1) = x1 + 33
    length_A(x1) = max{19, x1 - 3}
    uTake1_A(x1) = x1
    uTake2_A(x1,x2,x3,x4) = max{58, x1 - 26, x2 + 33, x4 + 44}
    uLength_A(x1,x2) = max{x1, x2 + 8}
    and_A(x1,x2) = max{17, x2 + 3}
    isNatList_A(x1) = max{16, x1 + 5}
    isNat_A(x1) = x1 + 40

The next rules are strictly ordered:

  p1, p2

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