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:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = ((0,1),(0,1)) x2
        tt_A() = (6,0)
        mark#_A(x1) = ((0,1),(0,1)) x1
        cons_A(x1,x2) = x1 + ((1,0),(1,1)) x2
        s_A(x1) = x1 + (4,5)
        uLength_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(0,1)) x2 + (1,2)
        a__uLength#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,1)) x2 + (1,0)
        mark_A(x1) = x1
        uTake2_A(x1,x2,x3,x4) = x1 + ((0,0),(1,1)) x2 + x3 + ((1,0),(1,1)) x4 + (8,9)
        a__uTake2#_A(x1,x2,x3,x4) = ((0,1),(0,1)) x3 + (9,0)
        uTake1_A(x1) = x1 + (7,6)
        take_A(x1,x2) = ((0,0),(1,1)) x1 + x2 + (8,1)
        a__take#_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,1),(0,1)) x2
        a__isNatIList#_A(x1) = (0,0)
        a__isNat#_A(x1) = (0,0)
        length_A(x1) = ((1,1),(0,1)) x1 + (2,3)
        a__isNatList#_A(x1) = (0,0)
        a__isNat_A(x1) = x1 + (1,0)
        a__isNatIList_A(x1) = x1 + (9,0)
        a__isNatList_A(x1) = x1 + (1,0)
        a__and_A(x1,x2) = x1 + x2
        |0|_A() = (5,0)
        a__length#_A(x1) = ((0,1),(0,1)) x1 + (1,0)
        isNat_A(x1) = x1 + (1,0)
        isNatList_A(x1) = x1 + (1,0)
        isNatIList_A(x1) = x1 + (9,0)
        and_A(x1,x2) = x1 + x2
        a__zeros_A() = (0,1)
        zeros_A() = (0,1)
        a__take_A(x1,x2) = ((0,0),(1,1)) x1 + x2 + (8,1)
        a__uTake1_A(x1) = x1 + (7,6)
        nil_A() = (6,1)
        a__uTake2_A(x1,x2,x3,x4) = x1 + ((0,0),(1,1)) x2 + x3 + ((1,0),(1,1)) x4 + (8,9)
        a__length_A(x1) = ((1,1),(0,1)) x1 + (2,3)
        a__uLength_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(0,1)) x2 + (1,2)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = (3,2)
        tt_A() = (1,0)
        mark#_A(x1) = (3,2)
        cons_A(x1,x2) = (1,1)
        s_A(x1) = (1,1)
        uLength_A(x1,x2) = (0,0)
        a__uLength#_A(x1,x2) = (0,1)
        mark_A(x1) = ((0,0),(1,0)) x1 + (5,2)
        uTake2_A(x1,x2,x3,x4) = (1,1)
        a__uTake2#_A(x1,x2,x3,x4) = (1,1)
        uTake1_A(x1) = (0,1)
        take_A(x1,x2) = (3,1)
        a__take#_A(x1,x2) = ((0,1),(1,0)) x1
        a__isNatIList#_A(x1) = (3,2)
        a__isNat#_A(x1) = (3,2)
        length_A(x1) = (1,1)
        a__isNatList#_A(x1) = (3,2)
        a__isNat_A(x1) = (5,3)
        a__isNatIList_A(x1) = (5,7)
        a__isNatList_A(x1) = (5,7)
        a__and_A(x1,x2) = ((0,0),(1,0)) x2 + (5,2)
        |0|_A() = (0,2)
        a__length#_A(x1) = (0,1)
        isNat_A(x1) = (1,0)
        isNatList_A(x1) = (5,1)
        isNatIList_A(x1) = (5,1)
        and_A(x1,x2) = (5,0)
        a__zeros_A() = (4,2)
        zeros_A() = (0,1)
        a__take_A(x1,x2) = (3,1)
        a__uTake1_A(x1) = (4,2)
        nil_A() = (1,3)
        a__uTake2_A(x1,x2,x3,x4) = (2,1)
        a__length_A(x1) = (2,1)
        a__uLength_A(x1,x2) = (1,0)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = (1,0)
        tt_A() = (2,1)
        mark#_A(x1) = (1,0)
        cons_A(x1,x2) = (3,1)
        s_A(x1) = (0,0)
        uLength_A(x1,x2) = (0,0)
        a__uLength#_A(x1,x2) = (1,0)
        mark_A(x1) = (3,1)
        uTake2_A(x1,x2,x3,x4) = (0,2)
        a__uTake2#_A(x1,x2,x3,x4) = (0,1)
        uTake1_A(x1) = (0,1)
        take_A(x1,x2) = (2,0)
        a__take#_A(x1,x2) = (0,1)
        a__isNatIList#_A(x1) = (1,0)
        a__isNat#_A(x1) = (1,0)
        length_A(x1) = (6,1)
        a__isNatList#_A(x1) = (1,0)
        a__isNat_A(x1) = (2,1)
        a__isNatIList_A(x1) = (3,1)
        a__isNatList_A(x1) = (3,1)
        a__and_A(x1,x2) = (3,1)
        |0|_A() = (1,1)
        a__length#_A(x1) = (1,0)
        isNat_A(x1) = (0,0)
        isNatList_A(x1) = (3,0)
        isNatIList_A(x1) = (0,1)
        and_A(x1,x2) = (0,0)
        a__zeros_A() = (0,1)
        zeros_A() = (0,2)
        a__take_A(x1,x2) = (2,1)
        a__uTake1_A(x1) = (4,2)
        nil_A() = (1,0)
        a__uTake2_A(x1,x2,x3,x4) = (1,1)
        a__length_A(x1) = (5,0)
        a__uLength_A(x1,x2) = (4,0)
    

The next rules are strictly ordered:

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


-- 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#(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:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = ((0,1),(0,0)) x2 + (1,0)
        tt_A() = (0,0)
        mark#_A(x1) = ((0,1),(0,0)) x1 + (1,0)
        and_A(x1,x2) = x1 + x2
        mark_A(x1) = ((0,1),(0,1)) x1
        isNatIList_A(x1) = x1 + (1,1)
        a__isNatIList#_A(x1) = ((0,1),(0,0)) x1 + (2,0)
        a__isNatList#_A(x1) = ((0,1),(0,0)) x1 + (2,0)
        cons_A(x1,x2) = x1 + ((0,0),(1,1)) x2
        a__isNat_A(x1) = ((0,1),(0,1)) x1
        a__isNatList_A(x1) = ((0,1),(0,1)) x1 + (1,1)
        a__isNat#_A(x1) = ((0,1),(0,0)) x1
        s_A(x1) = x1
        length_A(x1) = x1 + (1,3)
        take_A(x1,x2) = x1 + x2 + (0,1)
        a__isNatIList_A(x1) = ((0,1),(0,1)) x1 + (1,1)
        isNatList_A(x1) = x1 + (1,1)
        isNat_A(x1) = x1
        a__and_A(x1,x2) = x1 + ((0,1),(0,1)) x2
        a__zeros_A() = (1,1)
        |0|_A() = (0,0)
        zeros_A() = (0,1)
        a__take_A(x1,x2) = x1 + x2 + (0,1)
        a__uTake1_A(x1) = (0,1)
        nil_A() = (0,1)
        a__uTake2_A(x1,x2,x3,x4) = x2 + x3 + ((0,0),(1,1)) x4 + (0,1)
        a__length_A(x1) = x1 + (2,3)
        a__uLength_A(x1,x2) = ((0,0),(1,1)) x2 + (2,3)
        uTake1_A(x1) = (0,1)
        uTake2_A(x1,x2,x3,x4) = x2 + x3 + ((0,0),(1,1)) x4 + (0,1)
        uLength_A(x1,x2) = ((0,0),(1,1)) x2 + (2,3)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = (1,0)
        tt_A() = (3,2)
        mark#_A(x1) = (1,0)
        and_A(x1,x2) = (1,0)
        mark_A(x1) = (4,2)
        isNatIList_A(x1) = (1,1)
        a__isNatIList#_A(x1) = (1,0)
        a__isNatList#_A(x1) = (1,0)
        cons_A(x1,x2) = (4,2)
        a__isNat_A(x1) = (4,2)
        a__isNatList_A(x1) = (4,2)
        a__isNat#_A(x1) = (0,1)
        s_A(x1) = (4,2)
        length_A(x1) = (0,4)
        take_A(x1,x2) = x2
        a__isNatIList_A(x1) = (4,2)
        isNatList_A(x1) = (0,1)
        isNat_A(x1) = (1,0)
        a__and_A(x1,x2) = ((0,1),(0,1)) x1 + (2,0)
        a__zeros_A() = (3,1)
        |0|_A() = (1,2)
        zeros_A() = (0,1)
        a__take_A(x1,x2) = x2
        a__uTake1_A(x1) = (0,0)
        nil_A() = (0,0)
        a__uTake2_A(x1,x2,x3,x4) = (4,2)
        a__length_A(x1) = (6,3)
        a__uLength_A(x1,x2) = (5,3)
        uTake1_A(x1) = (0,0)
        uTake2_A(x1,x2,x3,x4) = (0,0)
        uLength_A(x1,x2) = (0,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = (1,0)
        tt_A() = (0,1)
        mark#_A(x1) = (1,0)
        and_A(x1,x2) = (1,1)
        mark_A(x1) = (3,1)
        isNatIList_A(x1) = (4,0)
        a__isNatIList#_A(x1) = (1,0)
        a__isNatList#_A(x1) = (1,0)
        cons_A(x1,x2) = (0,0)
        a__isNat_A(x1) = (0,1)
        a__isNatList_A(x1) = (3,1)
        a__isNat#_A(x1) = (0,1)
        s_A(x1) = (3,0)
        length_A(x1) = (0,2)
        take_A(x1,x2) = x2
        a__isNatIList_A(x1) = (3,1)
        isNatList_A(x1) = (4,2)
        isNat_A(x1) = (1,0)
        a__and_A(x1,x2) = (3,1)
        a__zeros_A() = (2,0)
        |0|_A() = (0,0)
        zeros_A() = (1,0)
        a__take_A(x1,x2) = x2
        a__uTake1_A(x1) = (0,0)
        nil_A() = (0,0)
        a__uTake2_A(x1,x2,x3,x4) = (0,0)
        a__length_A(x1) = (2,1)
        a__uLength_A(x1,x2) = (1,0)
        uTake1_A(x1) = (0,0)
        uTake2_A(x1,x2,x3,x4) = (0,0)
        uLength_A(x1,x2) = (0,0)
    

The next rules are strictly ordered:

  p8, p10, p12, p13, p14, p16, p19

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__isNatIList#(IL) -> a__isNatList#(IL)
p7: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p8: a__isNat#(s(N)) -> a__isNat#(N)
p9: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p10: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p11: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p12: mark#(isNatList(X)) -> a__isNatList#(X)
p13: 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, p9, p10, p11, p12, p13}
  {p8}


-- 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#(isNatList(X)) -> a__isNatList#(X)
p4: a__isNatList#(cons(N,L)) -> a__isNatList#(L)
p5: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L))
p6: mark#(isNatIList(X)) -> a__isNatIList#(X)
p7: a__isNatIList#(cons(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: mark#(and(X1,X2)) -> mark#(X2)
p11: mark#(and(X1,X2)) -> mark#(X1)
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 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:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = ((1,1),(0,0)) x2
        tt_A() = (0,0)
        mark#_A(x1) = ((1,1),(0,0)) x1
        cons_A(x1,x2) = ((0,1),(1,0)) x1 + (1,1)
        isNatList_A(x1) = (0,0)
        a__isNatList#_A(x1) = (0,0)
        a__isNat_A(x1) = (0,0)
        a__isNatList_A(x1) = (0,0)
        isNatIList_A(x1) = (0,0)
        a__isNatIList#_A(x1) = (0,0)
        a__isNatIList_A(x1) = (0,0)
        and_A(x1,x2) = x1 + ((1,1),(1,1)) x2
        mark_A(x1) = ((1,1),(1,1)) x1
        a__and_A(x1,x2) = x1 + ((1,1),(1,1)) x2
        a__zeros_A() = (3,2)
        |0|_A() = (1,1)
        zeros_A() = (2,2)
        a__take_A(x1,x2) = ((0,1),(1,0)) x1 + ((1,1),(1,1)) x2 + (2,1)
        a__uTake1_A(x1) = (2,2)
        nil_A() = (2,0)
        s_A(x1) = ((0,1),(1,0)) x1
        a__uTake2_A(x1,x2,x3,x4) = ((1,1),(1,1)) x3 + (2,1)
        take_A(x1,x2) = ((0,1),(1,0)) x1 + ((1,1),(1,1)) x2 + (2,0)
        a__length_A(x1) = (2,2)
        a__uLength_A(x1,x2) = (2,2)
        length_A(x1) = (1,2)
        uTake1_A(x1) = (1,2)
        uTake2_A(x1,x2,x3,x4) = ((1,0),(1,1)) x3 + (2,1)
        uLength_A(x1,x2) = (2,0)
        isNat_A(x1) = (0,0)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = (0,0)
        tt_A() = (1,1)
        mark#_A(x1) = (0,0)
        cons_A(x1,x2) = (0,3)
        isNatList_A(x1) = (2,0)
        a__isNatList#_A(x1) = (0,0)
        a__isNat_A(x1) = (3,1)
        a__isNatList_A(x1) = (2,1)
        isNatIList_A(x1) = (2,1)
        a__isNatIList#_A(x1) = (0,0)
        a__isNatIList_A(x1) = (2,1)
        and_A(x1,x2) = x2 + (0,1)
        mark_A(x1) = x1 + (0,1)
        a__and_A(x1,x2) = x2 + (0,1)
        a__zeros_A() = (2,2)
        |0|_A() = (1,0)
        zeros_A() = (1,0)
        a__take_A(x1,x2) = (1,0)
        a__uTake1_A(x1) = (0,2)
        nil_A() = (0,0)
        s_A(x1) = (0,0)
        a__uTake2_A(x1,x2,x3,x4) = (2,2)
        take_A(x1,x2) = (1,0)
        a__length_A(x1) = (1,2)
        a__uLength_A(x1,x2) = (0,0)
        length_A(x1) = (0,0)
        uTake1_A(x1) = (0,0)
        uTake2_A(x1,x2,x3,x4) = (0,0)
        uLength_A(x1,x2) = (0,0)
        isNat_A(x1) = (3,0)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = (0,0)
        tt_A() = (3,2)
        mark#_A(x1) = (0,0)
        cons_A(x1,x2) = (0,3)
        isNatList_A(x1) = (1,0)
        a__isNatList#_A(x1) = (0,0)
        a__isNat_A(x1) = (0,1)
        a__isNatList_A(x1) = (2,0)
        isNatIList_A(x1) = (1,0)
        a__isNatIList#_A(x1) = (0,0)
        a__isNatIList_A(x1) = (2,0)
        and_A(x1,x2) = ((0,1),(0,1)) x2 + (1,0)
        mark_A(x1) = ((0,1),(0,1)) x1 + (2,0)
        a__and_A(x1,x2) = ((0,1),(0,1)) x2 + (2,0)
        a__zeros_A() = (4,2)
        |0|_A() = (4,1)
        zeros_A() = (1,1)
        a__take_A(x1,x2) = (1,0)
        a__uTake1_A(x1) = (3,1)
        nil_A() = (2,1)
        s_A(x1) = (2,1)
        a__uTake2_A(x1,x2,x3,x4) = (4,2)
        take_A(x1,x2) = (0,0)
        a__length_A(x1) = (2,1)
        a__uLength_A(x1,x2) = (2,1)
        length_A(x1) = (3,2)
        uTake1_A(x1) = (4,0)
        uTake2_A(x1,x2,x3,x4) = (1,1)
        uLength_A(x1,x2) = (1,1)
        isNat_A(x1) = (0,1)
    

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

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


-- 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__isNatList#(L)
p9: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p10: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p11: mark#(isNatList(X)) -> a__isNatList#(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:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = x1 + ((0,1),(0,0)) x2
        tt_A() = (4,0)
        mark#_A(x1) = ((0,1),(0,0)) x1 + (4,0)
        and_A(x1,x2) = x1 + x2 + (1,0)
        mark_A(x1) = x1 + (4,0)
        isNatIList_A(x1) = x1 + (1,2)
        a__isNatIList#_A(x1) = ((0,1),(0,0)) x1 + (6,0)
        a__isNatList#_A(x1) = ((0,1),(0,0)) x1 + (5,0)
        cons_A(x1,x2) = x1 + x2 + (1,0)
        a__isNat_A(x1) = x1 + (4,0)
        a__isNatList_A(x1) = x1 + (4,1)
        a__isNatIList_A(x1) = x1 + (4,2)
        isNatList_A(x1) = x1 + (3,1)
        a__and_A(x1,x2) = x1 + x2 + (4,0)
        a__zeros_A() = (2,1)
        |0|_A() = (1,0)
        zeros_A() = (1,1)
        a__take_A(x1,x2) = x1 + x2 + (2,3)
        a__uTake1_A(x1) = x1 + (2,1)
        nil_A() = (2,1)
        s_A(x1) = x1 + (2,0)
        a__uTake2_A(x1,x2,x3,x4) = x2 + x3 + x4 + (2,3)
        take_A(x1,x2) = x1 + x2 + (1,3)
        a__length_A(x1) = x1 + (3,1)
        a__uLength_A(x1,x2) = x2 + (2,1)
        length_A(x1) = x1 + (1,1)
        uTake1_A(x1) = x1 + (2,1)
        uTake2_A(x1,x2,x3,x4) = x2 + x3 + x4 + (1,3)
        uLength_A(x1,x2) = x2 + (1,1)
        isNat_A(x1) = x1 + (1,0)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = ((0,1),(0,0)) x1
        tt_A() = (0,1)
        mark#_A(x1) = (1,0)
        and_A(x1,x2) = (1,2)
        mark_A(x1) = (0,1)
        isNatIList_A(x1) = (1,2)
        a__isNatIList#_A(x1) = (1,0)
        a__isNatList#_A(x1) = (1,0)
        cons_A(x1,x2) = (3,3)
        a__isNat_A(x1) = (0,1)
        a__isNatList_A(x1) = (0,1)
        a__isNatIList_A(x1) = (0,1)
        isNatList_A(x1) = (1,2)
        a__and_A(x1,x2) = (0,1)
        a__zeros_A() = (2,2)
        |0|_A() = (0,2)
        zeros_A() = (1,0)
        a__take_A(x1,x2) = (0,2)
        a__uTake1_A(x1) = (0,1)
        nil_A() = (0,0)
        s_A(x1) = (1,2)
        a__uTake2_A(x1,x2,x3,x4) = (0,2)
        take_A(x1,x2) = (0,0)
        a__length_A(x1) = (0,2)
        a__uLength_A(x1,x2) = (2,2)
        length_A(x1) = (0,0)
        uTake1_A(x1) = (0,1)
        uTake2_A(x1,x2,x3,x4) = (0,3)
        uLength_A(x1,x2) = (3,3)
        isNat_A(x1) = (1,2)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = (0,0)
        tt_A() = (0,0)
        mark#_A(x1) = (0,0)
        and_A(x1,x2) = (0,0)
        mark_A(x1) = (0,0)
        isNatIList_A(x1) = (1,0)
        a__isNatIList#_A(x1) = (0,0)
        a__isNatList#_A(x1) = (0,0)
        cons_A(x1,x2) = (0,1)
        a__isNat_A(x1) = (0,0)
        a__isNatList_A(x1) = (0,0)
        a__isNatIList_A(x1) = (0,0)
        isNatList_A(x1) = (1,1)
        a__and_A(x1,x2) = (0,0)
        a__zeros_A() = (1,1)
        |0|_A() = (1,0)
        zeros_A() = (2,2)
        a__take_A(x1,x2) = (3,1)
        a__uTake1_A(x1) = (2,1)
        nil_A() = (1,0)
        s_A(x1) = (2,2)
        a__uTake2_A(x1,x2,x3,x4) = (0,0)
        take_A(x1,x2) = (0,1)
        a__length_A(x1) = (0,0)
        a__uLength_A(x1,x2) = (1,1)
        length_A(x1) = (1,0)
        uTake1_A(x1) = (0,0)
        uTake2_A(x1,x2,x3,x4) = (0,1)
        uLength_A(x1,x2) = (0,0)
        isNat_A(x1) = (0,0)
    

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: 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__isNatList#(L)
p8: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p9: a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL)
p10: mark#(isNatList(X)) -> a__isNatList#(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}


-- Reduction pair.

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

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

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = x1 + ((1,0),(1,0)) x2
        tt_A() = (0,5)
        mark#_A(x1) = ((1,0),(1,0)) x1
        isNatList_A(x1) = x1 + (1,1)
        a__isNatList#_A(x1) = ((1,0),(1,0)) x1 + (1,1)
        cons_A(x1,x2) = ((1,1),(1,0)) x1 + ((1,1),(0,1)) x2
        a__isNat_A(x1) = ((1,1),(1,1)) x1 + (0,5)
        a__isNatList_A(x1) = x1 + (1,1)
        isNatIList_A(x1) = x1 + (2,5)
        a__isNatIList#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
        a__isNatIList_A(x1) = x1 + (2,5)
        and_A(x1,x2) = x1 + x2
        mark_A(x1) = x1
        a__and_A(x1,x2) = x1 + x2
        a__zeros_A() = (1,0)
        |0|_A() = (0,0)
        zeros_A() = (1,0)
        a__take_A(x1,x2) = ((1,1),(0,0)) x1 + x2 + (3,4)
        a__uTake1_A(x1) = (2,4)
        nil_A() = (1,4)
        s_A(x1) = ((0,1),(1,0)) x1 + (0,6)
        a__uTake2_A(x1,x2,x3,x4) = ((1,1),(0,0)) x2 + ((1,1),(1,0)) x3 + ((1,1),(0,1)) x4 + (8,4)
        take_A(x1,x2) = ((1,1),(0,0)) x1 + x2 + (3,4)
        a__length_A(x1) = ((1,1),(1,1)) x1 + (3,5)
        a__uLength_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (1,4)
        length_A(x1) = ((1,1),(1,1)) x1 + (3,5)
        uTake1_A(x1) = (2,4)
        uTake2_A(x1,x2,x3,x4) = ((1,1),(0,0)) x2 + ((1,1),(1,0)) x3 + ((1,1),(0,1)) x4 + (8,4)
        uLength_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (1,4)
        isNat_A(x1) = ((1,1),(1,1)) x1 + (0,5)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (8,0)
        tt_A() = (1,7)
        mark#_A(x1) = x1 + (9,0)
        isNatList_A(x1) = ((1,1),(1,0)) x1 + (0,4)
        a__isNatList#_A(x1) = ((1,1),(0,0)) x1 + (8,0)
        cons_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,0),(1,1)) x2 + (14,0)
        a__isNat_A(x1) = x1 + (2,3)
        a__isNatList_A(x1) = ((1,1),(1,0)) x1 + (3,4)
        isNatIList_A(x1) = ((1,1),(1,0)) x1 + (2,3)
        a__isNatIList#_A(x1) = ((1,1),(0,0)) x1
        a__isNatIList_A(x1) = ((1,1),(1,0)) x1 + (3,3)
        and_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,1)) x2 + (0,6)
        mark_A(x1) = ((1,1),(0,1)) x1
        a__and_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,1)) x2 + (5,6)
        a__zeros_A() = (18,19)
        |0|_A() = (1,4)
        zeros_A() = (0,19)
        a__take_A(x1,x2) = ((0,1),(0,1)) x1 + (6,7)
        a__uTake1_A(x1) = (2,2)
        nil_A() = (2,1)
        s_A(x1) = (1,3)
        a__uTake2_A(x1,x2,x3,x4) = ((0,0),(1,1)) x2 + ((1,1),(1,1)) x3 + ((1,0),(1,0)) x4 + (10,11)
        take_A(x1,x2) = ((0,1),(0,1)) x1 + (5,7)
        a__length_A(x1) = (1,0)
        a__uLength_A(x1,x2) = ((1,0),(1,1)) x2 + (1,2)
        length_A(x1) = (1,0)
        uTake1_A(x1) = (1,2)
        uTake2_A(x1,x2,x3,x4) = ((0,0),(1,1)) x2 + ((1,0),(1,1)) x3 + ((0,0),(1,0)) x4 + (0,11)
        uLength_A(x1,x2) = ((0,0),(1,1)) x2 + (0,2)
        isNat_A(x1) = x1 + (1,3)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = ((1,1),(0,0)) x2 + (1,1)
        tt_A() = (0,0)
        mark#_A(x1) = x1
        isNatList_A(x1) = ((0,0),(1,0)) x1 + (5,1)
        a__isNatList#_A(x1) = ((1,1),(0,0)) x1 + (10,1)
        cons_A(x1,x2) = ((1,1),(1,1)) x2 + (2,1)
        a__isNat_A(x1) = (4,7)
        a__isNatList_A(x1) = (4,7)
        isNatIList_A(x1) = ((1,1),(0,0)) x1 + (2,4)
        a__isNatIList#_A(x1) = ((1,1),(1,1)) x1 + (2,0)
        a__isNatIList_A(x1) = ((1,0),(1,1)) x1 + (1,3)
        and_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (0,8)
        mark_A(x1) = ((1,0),(1,0)) x1
        a__and_A(x1,x2) = (4,7)
        a__zeros_A() = (5,3)
        |0|_A() = (1,0)
        zeros_A() = (1,1)
        a__take_A(x1,x2) = (0,0)
        a__uTake1_A(x1) = (1,1)
        nil_A() = (2,0)
        s_A(x1) = (0,0)
        a__uTake2_A(x1,x2,x3,x4) = x4
        take_A(x1,x2) = (0,0)
        a__length_A(x1) = (0,0)
        a__uLength_A(x1,x2) = ((1,1),(1,0)) x2
        length_A(x1) = (0,0)
        uTake1_A(x1) = (0,2)
        uTake2_A(x1,x2,x3,x4) = (0,0)
        uLength_A(x1,x2) = (0,0)
        isNat_A(x1) = (0,0)
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p10

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL))
p2: mark#(and(X1,X2)) -> mark#(X2)
p3: mark#(and(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:

  {p2, p3}


-- Reduction pair.

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

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        mark#_A(x1) = ((1,1),(0,0)) x1
        and_A(x1,x2) = ((0,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        mark#_A(x1) = ((1,1),(1,1)) x1
        and_A(x1,x2) = ((1,1),(1,1)) x2 + (1,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        mark#_A(x1) = ((1,1),(1,1)) x1
        and_A(x1,x2) = ((1,1),(1,1)) x2 + (1,1)
    

The next rules are strictly ordered:

  p1, p2

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

-- Reduction pair.

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

p1: 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 reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__isNat#_A(x1) = ((1,1),(1,1)) x1
        s_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__isNat#_A(x1) = x1
        s_A(x1) = x1 + (1,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__isNat#_A(x1) = ((0,1),(1,1)) x1
        s_A(x1) = ((0,1),(1,1)) x1 + (1,1)
    

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:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__length#_A(x1) = x1 + (2,0)
        cons_A(x1,x2) = ((1,1),(1,1)) x2 + (2,0)
        a__uLength#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2
        a__and_A(x1,x2) = ((0,1),(0,1)) x2 + (3,0)
        a__isNat_A(x1) = ((1,0),(1,1)) x1 + (0,4)
        a__isNatList_A(x1) = ((1,0),(1,0)) x1 + (4,1)
        tt_A() = (0,5)
        mark_A(x1) = ((0,1),(0,1)) x1 + (3,0)
        a__isNatIList_A(x1) = ((1,0),(1,0)) x1 + (7,5)
        zeros_A() = (0,1)
        a__zeros_A() = (3,1)
        |0|_A() = (0,1)
        a__take_A(x1,x2) = ((0,1),(0,1)) x1 + x2 + (5,5)
        a__uTake1_A(x1) = (6,4)
        nil_A() = (4,2)
        s_A(x1) = x1 + (1,4)
        a__uTake2_A(x1,x2,x3,x4) = ((0,1),(0,1)) x2 + ((1,1),(1,1)) x4 + (11,9)
        take_A(x1,x2) = x1 + x2 + (4,5)
        a__length_A(x1) = x1 + (6,7)
        a__uLength_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (6,6)
        isNatIList_A(x1) = ((0,0),(1,0)) x1 + (1,5)
        length_A(x1) = x1 + (5,7)
        uTake1_A(x1) = (6,4)
        uTake2_A(x1,x2,x3,x4) = x2 + ((1,0),(1,1)) x4 + (1,9)
        uLength_A(x1,x2) = x1 + ((0,1),(0,1)) x2 + (1,6)
        and_A(x1,x2) = x2 + (2,0)
        isNatList_A(x1) = ((0,0),(1,0)) x1 + (1,1)
        isNat_A(x1) = ((0,0),(1,1)) x1 + (0,4)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__length#_A(x1) = ((0,1),(0,1)) x1
        cons_A(x1,x2) = ((1,0),(1,1)) x2 + (2,1)
        a__uLength#_A(x1,x2) = (1,1)
        a__and_A(x1,x2) = (0,0)
        a__isNat_A(x1) = ((1,1),(1,1)) x1
        a__isNatList_A(x1) = (0,0)
        tt_A() = (4,4)
        mark_A(x1) = (0,0)
        a__isNatIList_A(x1) = ((1,1),(1,1)) x1 + (1,1)
        zeros_A() = (1,1)
        a__zeros_A() = (4,3)
        |0|_A() = (1,3)
        a__take_A(x1,x2) = ((1,1),(1,0)) x2
        a__uTake1_A(x1) = (0,0)
        nil_A() = (1,1)
        s_A(x1) = (1,3)
        a__uTake2_A(x1,x2,x3,x4) = (3,2)
        take_A(x1,x2) = (0,1)
        a__length_A(x1) = (2,1)
        a__uLength_A(x1,x2) = (2,2)
        isNatIList_A(x1) = (1,0)
        length_A(x1) = x1 + (1,1)
        uTake1_A(x1) = (0,0)
        uTake2_A(x1,x2,x3,x4) = ((1,0),(1,1)) x4
        uLength_A(x1,x2) = (0,0)
        and_A(x1,x2) = (1,1)
        isNatList_A(x1) = (0,1)
        isNat_A(x1) = (0,0)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__length#_A(x1) = (1,0)
        cons_A(x1,x2) = ((0,0),(1,1)) x2 + (6,1)
        a__uLength#_A(x1,x2) = (0,0)
        a__and_A(x1,x2) = (5,3)
        a__isNat_A(x1) = ((1,1),(1,1)) x1 + (0,1)
        a__isNatList_A(x1) = (5,3)
        tt_A() = (5,1)
        mark_A(x1) = (5,3)
        a__isNatIList_A(x1) = ((0,0),(1,1)) x1 + (5,3)
        zeros_A() = (1,1)
        a__zeros_A() = (5,3)
        |0|_A() = (1,4)
        a__take_A(x1,x2) = (2,3)
        a__uTake1_A(x1) = (0,3)
        nil_A() = (0,0)
        s_A(x1) = (1,1)
        a__uTake2_A(x1,x2,x3,x4) = (1,3)
        take_A(x1,x2) = (3,4)
        a__length_A(x1) = (5,1)
        a__uLength_A(x1,x2) = (2,4)
        isNatIList_A(x1) = (0,1)
        length_A(x1) = (4,1)
        uTake1_A(x1) = (0,0)
        uTake2_A(x1,x2,x3,x4) = ((0,1),(1,0)) x4 + (0,4)
        uLength_A(x1,x2) = (0,5)
        and_A(x1,x2) = (0,1)
        isNatList_A(x1) = (0,0)
        isNat_A(x1) = (0,1)
    

The next rules are strictly ordered:

  p1, p2

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