YES

We show the termination of the TRS R:

  a__zeros() -> cons(|0|(),zeros())
  a__U11(tt(),L) -> s(a__length(mark(L)))
  a__and(tt(),X) -> mark(X)
  a__isNat(|0|()) -> tt()
  a__isNat(length(V1)) -> a__isNatList(V1)
  a__isNat(s(V1)) -> a__isNat(V1)
  a__isNatIList(V) -> a__isNatList(V)
  a__isNatIList(zeros()) -> tt()
  a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
  a__isNatList(nil()) -> tt()
  a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
  a__length(nil()) -> |0|()
  a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
  mark(zeros()) -> a__zeros()
  mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
  mark(length(X)) -> a__length(mark(X))
  mark(and(X1,X2)) -> a__and(mark(X1),X2)
  mark(isNat(X)) -> a__isNat(X)
  mark(isNatList(X)) -> a__isNatList(X)
  mark(isNatIList(X)) -> a__isNatIList(X)
  mark(cons(X1,X2)) -> cons(mark(X1),X2)
  mark(|0|()) -> |0|()
  mark(tt()) -> tt()
  mark(s(X)) -> s(mark(X))
  mark(nil()) -> nil()
  a__zeros() -> zeros()
  a__U11(X1,X2) -> U11(X1,X2)
  a__length(X) -> length(X)
  a__and(X1,X2) -> and(X1,X2)
  a__isNat(X) -> isNat(X)
  a__isNatList(X) -> isNatList(X)
  a__isNatIList(X) -> isNatIList(X)

-- SCC decomposition.

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

p1: a__U11#(tt(),L) -> a__length#(mark(L))
p2: a__U11#(tt(),L) -> mark#(L)
p3: a__and#(tt(),X) -> mark#(X)
p4: a__isNat#(length(V1)) -> a__isNatList#(V1)
p5: a__isNat#(s(V1)) -> a__isNat#(V1)
p6: a__isNatIList#(V) -> a__isNatList#(V)
p7: a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatIList(V2))
p8: a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1)
p9: a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2))
p10: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p11: a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L)
p12: a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNat(N))
p13: a__length#(cons(N,L)) -> a__isNatList#(L)
p14: mark#(zeros()) -> a__zeros#()
p15: mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2)
p16: mark#(U11(X1,X2)) -> mark#(X1)
p17: mark#(length(X)) -> a__length#(mark(X))
p18: mark#(length(X)) -> mark#(X)
p19: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p20: mark#(and(X1,X2)) -> mark#(X1)
p21: mark#(isNat(X)) -> a__isNat#(X)
p22: mark#(isNatList(X)) -> a__isNatList#(X)
p23: mark#(isNatIList(X)) -> a__isNatIList#(X)
p24: mark#(cons(X1,X2)) -> mark#(X1)
p25: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),L) -> s(a__length(mark(L)))
r3: a__and(tt(),X) -> mark(X)
r4: a__isNat(|0|()) -> tt()
r5: a__isNat(length(V1)) -> a__isNatList(V1)
r6: a__isNat(s(V1)) -> a__isNat(V1)
r7: a__isNatIList(V) -> a__isNatList(V)
r8: a__isNatIList(zeros()) -> tt()
r9: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
r10: a__isNatList(nil()) -> tt()
r11: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
r12: a__length(nil()) -> |0|()
r13: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
r14: mark(zeros()) -> a__zeros()
r15: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r16: mark(length(X)) -> a__length(mark(X))
r17: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r18: mark(isNat(X)) -> a__isNat(X)
r19: mark(isNatList(X)) -> a__isNatList(X)
r20: mark(isNatIList(X)) -> a__isNatIList(X)
r21: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r22: mark(|0|()) -> |0|()
r23: mark(tt()) -> tt()
r24: mark(s(X)) -> s(mark(X))
r25: mark(nil()) -> nil()
r26: a__zeros() -> zeros()
r27: a__U11(X1,X2) -> U11(X1,X2)
r28: a__length(X) -> length(X)
r29: a__and(X1,X2) -> and(X1,X2)
r30: a__isNat(X) -> isNat(X)
r31: a__isNatList(X) -> isNatList(X)
r32: a__isNatIList(X) -> isNatIList(X)

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}


-- Reduction pair.

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

p1: a__U11#(tt(),L) -> a__length#(mark(L))
p2: a__length#(cons(N,L)) -> a__isNatList#(L)
p3: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p4: a__isNat#(s(V1)) -> a__isNat#(V1)
p5: a__isNat#(length(V1)) -> a__isNatList#(V1)
p6: a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2))
p7: a__and#(tt(),X) -> mark#(X)
p8: mark#(s(X)) -> mark#(X)
p9: mark#(cons(X1,X2)) -> mark#(X1)
p10: mark#(isNatIList(X)) -> a__isNatIList#(X)
p11: a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1)
p12: a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatIList(V2))
p13: a__isNatIList#(V) -> a__isNatList#(V)
p14: mark#(isNatList(X)) -> a__isNatList#(X)
p15: mark#(isNat(X)) -> a__isNat#(X)
p16: mark#(and(X1,X2)) -> mark#(X1)
p17: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p18: mark#(length(X)) -> mark#(X)
p19: mark#(length(X)) -> a__length#(mark(X))
p20: a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNat(N))
p21: a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L)
p22: a__U11#(tt(),L) -> mark#(L)
p23: mark#(U11(X1,X2)) -> mark#(X1)
p24: mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),L) -> s(a__length(mark(L)))
r3: a__and(tt(),X) -> mark(X)
r4: a__isNat(|0|()) -> tt()
r5: a__isNat(length(V1)) -> a__isNatList(V1)
r6: a__isNat(s(V1)) -> a__isNat(V1)
r7: a__isNatIList(V) -> a__isNatList(V)
r8: a__isNatIList(zeros()) -> tt()
r9: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
r10: a__isNatList(nil()) -> tt()
r11: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
r12: a__length(nil()) -> |0|()
r13: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
r14: mark(zeros()) -> a__zeros()
r15: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r16: mark(length(X)) -> a__length(mark(X))
r17: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r18: mark(isNat(X)) -> a__isNat(X)
r19: mark(isNatList(X)) -> a__isNatList(X)
r20: mark(isNatIList(X)) -> a__isNatIList(X)
r21: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r22: mark(|0|()) -> |0|()
r23: mark(tt()) -> tt()
r24: mark(s(X)) -> s(mark(X))
r25: mark(nil()) -> nil()
r26: a__zeros() -> zeros()
r27: a__U11(X1,X2) -> U11(X1,X2)
r28: a__length(X) -> length(X)
r29: a__and(X1,X2) -> and(X1,X2)
r30: a__isNat(X) -> isNat(X)
r31: a__isNatList(X) -> isNatList(X)
r32: a__isNatIList(X) -> isNatIList(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__U11#_A(x1,x2) = ((0,0),(1,0)) x2 + (0,4)
            tt_A() = (4,0)
            a__length#_A(x1) = ((0,0),(1,0)) x1
            mark_A(x1) = ((1,0),(1,0)) x1 + (4,0)
            cons_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (4,0)
            a__isNatList#_A(x1) = (0,4)
            a__isNat#_A(x1) = (0,4)
            s_A(x1) = ((1,0),(0,0)) x1
            length_A(x1) = ((1,0),(0,0)) x1 + (5,0)
            a__and#_A(x1,x2) = ((0,0),(1,0)) x2 + (0,4)
            a__isNat_A(x1) = (4,0)
            isNatList_A(x1) = (0,5)
            mark#_A(x1) = ((0,0),(1,0)) x1 + (0,4)
            isNatIList_A(x1) = x1 + (5,1)
            a__isNatIList#_A(x1) = ((0,0),(1,0)) x1 + (0,5)
            isNat_A(x1) = (0,4)
            and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2
            a__isNatList_A(x1) = (4,0)
            a__and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2
            U11_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (5,0)
            a__zeros_A() = (7,2)
            |0|_A() = (0,1)
            zeros_A() = (3,1)
            a__U11_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (5,0)
            a__length_A(x1) = x1 + (5,0)
            a__isNatIList_A(x1) = ((1,0),(1,1)) x1 + (6,5)
            nil_A() = (3,2)
    
      precedence:
      
        a__U11# = a__length# = cons = a__isNatIList# > a__isNatIList > isNat > isNatIList > mark = a__isNat = a__isNatList = a__and = a__zeros > isNatList > tt = a__isNatList# = a__isNat# = s = length = a__and# = mark# = and = U11 = zeros = a__U11 = a__length > nil > |0|
      
      partial status:
    
        pi(a__U11#) = []
        pi(tt) = []
        pi(a__length#) = []
        pi(mark) = []
        pi(cons) = []
        pi(a__isNatList#) = []
        pi(a__isNat#) = []
        pi(s) = []
        pi(length) = []
        pi(a__and#) = []
        pi(a__isNat) = []
        pi(isNatList) = []
        pi(mark#) = []
        pi(isNatIList) = []
        pi(a__isNatIList#) = []
        pi(isNat) = []
        pi(and) = []
        pi(a__isNatList) = []
        pi(a__and) = []
        pi(U11) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = [1]
        pi(a__length) = [1]
        pi(a__isNatIList) = []
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__U11#_A(x1,x2) = (5,12)
            tt_A() = (11,7)
            a__length#_A(x1) = (5,12)
            mark_A(x1) = (8,11)
            cons_A(x1,x2) = (6,10)
            a__isNatList#_A(x1) = (4,13)
            a__isNat#_A(x1) = (4,13)
            s_A(x1) = (4,0)
            length_A(x1) = (1,0)
            a__and#_A(x1,x2) = (4,13)
            a__isNat_A(x1) = (8,11)
            isNatList_A(x1) = (5,0)
            mark#_A(x1) = (4,13)
            isNatIList_A(x1) = (13,0)
            a__isNatIList#_A(x1) = (9,14)
            isNat_A(x1) = (0,0)
            and_A(x1,x2) = (0,10)
            a__isNatList_A(x1) = (8,11)
            a__and_A(x1,x2) = (8,11)
            U11_A(x1,x2) = (3,0)
            a__zeros_A() = (7,12)
            |0|_A() = (0,0)
            zeros_A() = (0,13)
            a__U11_A(x1,x2) = ((0,0),(1,0)) x1 + (3,1)
            a__length_A(x1) = (2,8)
            a__isNatIList_A(x1) = (12,12)
            nil_A() = (0,0)
    
      precedence:
      
        a__isNatIList = nil > |0| > a__zeros > a__U11# = a__length# = cons > isNat > isNatIList = a__isNatIList# > a__isNatList# = a__isNat# = a__and# = mark# > isNatList > s = a__U11 = a__length > length > tt = mark = a__isNat = a__isNatList = a__and > and > U11 > zeros
      
      partial status:
    
        pi(a__U11#) = []
        pi(tt) = []
        pi(a__length#) = []
        pi(mark) = []
        pi(cons) = []
        pi(a__isNatList#) = []
        pi(a__isNat#) = []
        pi(s) = []
        pi(length) = []
        pi(a__and#) = []
        pi(a__isNat) = []
        pi(isNatList) = []
        pi(mark#) = []
        pi(isNatIList) = []
        pi(a__isNatIList#) = []
        pi(isNat) = []
        pi(and) = []
        pi(a__isNatList) = []
        pi(a__and) = []
        pi(U11) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__length) = []
        pi(a__isNatIList) = []
        pi(nil) = []
    

The next rules are strictly ordered:

  p2, p10, p11, p18, p19, p20, p22, p23, p24

We remove them from the problem.

-- SCC decomposition.

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

p1: a__U11#(tt(),L) -> a__length#(mark(L))
p2: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p3: a__isNat#(s(V1)) -> a__isNat#(V1)
p4: a__isNat#(length(V1)) -> a__isNatList#(V1)
p5: a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2))
p6: a__and#(tt(),X) -> mark#(X)
p7: mark#(s(X)) -> mark#(X)
p8: mark#(cons(X1,X2)) -> mark#(X1)
p9: a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatIList(V2))
p10: a__isNatIList#(V) -> a__isNatList#(V)
p11: mark#(isNatList(X)) -> a__isNatList#(X)
p12: mark#(isNat(X)) -> a__isNat#(X)
p13: mark#(and(X1,X2)) -> mark#(X1)
p14: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p15: a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),L) -> s(a__length(mark(L)))
r3: a__and(tt(),X) -> mark(X)
r4: a__isNat(|0|()) -> tt()
r5: a__isNat(length(V1)) -> a__isNatList(V1)
r6: a__isNat(s(V1)) -> a__isNat(V1)
r7: a__isNatIList(V) -> a__isNatList(V)
r8: a__isNatIList(zeros()) -> tt()
r9: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
r10: a__isNatList(nil()) -> tt()
r11: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
r12: a__length(nil()) -> |0|()
r13: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
r14: mark(zeros()) -> a__zeros()
r15: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r16: mark(length(X)) -> a__length(mark(X))
r17: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r18: mark(isNat(X)) -> a__isNat(X)
r19: mark(isNatList(X)) -> a__isNatList(X)
r20: mark(isNatIList(X)) -> a__isNatIList(X)
r21: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r22: mark(|0|()) -> |0|()
r23: mark(tt()) -> tt()
r24: mark(s(X)) -> s(mark(X))
r25: mark(nil()) -> nil()
r26: a__zeros() -> zeros()
r27: a__U11(X1,X2) -> U11(X1,X2)
r28: a__length(X) -> length(X)
r29: a__and(X1,X2) -> and(X1,X2)
r30: a__isNat(X) -> isNat(X)
r31: a__isNatList(X) -> isNatList(X)
r32: a__isNatIList(X) -> isNatIList(X)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__U11#(tt(),L) -> a__length#(mark(L))
p2: a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),L) -> s(a__length(mark(L)))
r3: a__and(tt(),X) -> mark(X)
r4: a__isNat(|0|()) -> tt()
r5: a__isNat(length(V1)) -> a__isNatList(V1)
r6: a__isNat(s(V1)) -> a__isNat(V1)
r7: a__isNatIList(V) -> a__isNatList(V)
r8: a__isNatIList(zeros()) -> tt()
r9: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
r10: a__isNatList(nil()) -> tt()
r11: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
r12: a__length(nil()) -> |0|()
r13: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
r14: mark(zeros()) -> a__zeros()
r15: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r16: mark(length(X)) -> a__length(mark(X))
r17: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r18: mark(isNat(X)) -> a__isNat(X)
r19: mark(isNatList(X)) -> a__isNatList(X)
r20: mark(isNatIList(X)) -> a__isNatIList(X)
r21: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r22: mark(|0|()) -> |0|()
r23: mark(tt()) -> tt()
r24: mark(s(X)) -> s(mark(X))
r25: mark(nil()) -> nil()
r26: a__zeros() -> zeros()
r27: a__U11(X1,X2) -> U11(X1,X2)
r28: a__length(X) -> length(X)
r29: a__and(X1,X2) -> and(X1,X2)
r30: a__isNat(X) -> isNat(X)
r31: a__isNatList(X) -> isNatList(X)
r32: a__isNatIList(X) -> isNatIList(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__U11#_A(x1,x2) = x1 + x2 + (1,19)
            tt_A() = (10,43)
            a__length#_A(x1) = x1 + (3,47)
            mark_A(x1) = x1 + (8,14)
            cons_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (8,0)
            a__and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (8,5)
            a__isNatList_A(x1) = ((1,0),(0,0)) x1 + (3,46)
            isNat_A(x1) = x1 + (2,19)
            a__zeros_A() = (9,15)
            |0|_A() = (0,12)
            zeros_A() = (1,1)
            a__U11_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (9,16)
            s_A(x1) = x1
            a__length_A(x1) = ((1,0),(1,0)) x1 + (1,18)
            a__isNat_A(x1) = x1 + (10,32)
            length_A(x1) = ((1,0),(1,0)) x1 + (1,13)
            a__isNatIList_A(x1) = x1 + (10,47)
            isNatIList_A(x1) = ((1,0),(1,1)) x1 + (3,1)
            nil_A() = (11,44)
            U11_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (9,15)
            isNatList_A(x1) = ((1,0),(0,0)) x1 + (2,1)
            and_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (7,1)
    
      precedence:
      
        a__U11# = tt = a__length# = mark = cons = a__and = a__isNatList = isNat = a__zeros = |0| = zeros = a__U11 = s = a__length = a__isNat = length = a__isNatIList = isNatIList = nil = U11 = isNatList = and
      
      partial status:
    
        pi(a__U11#) = []
        pi(tt) = []
        pi(a__length#) = []
        pi(mark) = []
        pi(cons) = []
        pi(a__and) = []
        pi(a__isNatList) = []
        pi(isNat) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(s) = []
        pi(a__length) = []
        pi(a__isNat) = []
        pi(length) = []
        pi(a__isNatIList) = []
        pi(isNatIList) = []
        pi(nil) = []
        pi(U11) = []
        pi(isNatList) = []
        pi(and) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__U11#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (1,0)
            tt_A() = (2,1)
            a__length#_A(x1) = ((1,0),(1,1)) x1
            mark_A(x1) = x1
            cons_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (7,6)
            a__and_A(x1,x2) = x2 + (2,1)
            a__isNatList_A(x1) = (3,17)
            isNat_A(x1) = x1 + (10,14)
            a__zeros_A() = (2,12)
            |0|_A() = (1,1)
            zeros_A() = (3,1)
            a__U11_A(x1,x2) = (8,17)
            s_A(x1) = (5,1)
            a__length_A(x1) = (9,16)
            a__isNat_A(x1) = (4,7)
            length_A(x1) = (8,1)
            a__isNatIList_A(x1) = ((1,0),(0,0)) x1 + (5,8)
            isNatIList_A(x1) = x1 + (4,9)
            nil_A() = (0,1)
            U11_A(x1,x2) = (7,1)
            isNatList_A(x1) = (4,7)
            and_A(x1,x2) = (1,2)
    
      precedence:
      
        a__isNat > a__isNatIList > a__and > a__zeros > nil > a__length > isNat > length > a__U11# = a__length# = a__isNatList = isNatList > tt = mark = cons = |0| = zeros = a__U11 = s = isNatIList = U11 = and
      
      partial status:
    
        pi(a__U11#) = [2]
        pi(tt) = []
        pi(a__length#) = [1]
        pi(mark) = []
        pi(cons) = []
        pi(a__and) = [2]
        pi(a__isNatList) = []
        pi(isNat) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(s) = []
        pi(a__length) = []
        pi(a__isNat) = []
        pi(length) = []
        pi(a__isNatIList) = []
        pi(isNatIList) = []
        pi(nil) = []
        pi(U11) = []
        pi(isNatList) = []
        pi(and) = []
    

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__U11#(tt(),L) -> a__length#(mark(L))

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),L) -> s(a__length(mark(L)))
r3: a__and(tt(),X) -> mark(X)
r4: a__isNat(|0|()) -> tt()
r5: a__isNat(length(V1)) -> a__isNatList(V1)
r6: a__isNat(s(V1)) -> a__isNat(V1)
r7: a__isNatIList(V) -> a__isNatList(V)
r8: a__isNatIList(zeros()) -> tt()
r9: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
r10: a__isNatList(nil()) -> tt()
r11: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
r12: a__length(nil()) -> |0|()
r13: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
r14: mark(zeros()) -> a__zeros()
r15: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r16: mark(length(X)) -> a__length(mark(X))
r17: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r18: mark(isNat(X)) -> a__isNat(X)
r19: mark(isNatList(X)) -> a__isNatList(X)
r20: mark(isNatIList(X)) -> a__isNatIList(X)
r21: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r22: mark(|0|()) -> |0|()
r23: mark(tt()) -> tt()
r24: mark(s(X)) -> s(mark(X))
r25: mark(nil()) -> nil()
r26: a__zeros() -> zeros()
r27: a__U11(X1,X2) -> U11(X1,X2)
r28: a__length(X) -> length(X)
r29: a__and(X1,X2) -> and(X1,X2)
r30: a__isNat(X) -> isNat(X)
r31: a__isNatList(X) -> isNatList(X)
r32: a__isNatIList(X) -> isNatIList(X)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p2: a__isNat#(length(V1)) -> a__isNatList#(V1)
p3: a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2))
p4: a__and#(tt(),X) -> mark#(X)
p5: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p6: mark#(and(X1,X2)) -> mark#(X1)
p7: mark#(isNat(X)) -> a__isNat#(X)
p8: a__isNat#(s(V1)) -> a__isNat#(V1)
p9: mark#(isNatList(X)) -> a__isNatList#(X)
p10: mark#(cons(X1,X2)) -> mark#(X1)
p11: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),L) -> s(a__length(mark(L)))
r3: a__and(tt(),X) -> mark(X)
r4: a__isNat(|0|()) -> tt()
r5: a__isNat(length(V1)) -> a__isNatList(V1)
r6: a__isNat(s(V1)) -> a__isNat(V1)
r7: a__isNatIList(V) -> a__isNatList(V)
r8: a__isNatIList(zeros()) -> tt()
r9: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
r10: a__isNatList(nil()) -> tt()
r11: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
r12: a__length(nil()) -> |0|()
r13: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
r14: mark(zeros()) -> a__zeros()
r15: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r16: mark(length(X)) -> a__length(mark(X))
r17: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r18: mark(isNat(X)) -> a__isNat(X)
r19: mark(isNatList(X)) -> a__isNatList(X)
r20: mark(isNatIList(X)) -> a__isNatIList(X)
r21: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r22: mark(|0|()) -> |0|()
r23: mark(tt()) -> tt()
r24: mark(s(X)) -> s(mark(X))
r25: mark(nil()) -> nil()
r26: a__zeros() -> zeros()
r27: a__U11(X1,X2) -> U11(X1,X2)
r28: a__length(X) -> length(X)
r29: a__and(X1,X2) -> and(X1,X2)
r30: a__isNat(X) -> isNat(X)
r31: a__isNatList(X) -> isNatList(X)
r32: a__isNatIList(X) -> isNatIList(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__isNatList#_A(x1) = ((1,0),(0,0)) x1 + (5,5)
            cons_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (12,8)
            a__isNat#_A(x1) = ((1,0),(0,0)) x1 + (0,5)
            length_A(x1) = x1 + (6,0)
            a__and#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (2,3)
            a__isNat_A(x1) = ((1,0),(0,0)) x1 + (12,4)
            isNatList_A(x1) = ((1,0),(1,1)) x1 + (2,1)
            tt_A() = (12,4)
            mark#_A(x1) = ((1,0),(0,0)) x1 + (13,7)
            and_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (10,2)
            mark_A(x1) = ((1,0),(1,1)) x1 + (12,0)
            isNat_A(x1) = ((1,0),(1,1)) x1 + (0,6)
            s_A(x1) = ((1,0),(0,0)) x1 + (0,6)
            a__zeros_A() = (25,14)
            |0|_A() = (0,15)
            zeros_A() = (13,5)
            a__U11_A(x1,x2) = x2 + (18,7)
            a__length_A(x1) = x1 + (6,0)
            a__and_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (10,12)
            a__isNatIList_A(x1) = x1 + (24,6)
            a__isNatList_A(x1) = x1 + (13,4)
            isNatIList_A(x1) = x1 + (13,1)
            nil_A() = (0,16)
            U11_A(x1,x2) = ((1,0),(1,1)) x2 + (13,8)
    
      precedence:
      
        a__isNat = and = mark = |0| = a__and = a__isNatList > a__isNatIList > cons = a__isNat# = length = a__and# = tt = mark# = s = a__zeros = zeros = a__U11 = a__length = isNatIList = nil = U11 > a__isNatList# = isNatList = isNat
      
      partial status:
    
        pi(a__isNatList#) = []
        pi(cons) = []
        pi(a__isNat#) = []
        pi(length) = []
        pi(a__and#) = [2]
        pi(a__isNat) = []
        pi(isNatList) = [1]
        pi(tt) = []
        pi(mark#) = []
        pi(and) = []
        pi(mark) = []
        pi(isNat) = []
        pi(s) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__length) = []
        pi(a__and) = []
        pi(a__isNatIList) = []
        pi(a__isNatList) = []
        pi(isNatIList) = []
        pi(nil) = []
        pi(U11) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__isNatList#_A(x1) = (3,0)
            cons_A(x1,x2) = (0,0)
            a__isNat#_A(x1) = (0,0)
            length_A(x1) = (0,3)
            a__and#_A(x1,x2) = x2 + (1,1)
            a__isNat_A(x1) = (0,3)
            isNatList_A(x1) = (1,3)
            tt_A() = (0,0)
            mark#_A(x1) = (0,0)
            and_A(x1,x2) = (0,0)
            mark_A(x1) = (0,5)
            isNat_A(x1) = (0,5)
            s_A(x1) = (0,1)
            a__zeros_A() = (0,2)
            |0|_A() = (0,1)
            zeros_A() = (1,3)
            a__U11_A(x1,x2) = (0,2)
            a__length_A(x1) = (0,4)
            a__and_A(x1,x2) = (0,1)
            a__isNatIList_A(x1) = (1,4)
            a__isNatList_A(x1) = (0,2)
            isNatIList_A(x1) = (0,0)
            nil_A() = (1,6)
            U11_A(x1,x2) = (0,0)
    
      precedence:
      
        s > |0| = a__U11 = nil > a__zeros > zeros > isNat = U11 > tt = mark > a__isNat = a__length = a__isNatList > cons = a__and > and > length > a__isNatList# = a__isNat# = a__and# = isNatList = mark# = a__isNatIList = isNatIList
      
      partial status:
    
        pi(a__isNatList#) = []
        pi(cons) = []
        pi(a__isNat#) = []
        pi(length) = []
        pi(a__and#) = [2]
        pi(a__isNat) = []
        pi(isNatList) = []
        pi(tt) = []
        pi(mark#) = []
        pi(and) = []
        pi(mark) = []
        pi(isNat) = []
        pi(s) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__length) = []
        pi(a__and) = []
        pi(a__isNatIList) = []
        pi(a__isNatList) = []
        pi(isNatIList) = []
        pi(nil) = []
        pi(U11) = []
    

The next rules are strictly ordered:

  p2, p4, p5, p9

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p2: a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2))
p3: mark#(and(X1,X2)) -> mark#(X1)
p4: mark#(isNat(X)) -> a__isNat#(X)
p5: a__isNat#(s(V1)) -> a__isNat#(V1)
p6: mark#(cons(X1,X2)) -> mark#(X1)
p7: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),L) -> s(a__length(mark(L)))
r3: a__and(tt(),X) -> mark(X)
r4: a__isNat(|0|()) -> tt()
r5: a__isNat(length(V1)) -> a__isNatList(V1)
r6: a__isNat(s(V1)) -> a__isNat(V1)
r7: a__isNatIList(V) -> a__isNatList(V)
r8: a__isNatIList(zeros()) -> tt()
r9: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
r10: a__isNatList(nil()) -> tt()
r11: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
r12: a__length(nil()) -> |0|()
r13: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
r14: mark(zeros()) -> a__zeros()
r15: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r16: mark(length(X)) -> a__length(mark(X))
r17: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r18: mark(isNat(X)) -> a__isNat(X)
r19: mark(isNatList(X)) -> a__isNatList(X)
r20: mark(isNatIList(X)) -> a__isNatIList(X)
r21: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r22: mark(|0|()) -> |0|()
r23: mark(tt()) -> tt()
r24: mark(s(X)) -> s(mark(X))
r25: mark(nil()) -> nil()
r26: a__zeros() -> zeros()
r27: a__U11(X1,X2) -> U11(X1,X2)
r28: a__length(X) -> length(X)
r29: a__and(X1,X2) -> and(X1,X2)
r30: a__isNat(X) -> isNat(X)
r31: a__isNatList(X) -> isNatList(X)
r32: a__isNatIList(X) -> isNatIList(X)

The estimated dependency graph contains the following SCCs:

  {p3, p6, p7}
  {p5}


-- Reduction pair.

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

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

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),L) -> s(a__length(mark(L)))
r3: a__and(tt(),X) -> mark(X)
r4: a__isNat(|0|()) -> tt()
r5: a__isNat(length(V1)) -> a__isNatList(V1)
r6: a__isNat(s(V1)) -> a__isNat(V1)
r7: a__isNatIList(V) -> a__isNatList(V)
r8: a__isNatIList(zeros()) -> tt()
r9: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
r10: a__isNatList(nil()) -> tt()
r11: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
r12: a__length(nil()) -> |0|()
r13: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
r14: mark(zeros()) -> a__zeros()
r15: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r16: mark(length(X)) -> a__length(mark(X))
r17: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r18: mark(isNat(X)) -> a__isNat(X)
r19: mark(isNatList(X)) -> a__isNatList(X)
r20: mark(isNatIList(X)) -> a__isNatIList(X)
r21: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r22: mark(|0|()) -> |0|()
r23: mark(tt()) -> tt()
r24: mark(s(X)) -> s(mark(X))
r25: mark(nil()) -> nil()
r26: a__zeros() -> zeros()
r27: a__U11(X1,X2) -> U11(X1,X2)
r28: a__length(X) -> length(X)
r29: a__and(X1,X2) -> and(X1,X2)
r30: a__isNat(X) -> isNat(X)
r31: a__isNatList(X) -> isNatList(X)
r32: a__isNatIList(X) -> isNatIList(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = x1 + (1,1)
            s_A(x1) = x1 + (2,1)
            cons_A(x1,x2) = x1 + x2 + (2,2)
            and_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,2)
    
      precedence:
      
        mark# = s = cons = and
      
      partial status:
    
        pi(mark#) = [1]
        pi(s) = [1]
        pi(cons) = [1, 2]
        pi(and) = [1, 2]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = x1 + (1,1)
            s_A(x1) = x1 + (2,2)
            cons_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (2,2)
            and_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,2)
    
      precedence:
      
        s > mark# = cons > and
      
      partial status:
    
        pi(mark#) = [1]
        pi(s) = []
        pi(cons) = []
        pi(and) = [1, 2]
    

The next rules are strictly ordered:

  p1, p2, p3

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(V1)) -> a__isNat#(V1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),L) -> s(a__length(mark(L)))
r3: a__and(tt(),X) -> mark(X)
r4: a__isNat(|0|()) -> tt()
r5: a__isNat(length(V1)) -> a__isNatList(V1)
r6: a__isNat(s(V1)) -> a__isNat(V1)
r7: a__isNatIList(V) -> a__isNatList(V)
r8: a__isNatIList(zeros()) -> tt()
r9: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
r10: a__isNatList(nil()) -> tt()
r11: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
r12: a__length(nil()) -> |0|()
r13: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
r14: mark(zeros()) -> a__zeros()
r15: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r16: mark(length(X)) -> a__length(mark(X))
r17: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r18: mark(isNat(X)) -> a__isNat(X)
r19: mark(isNatList(X)) -> a__isNatList(X)
r20: mark(isNatIList(X)) -> a__isNatIList(X)
r21: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r22: mark(|0|()) -> |0|()
r23: mark(tt()) -> tt()
r24: mark(s(X)) -> s(mark(X))
r25: mark(nil()) -> nil()
r26: a__zeros() -> zeros()
r27: a__U11(X1,X2) -> U11(X1,X2)
r28: a__length(X) -> length(X)
r29: a__and(X1,X2) -> and(X1,X2)
r30: a__isNat(X) -> isNat(X)
r31: a__isNatList(X) -> isNatList(X)
r32: a__isNatIList(X) -> isNatIList(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__isNat#_A(x1) = ((1,0),(1,1)) x1 + (2,2)
            s_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        a__isNat# = s
      
      partial status:
    
        pi(a__isNat#) = []
        pi(s) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__isNat#_A(x1) = ((1,0),(0,0)) x1
            s_A(x1) = (1,1)
    
      precedence:
      
        s > a__isNat#
      
      partial status:
    
        pi(a__isNat#) = []
        pi(s) = []
    

The next rules are strictly ordered:

  p1

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