/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSToCSRProof [EQUIVALENT, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 61 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 10 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) CSRRRRProof [EQUIVALENT, 0 ms]
(10) CSR
(11) CSRInnermostProof [EQUIVALENT, 0 ms]
(12) CSR
(13) CSDependencyPairsProof [EQUIVALENT, 0 ms]
(14) QCSDP
(15) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(16) AND
    (17) QCSDP
        (18) QCSDPSubtermProof [EQUIVALENT, 9 ms]
        (19) QCSDP
        (20) PIsEmptyProof [EQUIVALENT, 0 ms]
        (21) YES
    (22) QCSDP
        (23) QCSUsableRulesProof [EQUIVALENT, 0 ms]
        (24) QCSDP
        (25) QCSDPMuMonotonicPoloProof [EQUIVALENT, 8 ms]
        (26) QCSDP
        (27) PIsEmptyProof [EQUIVALENT, 0 ms]
        (28) YES
    (29) QCSDP
        (30) QCSDPReductionPairProof [EQUIVALENT, 19 ms]
        (31) QCSDP
        (32) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
        (33) TRUE


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(isNatList(V1))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(V)) -> mark(isNatList(V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

----------------------------------------

(1) QTRSToCSRProof (EQUIVALENT)
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(isNatList(V1))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(V)) -> mark(isNatList(V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set
nil: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
----------------------------------------

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(length(V1)) -> isNatList(V1)
   isNat(s(V1)) -> isNat(V1)
   isNatIList(V) -> isNatList(V)
   isNatIList(zeros) -> tt
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(nil) -> tt
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set
nil: empty set

----------------------------------------

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(length(V1)) -> isNatList(V1)
   isNat(s(V1)) -> isNat(V1)
   isNatIList(V) -> isNatList(V)
   isNatIList(zeros) -> tt
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(nil) -> tt
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2)) = x_1 + 2*x_2
   POL(and(x_1, x_2)) = x_1 + 2*x_2
   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNat(x_1)) = x_1
   POL(isNatIList(x_1)) = 2*x_1
   POL(isNatList(x_1)) = 2*x_1
   POL(length(x_1)) = 2*x_1
   POL(nil) = 2
   POL(s(x_1)) = x_1
   POL(tt) = 0
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   isNatList(nil) -> tt
   length(nil) -> 0




----------------------------------------

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(length(V1)) -> isNatList(V1)
   isNat(s(V1)) -> isNat(V1)
   isNatIList(V) -> isNatList(V)
   isNatIList(zeros) -> tt
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set

----------------------------------------

(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(length(V1)) -> isNatList(V1)
   isNat(s(V1)) -> isNat(V1)
   isNatIList(V) -> isNatList(V)
   isNatIList(zeros) -> tt
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2)) = 1 + x_1 + x_2
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = x_1
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 1 + x_1
   POL(s(x_1)) = x_1
   POL(tt) = 0
   POL(zeros) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   isNatIList(zeros) -> tt




----------------------------------------

(6)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(length(V1)) -> isNatList(V1)
   isNat(s(V1)) -> isNat(V1)
   isNatIList(V) -> isNatList(V)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set

----------------------------------------

(7) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(length(V1)) -> isNatList(V1)
   isNat(s(V1)) -> isNat(V1)
   isNatIList(V) -> isNatList(V)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(and(x_1, x_2)) = 2*x_1 + x_2
   POL(cons(x_1, x_2)) = 2*x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 1 + 2*x_1
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 2*x_1
   POL(s(x_1)) = x_1
   POL(tt) = 0
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   isNatIList(V) -> isNatList(V)




----------------------------------------

(8)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(length(V1)) -> isNatList(V1)
   isNat(s(V1)) -> isNat(V1)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set

----------------------------------------

(9) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(length(V1)) -> isNatList(V1)
   isNat(s(V1)) -> isNat(V1)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(and(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNat(x_1)) = x_1
   POL(isNatIList(x_1)) = x_1
   POL(isNatList(x_1)) = x_1
   POL(length(x_1)) = 1 + 2*x_1
   POL(s(x_1)) = x_1
   POL(tt) = 0
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   isNat(length(V1)) -> isNatList(V1)




----------------------------------------

(10)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set

----------------------------------------

(11) CSRInnermostProof (EQUIVALENT)
The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
----------------------------------------

(12)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set


Innermost Strategy.

----------------------------------------

(13) CSDependencyPairsProof (EQUIVALENT)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
----------------------------------------

(14)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, length_1, LENGTH_1} are replacing on all positions.
For all symbols f in {cons_2, U11_2, and_2, U11'_2, AND_2} we have mu(f) = {1}.
The symbols in {isNat_1, isNatIList_1, isNatList_1, ISNAT_1, ISNATILIST_1, ISNATLIST_1, U_1} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:
   U11'(tt, L) -> LENGTH(L)
   ISNAT(s(V1)) -> ISNAT(V1)
   ISNATILIST(cons(V1, V2)) -> AND(isNat(V1), isNatIList(V2))
   ISNATILIST(cons(V1, V2)) -> ISNAT(V1)
   ISNATLIST(cons(V1, V2)) -> AND(isNat(V1), isNatList(V2))
   ISNATLIST(cons(V1, V2)) -> ISNAT(V1)
   LENGTH(cons(N, L)) -> U11'(and(isNatList(L), isNat(N)), L)
   LENGTH(cons(N, L)) -> AND(isNatList(L), isNat(N))
   LENGTH(cons(N, L)) -> ISNATLIST(L)

The collapsing dependency pairs are DP_c:
   U11'(tt, L) -> L
   AND(tt, X) -> X


The hidden terms of R are:

   zeros
   isNatIList(x0)
   isNatList(x0)
   isNat(x0)

Every hiding context is built from:none

Hence, the new unhiding pairs DP_u are :
   U11'(tt, L) -> U(L)
   AND(tt, X) -> U(X)
   U(zeros) -> ZEROS
   U(isNatIList(x0)) -> ISNATILIST(x0)
   U(isNatList(x0)) -> ISNATLIST(x0)
   U(isNat(x0)) -> ISNAT(x0)

The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The set Q consists of the following terms:

   zeros
   U11(tt, x0)
   and(tt, x0)
   isNat(0)
   isNat(s(x0))
   isNatIList(cons(x0, x1))
   isNatList(cons(x0, x1))
   length(cons(x0, x1))


----------------------------------------

(15) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 7 less nodes.

----------------------------------------

(16)
Complex Obligation (AND)

----------------------------------------

(17)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, length_1} are replacing on all positions.
For all symbols f in {cons_2, U11_2, and_2} we have mu(f) = {1}.
The symbols in {isNat_1, isNatIList_1, isNatList_1, ISNAT_1} are not replacing on any position.

The TRS P consists of the following rules:

   ISNAT(s(V1)) -> ISNAT(V1)

The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The set Q consists of the following terms:

   zeros
   U11(tt, x0)
   and(tt, x0)
   isNat(0)
   isNat(s(x0))
   isNatIList(cons(x0, x1))
   isNatList(cons(x0, x1))
   length(cons(x0, x1))


----------------------------------------

(18) QCSDPSubtermProof (EQUIVALENT)
We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.

   ISNAT(s(V1)) -> ISNAT(V1)
The remaining pairs can at least be oriented weakly.
none
Used ordering:  Combined order from the following AFS and order.
ISNAT(x1)  =  x1


Subterm Order

----------------------------------------

(19)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, length_1} are replacing on all positions.
For all symbols f in {cons_2, U11_2, and_2} we have mu(f) = {1}.
The symbols in {isNat_1, isNatIList_1, isNatList_1} are not replacing on any position.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The set Q consists of the following terms:

   zeros
   U11(tt, x0)
   and(tt, x0)
   isNat(0)
   isNat(s(x0))
   isNatIList(cons(x0, x1))
   isNatList(cons(x0, x1))
   length(cons(x0, x1))


----------------------------------------

(20) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
----------------------------------------

(21)
YES

----------------------------------------

(22)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, length_1} are replacing on all positions.
For all symbols f in {cons_2, U11_2, and_2, AND_2} we have mu(f) = {1}.
The symbols in {isNat_1, isNatIList_1, isNatList_1, ISNATILIST_1, U_1, ISNATLIST_1} are not replacing on any position.

The TRS P consists of the following rules:

   ISNATILIST(cons(V1, V2)) -> AND(isNat(V1), isNatIList(V2))
   AND(tt, X) -> U(X)
   U(isNatIList(x0)) -> ISNATILIST(x0)
   U(isNatList(x0)) -> ISNATLIST(x0)
   ISNATLIST(cons(V1, V2)) -> AND(isNat(V1), isNatList(V2))

The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The set Q consists of the following terms:

   zeros
   U11(tt, x0)
   and(tt, x0)
   isNat(0)
   isNat(s(x0))
   isNatIList(cons(x0, x1))
   isNatList(cons(x0, x1))
   length(cons(x0, x1))


----------------------------------------

(23) QCSUsableRulesProof (EQUIVALENT)
The following rules are not useable [DA_EMMES] and can be deleted:

   zeros -> cons(0, zeros)
   U11(tt, x0) -> s(length(x0))
   and(tt, x0) -> x0
   isNatIList(cons(x0, x1)) -> and(isNat(x0), isNatIList(x1))
   isNatList(cons(x0, x1)) -> and(isNat(x0), isNatList(x1))
   length(cons(x0, x1)) -> U11(and(isNatList(x1), isNat(x0)), x1)

----------------------------------------

(24)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, length_1} are replacing on all positions.
For all symbols f in {cons_2, AND_2, U11_2, and_2} we have mu(f) = {1}.
The symbols in {isNat_1, isNatIList_1, isNatList_1, ISNATILIST_1, U_1, ISNATLIST_1} are not replacing on any position.

The TRS P consists of the following rules:

   ISNATILIST(cons(V1, V2)) -> AND(isNat(V1), isNatIList(V2))
   AND(tt, X) -> U(X)
   U(isNatIList(x0)) -> ISNATILIST(x0)
   U(isNatList(x0)) -> ISNATLIST(x0)
   ISNATLIST(cons(V1, V2)) -> AND(isNat(V1), isNatList(V2))

The TRS R consists of the following rules:

   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)

The set Q consists of the following terms:

   zeros
   U11(tt, x0)
   and(tt, x0)
   isNat(0)
   isNat(s(x0))
   isNatIList(cons(x0, x1))
   isNatList(cons(x0, x1))
   length(cons(x0, x1))


----------------------------------------

(25) QCSDPMuMonotonicPoloProof (EQUIVALENT)
By using the following mu-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted.

Strictly oriented dependency pairs:

   ISNATILIST(cons(V1, V2)) -> AND(isNat(V1), isNatIList(V2))
   AND(tt, X) -> U(X)
   U(isNatIList(x0)) -> ISNATILIST(x0)
   U(isNatList(x0)) -> ISNATLIST(x0)
   ISNATLIST(cons(V1, V2)) -> AND(isNat(V1), isNatList(V2))

Strictly oriented rules of the TRS R:

   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)

Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(AND(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(ISNATILIST(x_1)) = 1 + 2*x_1
   POL(ISNATLIST(x_1)) = 1 + 2*x_1
   POL(U(x_1)) = 2 + 2*x_1
   POL(cons(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(isNat(x_1)) = 2*x_1
   POL(isNatIList(x_1)) = 2*x_1
   POL(isNatList(x_1)) = 2*x_1
   POL(s(x_1)) = 1 + 2*x_1
   POL(tt) = 1


----------------------------------------

(26)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, length_1} are replacing on all positions.
For all symbols f in {U11_2, and_2, cons_2} we have mu(f) = {1}.
The symbols in {isNat_1, isNatIList_1, isNatList_1} are not replacing on any position.

The TRS P consists of the following rules:
none

R is empty.
The set Q consists of the following terms:

   zeros
   U11(tt, x0)
   and(tt, x0)
   isNat(0)
   isNat(s(x0))
   isNatIList(cons(x0, x1))
   isNatList(cons(x0, x1))
   length(cons(x0, x1))


----------------------------------------

(27) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
----------------------------------------

(28)
YES

----------------------------------------

(29)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, length_1, LENGTH_1} are replacing on all positions.
For all symbols f in {cons_2, U11_2, and_2, U11'_2} we have mu(f) = {1}.
The symbols in {isNat_1, isNatIList_1, isNatList_1} are not replacing on any position.

The TRS P consists of the following rules:

   LENGTH(cons(N, L)) -> U11'(and(isNatList(L), isNat(N)), L)
   U11'(tt, L) -> LENGTH(L)

The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The set Q consists of the following terms:

   zeros
   U11(tt, x0)
   and(tt, x0)
   isNat(0)
   isNat(s(x0))
   isNatIList(cons(x0, x1))
   isNatList(cons(x0, x1))
   length(cons(x0, x1))


----------------------------------------

(30) QCSDPReductionPairProof (EQUIVALENT)
Using the order

Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( tt ) = 1
POL( isNatList_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, -1}
POL( isNat_1(x_1) ) = 1
POL( 0 ) = 0
POL( s_1(x_1) ) = x_1 + 2
POL( zeros ) = 0
POL( isNatIList_1(x_1) ) = 1
POL( LENGTH_1(x_1) ) = 0
POL( U11'_2(x_1, x_2) ) = max{0, 2x_1 - 1}


the following usable rules


   and(tt, X) -> X
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)
   zeros -> cons(0, zeros)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))


could all be oriented weakly.

Furthermore, the pairs


   U11'(tt, L) -> LENGTH(L)


could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].


----------------------------------------

(31)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, length_1, LENGTH_1} are replacing on all positions.
For all symbols f in {cons_2, U11_2, and_2, U11'_2} we have mu(f) = {1}.
The symbols in {isNat_1, isNatIList_1, isNatList_1} are not replacing on any position.

The TRS P consists of the following rules:

   LENGTH(cons(N, L)) -> U11'(and(isNatList(L), isNat(N)), L)

The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt, L) -> s(length(L))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(s(V1)) -> isNat(V1)
   isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
   isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
   length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)

The set Q consists of the following terms:

   zeros
   U11(tt, x0)
   and(tt, x0)
   isNat(0)
   isNat(s(x0))
   isNatIList(cons(x0, x1))
   isNatList(cons(x0, x1))
   length(cons(x0, x1))


----------------------------------------

(32) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.

----------------------------------------

(33)
TRUE