YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRSToCSRProof [SOUND, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 46 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 20 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 21 ms]
(8) CSR
(9) CSRRRRProof [EQUIVALENT, 0 ms]
(10) CSR
(11) CSRRRRProof [EQUIVALENT, 8 ms]
(12) CSR
(13) CSRRRRProof [EQUIVALENT, 0 ms]
(14) CSR
(15) CSRRRRProof [EQUIVALENT, 0 ms]
(16) CSR
(17) CSRRRRProof [EQUIVALENT, 0 ms]
(18) CSR
(19) CSRRRRProof [EQUIVALENT, 0 ms]
(20) CSR
(21) RisEmptyProof [EQUIVALENT, 0 ms]
(22) YES


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

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

   active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
   active(U12(tt, V2)) -> mark(U13(isNat(V2)))
   active(U13(tt)) -> mark(tt)
   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
   active(U22(tt)) -> mark(tt)
   active(U31(tt, N)) -> mark(N)
   active(U41(tt, M, N)) -> mark(s(plus(N, M)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
   active(plus(N, 0)) -> mark(U31(and(isNat(N), isNatKind(N)), N))
   active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(U13(X)) -> U13(active(X))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(isNatKind(X)) -> isNatKind(proper(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   U13(ok(X)) -> ok(U13(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNatKind(ok(X)) -> ok(isNatKind(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(U11(x0, x1, x2))
   active(U12(x0, x1))
   active(U13(x0))
   active(U21(x0, x1))
   active(U22(x0))
   active(U31(x0, x1))
   active(U41(x0, x1, x2))
   active(s(x0))
   active(plus(x0, x1))
   active(and(x0, x1))
   U11(mark(x0), x1, x2)
   U12(mark(x0), x1)
   U13(mark(x0))
   U21(mark(x0), x1)
   U22(mark(x0))
   U31(mark(x0), x1)
   U41(mark(x0), x1, x2)
   s(mark(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   and(mark(x0), x1)
   proper(U11(x0, x1, x2))
   proper(tt)
   proper(U12(x0, x1))
   proper(isNat(x0))
   proper(U13(x0))
   proper(U21(x0, x1))
   proper(U22(x0))
   proper(U31(x0, x1))
   proper(U41(x0, x1, x2))
   proper(s(x0))
   proper(plus(x0, x1))
   proper(and(x0, x1))
   proper(0)
   proper(isNatKind(x0))
   U11(ok(x0), ok(x1), ok(x2))
   U12(ok(x0), ok(x1))
   isNat(ok(x0))
   U13(ok(x0))
   U21(ok(x0), ok(x1))
   U22(ok(x0))
   U31(ok(x0), ok(x1))
   U41(ok(x0), ok(x1), ok(x2))
   s(ok(x0))
   plus(ok(x0), ok(x1))
   and(ok(x0), ok(x1))
   isNatKind(ok(x0))
   top(mark(x0))
   top(ok(x0))


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

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

   active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
   active(U12(tt, V2)) -> mark(U13(isNat(V2)))
   active(U13(tt)) -> mark(tt)
   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
   active(U22(tt)) -> mark(tt)
   active(U31(tt, N)) -> mark(N)
   active(U41(tt, M, N)) -> mark(s(plus(N, M)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
   active(plus(N, 0)) -> mark(U31(and(isNat(N), isNatKind(N)), N))
   active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(U13(X)) -> U13(active(X))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(isNatKind(X)) -> isNatKind(proper(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   U13(ok(X)) -> ok(U13(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNatKind(ok(X)) -> ok(isNatKind(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(U11(x0, x1, x2))
   active(U12(x0, x1))
   active(U13(x0))
   active(U21(x0, x1))
   active(U22(x0))
   active(U31(x0, x1))
   active(U41(x0, x1, x2))
   active(s(x0))
   active(plus(x0, x1))
   active(and(x0, x1))
   U11(mark(x0), x1, x2)
   U12(mark(x0), x1)
   U13(mark(x0))
   U21(mark(x0), x1)
   U22(mark(x0))
   U31(mark(x0), x1)
   U41(mark(x0), x1, x2)
   s(mark(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   and(mark(x0), x1)
   proper(U11(x0, x1, x2))
   proper(tt)
   proper(U12(x0, x1))
   proper(isNat(x0))
   proper(U13(x0))
   proper(U21(x0, x1))
   proper(U22(x0))
   proper(U31(x0, x1))
   proper(U41(x0, x1, x2))
   proper(s(x0))
   proper(plus(x0, x1))
   proper(and(x0, x1))
   proper(0)
   proper(isNatKind(x0))
   U11(ok(x0), ok(x1), ok(x2))
   U12(ok(x0), ok(x1))
   isNat(ok(x0))
   U13(ok(x0))
   U21(ok(x0), ok(x1))
   U22(ok(x0))
   U31(ok(x0), ok(x1))
   U41(ok(x0), ok(x1), ok(x2))
   s(ok(x0))
   plus(ok(x0), ok(x1))
   and(ok(x0), ok(x1))
   isNatKind(ok(x0))
   top(mark(x0))
   top(ok(x0))

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:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U31: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set
The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
----------------------------------------

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U13(tt) -> tt
   U21(tt, V1) -> U22(isNat(V1))
   U22(tt) -> tt
   U31(tt, N) -> N
   U41(tt, M, N) -> s(plus(N, M))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)
   plus(N, 0) -> U31(and(isNat(N), isNatKind(N)), N)
   plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U31: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set

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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U13(tt) -> tt
   U21(tt, V1) -> U22(isNat(V1))
   U22(tt) -> tt
   U31(tt, N) -> N
   U41(tt, M, N) -> s(plus(N, M))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)
   plus(N, 0) -> U31(and(isNat(N), isNatKind(N)), N)
   plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U31: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2, x_3)) = x_1
   POL(U12(x_1, x_2)) = x_1
   POL(U13(x_1)) = x_1
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1)) = x_1
   POL(U31(x_1, x_2)) = 1 + x_1 + x_2
   POL(U41(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
   POL(s(x_1)) = x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U31(tt, N) -> N




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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U13(tt) -> tt
   U21(tt, V1) -> U22(isNat(V1))
   U22(tt) -> tt
   U41(tt, M, N) -> s(plus(N, M))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)
   plus(N, 0) -> U31(and(isNat(N), isNatKind(N)), N)
   plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U31: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set

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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U13(tt) -> tt
   U21(tt, V1) -> U22(isNat(V1))
   U22(tt) -> tt
   U41(tt, M, N) -> s(plus(N, M))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)
   plus(N, 0) -> U31(and(isNat(N), isNatKind(N)), N)
   plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U31: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 2
   POL(U11(x_1, x_2, x_3)) = x_1
   POL(U12(x_1, x_2)) = 2*x_1
   POL(U13(x_1)) = 2*x_1
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1)) = x_1
   POL(U31(x_1, x_2)) = 2*x_1
   POL(U41(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3
   POL(and(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNat(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(plus(x_1, x_2)) = x_1 + 2*x_2
   POL(s(x_1)) = x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   plus(N, 0) -> U31(and(isNat(N), isNatKind(N)), N)




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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U13(tt) -> tt
   U21(tt, V1) -> U22(isNat(V1))
   U22(tt) -> tt
   U41(tt, M, N) -> s(plus(N, M))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)
   plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set

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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U13(tt) -> tt
   U21(tt, V1) -> U22(isNat(V1))
   U22(tt) -> tt
   U41(tt, M, N) -> s(plus(N, M))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)
   plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2, x_3)) = 2*x_1
   POL(U12(x_1, x_2)) = x_1
   POL(U13(x_1)) = x_1
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3
   POL(and(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNat(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(plus(x_1, x_2)) = x_1 + 2*x_2
   POL(s(x_1)) = 1 + x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)




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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U13(tt) -> tt
   U21(tt, V1) -> U22(isNat(V1))
   U22(tt) -> tt
   U41(tt, M, N) -> s(plus(N, M))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set

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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U13(tt) -> tt
   U21(tt, V1) -> U22(isNat(V1))
   U22(tt) -> tt
   U41(tt, M, N) -> s(plus(N, M))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2, x_3)) = 2*x_1
   POL(U12(x_1, x_2)) = 2*x_1
   POL(U13(x_1)) = 2*x_1
   POL(U21(x_1, x_2)) = 2*x_1
   POL(U22(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(and(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNat(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(plus(x_1, x_2)) = 2*x_1 + x_2
   POL(s(x_1)) = x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U41(tt, M, N) -> s(plus(N, M))




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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U13(tt) -> tt
   U21(tt, V1) -> U22(isNat(V1))
   U22(tt) -> tt
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set

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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U13(tt) -> tt
   U21(tt, V1) -> U22(isNat(V1))
   U22(tt) -> tt
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U12(x_1, x_2)) = 1 + x_1 + x_2
   POL(U13(x_1)) = 1 + x_1
   POL(U21(x_1, x_2)) = 1 + x_1 + x_2
   POL(U22(x_1)) = 1 + x_1
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = x_1
   POL(isNatKind(x_1)) = 0
   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
   POL(s(x_1)) = 1 + x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U13(tt) -> tt
   U22(tt) -> tt




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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U21(tt, V1) -> U22(isNat(V1))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set

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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U12(tt, V2) -> U13(isNat(V2))
   U21(tt, V1) -> U22(isNat(V1))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U12(x_1, x_2)) = 1 + x_1 + x_2
   POL(U13(x_1)) = x_1
   POL(U21(x_1, x_2)) = 1 + x_1 + x_2
   POL(U22(x_1)) = 1 + x_1
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = x_1
   POL(isNatKind(x_1)) = 0
   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
   POL(s(x_1)) = 1 + x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U12(tt, V2) -> U13(isNat(V2))




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

(14)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U21(tt, V1) -> U22(isNat(V1))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U22: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set

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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U21(tt, V1) -> U22(isNat(V1))
   and(tt, X) -> X
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U22: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2, x_3)) = x_1 + 2*x_2
   POL(U12(x_1, x_2)) = x_1
   POL(U21(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U22(x_1)) = 1 + x_1
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 2*x_1
   POL(isNatKind(x_1)) = 1 + 2*x_1
   POL(plus(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(s(x_1)) = 2 + 2*x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
   isNat(s(V1)) -> U21(isNatKind(V1), V1)
   isNatKind(0) -> tt
   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
   isNatKind(s(V1)) -> isNatKind(V1)




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

(16)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U21(tt, V1) -> U22(isNat(V1))
   and(tt, X) -> X
   isNat(0) -> tt

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U22: {1}
and: {1}
0: empty set

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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U21(tt, V1) -> U22(isNat(V1))
   and(tt, X) -> X
   isNat(0) -> tt

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U22: {1}
and: {1}
0: empty set
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 0], [1, 0]] 	* 	x_2 	 +  	[[1, 1], [1, 1]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[1], [1]]
>>>

   <<<
 POL(U12(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1 	 +  	[[1, 0], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(and(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 0], [0, 1]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[1], [1]]
>>>

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   and(tt, X) -> X
   isNat(0) -> tt




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

(18)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U21(tt, V1) -> U22(isNat(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U22: {1}

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

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

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U21(tt, V1) -> U22(isNat(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U22: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3
   POL(U12(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(U21(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(U22(x_1)) = 2*x_1
   POL(isNat(x_1)) = x_1
   POL(tt) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U11(tt, V1, V2) -> U12(isNat(V1), V2)
   U21(tt, V1) -> U22(isNat(V1))




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

(20)
Obligation:
Context-sensitive rewrite system:
R is empty.

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

(21) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
----------------------------------------

(22)
YES