(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U11(tt, V2) → A__U12(a__isNat(V2))
A__U11(tt, V2) → A__ISNAT(V2)
A__U31(tt, V2) → A__U32(a__isNat(V2))
A__U31(tt, V2) → A__ISNAT(V2)
A__U41(tt, N) → MARK(N)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U51(tt, M, N) → A__ISNAT(N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U52(tt, M, N) → MARK(N)
A__U52(tt, M, N) → MARK(M)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U71(tt, M, N) → A__ISNAT(N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__U72(tt, M, N) → MARK(N)
A__U72(tt, M, N) → MARK(M)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, 0) → A__U61(a__isNat(N))
A__X(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → A__U12(mark(X))
MARK(U12(X)) → MARK(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U21(X)) → A__U21(mark(X))
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → A__U61(mark(X))
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 17 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U11(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__U31(tt, V2) → A__ISNAT(V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__U31(tt, V2) → A__ISNAT(V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U11(x1, x2)  =  A__U11(x2)
tt  =  tt
A__ISNAT(x1)  =  A__ISNAT(x1)
plus(x1, x2)  =  plus(x1, x2)
a__isNat(x1)  =  a__isNat
s(x1)  =  x1
x(x1, x2)  =  x(x1, x2)
A__U31(x1, x2)  =  A__U31(x1, x2)
0  =  0
a__U11(x1, x2)  =  a__U11
a__U21(x1)  =  a__U21
a__U31(x1, x2)  =  a__U31
isNat(x1)  =  isNat
U11(x1, x2)  =  U11
U21(x1)  =  U21
U31(x1, x2)  =  U31
a__U32(x1)  =  a__U32
U32(x1)  =  U32
a__U12(x1)  =  a__U12
U12(x1)  =  U12

Recursive path order with status [RPO].
Quasi-Precedence:
plus2 > [AU111, tt, AISNAT1, aisNat, AU312, aU11, aU21, aU31, isNat, U11, U21, U31, aU32, U32, aU12, U12]
x2 > [AU111, tt, AISNAT1, aisNat, AU312, aU11, aU21, aU31, isNat, U11, U21, U31, aU32, U32, aU12, U12]
0 > [AU111, tt, AISNAT1, aisNat, AU312, aU11, aU21, aU31, isNat, U11, U21, U31, aU32, U32, aU12, U12]

Status:
AU111: multiset
tt: multiset
AISNAT1: multiset
plus2: multiset
aisNat: multiset
x2: multiset
AU312: multiset
0: multiset
aU11: multiset
aU21: multiset
aU31: multiset
isNat: multiset
U11: multiset
U21: multiset
U31: multiset
aU32: multiset
U32: multiset
aU12: multiset
U12: multiset


The following usable rules [FROCOS05] were oriented:

a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__isNat(X) → isNat(X)
a__U11(X1, X2) → U11(X1, X2)
a__U21(tt) → tt
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U12(X) → U12(X)

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U11(tt, V2) → A__ISNAT(V2)
A__ISNAT(s(V1)) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(9) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(s(V1)) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISNAT(s(V1)) → A__ISNAT(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__ISNAT(x1)  =  x1
s(x1)  =  s(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
s1: multiset


The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(13) TRUE

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(M)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__U72(tt, M, N) → MARK(N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
A__U72(tt, M, N) → MARK(M)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(M)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → MARK(X)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__U72(tt, M, N) → MARK(N)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__U72(tt, M, N) → MARK(M)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2)  =  x1
U12(x1)  =  x1
U21(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
A__U41(x1, x2)  =  x2
mark(x1)  =  x1
tt  =  tt
U51(x1, x2, x3)  =  U51(x1, x2, x3)
A__U51(x1, x2, x3)  =  A__U51(x2, x3)
A__U52(x1, x2, x3)  =  A__U52(x2, x3)
a__isNat(x1)  =  a__isNat
A__PLUS(x1, x2)  =  A__PLUS(x1, x2)
0  =  0
s(x1)  =  s(x1)
U52(x1, x2, x3)  =  U52(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
U61(x1)  =  U61(x1)
U71(x1, x2, x3)  =  U71(x1, x2, x3)
A__U71(x1, x2, x3)  =  A__U71(x1, x2, x3)
A__U72(x1, x2, x3)  =  A__U72(x1, x2, x3)
a__x(x1, x2)  =  a__x(x1, x2)
A__X(x1, x2)  =  A__X(x1, x2)
U72(x1, x2, x3)  =  U72(x1, x2, x3)
x(x1, x2)  =  x(x1, x2)
a__U11(x1, x2)  =  x1
a__U12(x1)  =  x1
isNat(x1)  =  isNat
a__U21(x1)  =  x1
a__U31(x1, x2)  =  x1
a__U32(x1)  =  x1
a__U41(x1, x2)  =  a__U41(x1, x2)
a__plus(x1, x2)  =  a__plus(x1, x2)
a__U71(x1, x2, x3)  =  a__U71(x1, x2, x3)
a__U72(x1, x2, x3)  =  a__U72(x1, x2, x3)
a__U51(x1, x2, x3)  =  a__U51(x1, x2, x3)
a__U52(x1, x2, x3)  =  a__U52(x1, x2, x3)
a__U61(x1)  =  a__U61(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U713, AU713, AU723, ax2, AX2, U723, x2, aU713, aU723] > [U513, AU512, AU522, APLUS2, U523, plus2, aplus2, aU513, aU523] > [tt, aisNat, s1, isNat] > [0, U611, aU611] > [U412, aU412]

Status:
U412: [2,1]
tt: multiset
U513: multiset
AU512: multiset
AU522: multiset
aisNat: []
APLUS2: multiset
0: multiset
s1: [1]
U523: multiset
plus2: multiset
U611: [1]
U713: [2,3,1]
AU713: [2,3,1]
AU723: [2,3,1]
ax2: [2,1]
AX2: [2,1]
U723: [2,3,1]
x2: [2,1]
isNat: []
aU412: [2,1]
aplus2: multiset
aU713: [2,3,1]
aU723: [2,3,1]
aU513: multiset
aU523: multiset
aU611: [1]


The following usable rules [FROCOS05] were oriented:

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
a__U41(tt, N) → mark(N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U41(a__isNat(N), N)
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__isNat(X) → isNat(X)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(X1, X2) → x(X1, X2)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(X1, X2) → plus(X1, X2)
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U11(X1, X2) → U11(X1, X2)
a__U12(tt) → tt
a__U12(X) → U12(X)
a__U21(tt) → tt
a__U21(X) → U21(X)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U61(tt) → 0
a__U61(X) → U61(X)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
A__U41(tt, N) → MARK(N)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U12(x1)  =  U12(x1)
U11(x1, x2)  =  U11(x1, x2)
U21(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U32(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[MARK1, U121, U112, U311]

Status:
MARK1: [1]
U121: multiset
U112: multiset
U311: multiset


The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U21(X)) → MARK(X)
MARK(U32(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U21(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U21(x1)  =  U21(x1)
U32(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
U211: multiset


The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U32(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U32(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U32(x1)  =  U32(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
U321: multiset


The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(25) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(26) TRUE