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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 PisEmptyProof (⇔)
↳8 TRUE

(0) Obligation:

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

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

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:

U111(tt, V2) → U121(isNat(activate(V2)))
U111(tt, V2) → ISNAT(activate(V2))
U111(tt, V2) → ACTIVATE(V2)
U311(tt, V2) → U321(isNat(activate(V2)))
U311(tt, V2) → ISNAT(activate(V2))
U311(tt, V2) → ACTIVATE(V2)
U411(tt, N) → ACTIVATE(N)
U511(tt, M, N) → U521(isNat(activate(N)), activate(M), activate(N))
U511(tt, M, N) → ISNAT(activate(N))
U511(tt, M, N) → ACTIVATE(N)
U511(tt, M, N) → ACTIVATE(M)
U521(tt, M, N) → S(plus(activate(N), activate(M)))
U521(tt, M, N) → PLUS(activate(N), activate(M))
U521(tt, M, N) → ACTIVATE(N)
U521(tt, M, N) → ACTIVATE(M)
U611(tt) → 01
U711(tt, M, N) → U721(isNat(activate(N)), activate(M), activate(N))
U711(tt, M, N) → ISNAT(activate(N))
U711(tt, M, N) → ACTIVATE(N)
U711(tt, M, N) → ACTIVATE(M)
U721(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U721(tt, M, N) → X(activate(N), activate(M))
U721(tt, M, N) → ACTIVATE(N)
U721(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → U111(isNat(activate(V1)), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U211(isNat(activate(V1)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U311(isNat(activate(V1)), activate(V2))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → U411(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U511(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
X(N, 0) → U611(isNat(N))
X(N, 0) → ISNAT(N)
X(N, s(M)) → U711(isNat(M), M, N)
X(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 01
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__x(X1, X2)) → X(X1, X2)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

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 1 SCC with 8 less nodes.

(4) Obligation:

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

U111(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U111(isNat(activate(V1)), activate(V2))
U111(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U411(isNat(N), N)
U411(tt, N) → ACTIVATE(N)
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
X(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U311(isNat(activate(V1)), activate(V2))
U311(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U311(tt, V2) → ACTIVATE(V2)
X(N, s(M)) → U711(isNat(M), M, N)
U711(tt, M, N) → U721(isNat(activate(N)), activate(M), activate(N))
U721(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U511(isNat(M), M, N)
U511(tt, M, N) → U521(isNat(activate(N)), activate(M), activate(N))
U521(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U521(tt, M, N) → ACTIVATE(N)
U521(tt, M, N) → ACTIVATE(M)
U511(tt, M, N) → ISNAT(activate(N))
U511(tt, M, N) → ACTIVATE(N)
U511(tt, M, N) → ACTIVATE(M)
U721(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)
U721(tt, M, N) → ACTIVATE(N)
U721(tt, M, N) → ACTIVATE(M)
U711(tt, M, N) → ISNAT(activate(N))
U711(tt, M, N) → ACTIVATE(N)
U711(tt, M, N) → ACTIVATE(M)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U111(isNat(activate(V1)), activate(V2))
U111(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U411(isNat(N), N)
U411(tt, N) → ACTIVATE(N)
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
X(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U311(isNat(activate(V1)), activate(V2))
U311(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U311(tt, V2) → ACTIVATE(V2)
X(N, s(M)) → U711(isNat(M), M, N)
U711(tt, M, N) → U721(isNat(activate(N)), activate(M), activate(N))
U721(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U511(isNat(M), M, N)
U511(tt, M, N) → U521(isNat(activate(N)), activate(M), activate(N))
U521(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U521(tt, M, N) → ACTIVATE(N)
U521(tt, M, N) → ACTIVATE(M)
U511(tt, M, N) → ISNAT(activate(N))
U511(tt, M, N) → ACTIVATE(N)
U511(tt, M, N) → ACTIVATE(M)
U721(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)
U721(tt, M, N) → ACTIVATE(N)
U721(tt, M, N) → ACTIVATE(M)
U711(tt, M, N) → ISNAT(activate(N))
U711(tt, M, N) → ACTIVATE(N)
U711(tt, M, N) → ACTIVATE(M)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U111(x0, x1, x2)  =  U111(x0)
ISNAT(x0, x1)  =  ISNAT(x0)
ACTIVATE(x0, x1)  =  ACTIVATE(x0, x1)
PLUS(x0, x1, x2)  =  PLUS(x0)
U411(x0, x1, x2)  =  U411(x0)
X(x0, x1, x2)  =  X(x0)
U311(x0, x1, x2)  =  U311(x0)
U711(x0, x1, x2, x3)  =  U711(x0)
U721(x0, x1, x2, x3)  =  U721(x0, x2)
U511(x0, x1, x2, x3)  =  U511(x0, x3)
U521(x0, x1, x2, x3)  =  U521(x0, x3)

Tags:
U111 has argument tags [16,47,2] and root tag 12
ISNAT has argument tags [16,16] and root tag 2
ACTIVATE has argument tags [8,16] and root tag 9
PLUS has argument tags [8,0,57] and root tag 0
U411 has argument tags [3,60,48] and root tag 6
X has argument tags [14,51,0] and root tag 12
U311 has argument tags [0,8,15] and root tag 0
U711 has argument tags [60,6,63,35] and root tag 8
U721 has argument tags [51,3,7,33] and root tag 0
U511 has argument tags [14,5,0,9] and root tag 0
U521 has argument tags [47,21,4,7] and root tag 3

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U111(x1, x2)  =  x2
tt  =  tt
ISNAT(x1)  =  x1
activate(x1)  =  x1
n__plus(x1, x2)  =  n__plus(x1, x2)
isNat(x1)  =  isNat
ACTIVATE(x1)  =  x1
PLUS(x1, x2)  =  PLUS(x1, x2)
0  =  0
U411(x1, x2)  =  U411(x1, x2)
n__x(x1, x2)  =  n__x(x1, x2)
X(x1, x2)  =  X(x1, x2)
n__s(x1)  =  n__s(x1)
U311(x1, x2)  =  U311(x2)
s(x1)  =  s(x1)
U711(x1, x2, x3)  =  U711(x1, x2, x3)
U721(x1, x2, x3)  =  U721(x1, x2, x3)
x(x1, x2)  =  x(x1, x2)
U511(x1, x2, x3)  =  U511(x1, x2, x3)
U521(x1, x2, x3)  =  U521(x2, x3)
n__0  =  n__0
plus(x1, x2)  =  plus(x1, x2)
U41(x1, x2)  =  U41(x1, x2)
U71(x1, x2, x3)  =  U71(x1, x2, x3)
U72(x1, x2, x3)  =  U72(x1, x2, x3)
U11(x1, x2)  =  x1
U21(x1)  =  U21
U31(x1, x2)  =  x1
U61(x1)  =  U61
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U12(x1)  =  x1
U32(x1)  =  U32
U52(x1, x2, x3)  =  U52(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:
[nx2, X2, U71^13, U72^13, x2, U713, U723] > [nplus2, plus2, U513, U523] > [ns1, s1] > [tt, isNat, 0, n0, U21, U61, U32] > [PLUS2, U41^12, U51^13, U52^12]
[nx2, X2, U71^13, U72^13, x2, U713, U723] > [nplus2, plus2, U513, U523] > [ns1, s1] > [tt, isNat, 0, n0, U21, U61, U32] > U412
[nx2, X2, U71^13, U72^13, x2, U713, U723] > U31^11

Status:
tt: multiset
nplus2: [1,2]
isNat: multiset
PLUS2: [1,2]
0: multiset
U41^12: [2,1]
nx2: multiset
X2: multiset
ns1: [1]
U31^11: multiset
s1: [1]
U71^13: multiset
U72^13: multiset
x2: multiset
U51^13: [3,2,1]
U52^12: [2,1]
n0: multiset
plus2: [1,2]
U412: [1,2]
U713: multiset
U723: multiset
U21: multiset
U61: multiset
U513: [3,2,1]
U32: multiset
U523: [3,2,1]


The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U41(isNat(N), N)
U41(tt, N) → activate(N)
activate(n__x(X1, X2)) → x(X1, X2)
x(N, s(M)) → U71(isNat(M), M, N)
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
activate(n__s(X)) → s(X)
activate(X) → X
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
x(N, 0) → U61(isNat(N))
x(X1, X2) → n__x(X1, X2)
plus(N, s(M)) → U51(isNat(M), M, N)
U11(tt, V2) → U12(isNat(activate(V2)))
U31(tt, V2) → U32(isNat(activate(V2)))
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
U21(tt) → tt
U32(tt) → tt
U12(tt) → tt
s(X) → n__s(X)
U61(tt) → 0
0 → n__0

(6) Obligation:

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

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

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

(7) PisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE