(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, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
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, N) → ACTIVATE(N)
U411(tt, M, N) → U421(isNat(activate(N)), activate(M), activate(N))
U411(tt, M, N) → ISNAT(activate(N))
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
U421(tt, M, N) → S(plus(activate(N), activate(M)))
U421(tt, M, N) → PLUS(activate(N), activate(M))
U421(tt, M, N) → ACTIVATE(N)
U421(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)
PLUS(N, 0) → U311(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U411(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 01
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__s(X)) → S(X)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
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 5 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) → U311(isNat(N), N)
U311(tt, N) → ACTIVATE(N)
PLUS(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)
PLUS(N, s(M)) → U411(isNat(M), M, N)
U411(tt, M, N) → U421(isNat(activate(N)), activate(M), activate(N))
U421(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U421(tt, M, N) → ACTIVATE(N)
U421(tt, M, N) → ACTIVATE(M)
U411(tt, M, N) → ISNAT(activate(N))
U411(tt, M, N) → ACTIVATE(N)
U411(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, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
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) → U311(isNat(N), N)
U311(tt, N) → ACTIVATE(N)
PLUS(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)
PLUS(N, s(M)) → ISNAT(M)
U421(tt, M, N) → ACTIVATE(N)
U421(tt, M, N) → ACTIVATE(M)
U411(tt, M, N) → ISNAT(activate(N))
U411(tt, M, N) → ACTIVATE(N)
U411(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(x2)
ISNAT(x0, x1)  =  ISNAT(x0, x1)
ACTIVATE(x0, x1)  =  ACTIVATE(x0)
PLUS(x0, x1, x2)  =  PLUS(x0, x1, x2)
U311(x0, x1, x2)  =  U311(x2)
U411(x0, x1, x2, x3)  =  U411(x0, x3)
U421(x0, x1, x2, x3)  =  U421(x2, x3)

Tags:
U111 has argument tags [27,5,19] and root tag 4
ISNAT has argument tags [6,19] and root tag 0
ACTIVATE has argument tags [19,6] and root tag 0
PLUS has argument tags [8,20,19] and root tag 4
U311 has argument tags [16,12,19] and root tag 1
U411 has argument tags [19,6,24,20] and root tag 4
U421 has argument tags [21,7,19,20] and root tag 4

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)  =  U111
tt  =  tt
ISNAT(x1)  =  ISNAT
activate(x1)  =  x1
n__plus(x1, x2)  =  n__plus(x1, x2)
isNat(x1)  =  isNat
ACTIVATE(x1)  =  x1
PLUS(x1, x2)  =  x1
0  =  0
U311(x1, x2)  =  U311(x1, x2)
n__s(x1)  =  x1
s(x1)  =  x1
U411(x1, x2, x3)  =  x2
U421(x1, x2, x3)  =  x3
n__0  =  n__0
plus(x1, x2)  =  plus(x1, x2)
U31(x1, x2)  =  x2
U11(x1, x2)  =  U11
U21(x1)  =  x1
U41(x1, x2, x3)  =  U41(x2, x3)
U12(x1)  =  U12
U42(x1, x2, x3)  =  U42(x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:
[nplus2, plus2, U412, U422] > U11 > U12 > [U11^1, tt, ISNAT, 0, n0]
isNat > U11 > U12 > [U11^1, tt, ISNAT, 0, n0]
U31^12 > [U11^1, tt, ISNAT, 0, n0]

Status:
U11^1: []
tt: multiset
ISNAT: multiset
nplus2: multiset
isNat: []
0: multiset
U31^12: multiset
n0: multiset
plus2: multiset
U11: multiset
U412: multiset
U12: multiset
U422: multiset


The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(isNat(N), N)
U31(tt, N) → 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)))
plus(N, s(M)) → U41(isNat(M), M, N)
U11(tt, V2) → U12(isNat(activate(V2)))
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
U21(tt) → tt
U12(tt) → tt
s(X) → n__s(X)
0n__0

(6) Obligation:

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, s(M)) → U411(isNat(M), M, N)
U411(tt, M, N) → U421(isNat(activate(N)), activate(M), activate(N))
U421(tt, M, N) → PLUS(activate(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, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(8) Complex Obligation (AND)

(9) Obligation:

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

U411(tt, M, N) → U421(isNat(activate(N)), activate(M), activate(N))
U421(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U411(isNat(M), M, N)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

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.


U421(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U411(isNat(M), M, N)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U411(x0, x1, x2, x3)  =  U411(x0, x1, x3)
U421(x0, x1, x2, x3)  =  U421(x0, x1, x2, x3)
PLUS(x0, x1, x2)  =  PLUS(x0, x2)

Tags:
U411 has argument tags [10,1,0,0] and root tag 3
U421 has argument tags [0,1,12,1] and root tag 3
PLUS has argument tags [0,1,13] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U411(x1, x2, x3)  =  U411(x2, x3)
tt  =  tt
U421(x1, x2, x3)  =  U421(x2, x3)
isNat(x1)  =  isNat
activate(x1)  =  x1
PLUS(x1, x2)  =  PLUS(x1, x2)
s(x1)  =  s(x1)
n__0  =  n__0
0  =  0
n__plus(x1, x2)  =  n__plus(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
U31(x1, x2)  =  U31(x1, x2)
n__s(x1)  =  n__s(x1)
U11(x1, x2)  =  x1
U21(x1)  =  U21
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U12(x1)  =  x1
U42(x1, x2, x3)  =  U42(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:
[n0, 0] > [tt, isNat, s1, ns1, U21] > [U41^12, U42^12, PLUS2]
[n0, 0] > U312 > [U41^12, U42^12, PLUS2]
[nplus2, plus2, U413, U423] > [tt, isNat, s1, ns1, U21] > [U41^12, U42^12, PLUS2]
[nplus2, plus2, U413, U423] > U312 > [U41^12, U42^12, PLUS2]

Status:
U41^12: [1,2]
tt: multiset
U42^12: [1,2]
isNat: multiset
PLUS2: [2,1]
s1: multiset
n0: multiset
0: multiset
nplus2: [2,1]
plus2: [2,1]
U312: [1,2]
ns1: multiset
U21: multiset
U413: [2,3,1]
U423: [2,3,1]


The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(isNat(N), N)
U31(tt, N) → 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)))
plus(N, s(M)) → U41(isNat(M), M, N)
U11(tt, V2) → U12(isNat(activate(V2)))
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
U21(tt) → tt
U12(tt) → tt
s(X) → n__s(X)
0n__0

(11) Obligation:

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

U411(tt, M, N) → U421(isNat(activate(N)), activate(M), activate(N))

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

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

(12) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(13) TRUE

(14) Obligation:

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

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.


ISNAT(n__s(V1)) → ISNAT(activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ISNAT(x0, x1)  =  ISNAT(x1)

Tags:
ISNAT has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ISNAT(x1)  =  ISNAT(x1)
n__s(x1)  =  n__s(x1)
activate(x1)  =  x1
n__0  =  n__0
0  =  0
n__plus(x1, x2)  =  n__plus(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
U31(x1, x2)  =  U31(x1, x2)
isNat(x1)  =  isNat
tt  =  tt
s(x1)  =  s(x1)
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U11(x1, x2)  =  U11
U21(x1)  =  x1
U42(x1, x2, x3)  =  U42(x1, x2, x3)
U12(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[nplus2, plus2, U413, U423] > [ns1, s1] > [n0, 0, isNat, tt, U11]
[nplus2, plus2, U413, U423] > U312

Status:
ISNAT1: multiset
ns1: [1]
n0: multiset
0: multiset
nplus2: multiset
plus2: multiset
U312: multiset
isNat: multiset
tt: multiset
s1: [1]
U413: multiset
U11: multiset
U423: multiset


The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(isNat(N), N)
U31(tt, N) → activate(N)
activate(n__s(X)) → s(X)
activate(X) → X
plus(N, s(M)) → U41(isNat(M), M, N)
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U11(tt, V2) → U12(isNat(activate(V2)))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
isNat(n__0) → tt
U21(tt) → tt
U12(tt) → tt
s(X) → n__s(X)
0n__0

(16) 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, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE