YES Termination w.r.t. Q proof of Transformed_CSR_04_MYNAT_nokinds_Z.ari

(0) Obligation:

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

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0
U41(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), n__isNat(N)), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
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__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
U11(x1, x2)  =  U11(x1, x2)
tt  =  tt
activate(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
U31(x1)  =  x1
0  =  0
U41(x1, x2, x3)  =  U41(x1, x2, x3)
x(x1, x2)  =  x(x1, x2)
and(x1, x2)  =  and(x1, x2)
isNat(x1)  =  isNat(x1)
n__0  =  n__0
n__plus(x1, x2)  =  n__plus(x1, x2)
n__isNat(x1)  =  n__isNat(x1)
n__s(x1)  =  n__s(x1)
n__x(x1, x2)  =  n__x(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:
[tt, 0, U413, x2, n0, nx2] > [U213, plus2, nplus2] > U112
[tt, 0, U413, x2, n0, nx2] > [U213, plus2, nplus2] > [s1, and2, isNat1, nisNat1, ns1]

Status:
U112: [2,1]
tt: multiset
U213: [2,3,1]
s1: multiset
plus2: [2,1]
0: multiset
U413: [2,3,1]
x2: [2,1]
and2: multiset
isNat1: multiset
n0: multiset
nplus2: [2,1]
nisNat1: multiset
ns1: multiset
nx2: [2,1]

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

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U41(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), n__isNat(N)), M, N)


(2) Obligation:

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

U31(tt) → 0
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
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__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Knuth-Bendix order [KBO] with precedence:
activate1 > plus2 > nplus2 > x2 > nx2 > tt > U311 > s1 > isNat1 > nisNat1 > ns1 > 0 > n0

and weight map:

tt=1
0=2
n__0=2
U31_1=1
isNat_1=1
n__isNat_1=1
s_1=1
n__s_1=1
activate_1=0
plus_2=0
n__plus_2=0
x_2=0
n__x_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U31(tt) → 0
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
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__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X


(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) YES