TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60001 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (548ms).
 | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (359ms), PolynomialLinearRange4iUR (timeout), DependencyGraph (252ms), PolynomialLinearRange8NegiUR (timeout), DependencyGraph (253ms), ReductionPairSAT (12309ms), DependencyGraph (257ms), SizeChangePrinciple (timeout)].

The following open problems remain:

Open Dependency Pair Problem 2

Dependency Pairs

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

Rewrite Rules

U11(tt, N) → activate(N)U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0U41(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)0 → n__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) → 0activate(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

Original Signature

Termination of terms over the following signature is verified: plus, n__isNat, n__plus, and, n__s, activate, isNat, 0, n__0, s, tt, U41, U11, U31, U21, n__x, x

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

U11(tt, N) → activate(N)U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0U41(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)0 → n__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) → 0activate(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

Original Signature

Termination of terms over the following signature is verified: plus, n__isNat, n__plus, and, n__s, activate, isNat, 0, n__0, s, tt, U41, U11, U31, U21, x, n__x

Strategy

The following SCCs where found

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