TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60012 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (1513ms).
 | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (853ms), PolynomialLinearRange4iUR (timeout), DependencyGraph (652ms), PolynomialLinearRange8NegiUR (timeout), DependencyGraph (657ms), ReductionPairSAT (timeout)].

The following open problems remain:

Open Dependency Pair Problem 2

Dependency Pairs

take#(s(M), cons(N, IL)) → isNatIList#(activate(IL))isNatIList#(n__cons(V1, V2)) → activate#(V2)
and#(tt, X) → activate#(X)activate#(n__isNatIList(X)) → isNatIList#(X)
isNatList#(n__cons(V1, V2)) → and#(isNat(activate(V1)), n__isNatList(activate(V2)))isNatList#(n__take(V1, V2)) → activate#(V1)
isNatIList#(n__cons(V1, V2)) → and#(isNat(activate(V1)), n__isNatIList(activate(V2)))length#(cons(N, L)) → and#(isNatList(activate(L)), n__isNat(N))
isNatList#(n__cons(V1, V2)) → isNat#(activate(V1))take#(s(M), cons(N, IL)) → and#(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
U31#(tt, IL, M, N) → activate#(N)U31#(tt, IL, M, N) → activate#(M)
U31#(tt, IL, M, N) → activate#(IL)activate#(n__isNat(X)) → isNat#(X)
isNatIList#(V) → activate#(V)isNat#(n__length(V1)) → activate#(V1)
isNat#(n__s(V1)) → activate#(V1)isNatList#(n__take(V1, V2)) → and#(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatIList#(V) → isNatList#(activate(V))isNatList#(n__cons(V1, V2)) → activate#(V2)
isNatList#(n__take(V1, V2)) → isNat#(activate(V1))U11#(tt, L) → activate#(L)
take#(s(M), cons(N, IL)) → U31#(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)activate#(n__isNatList(X)) → isNatList#(X)
length#(cons(N, L)) → isNatList#(activate(L))activate#(n__length(X)) → length#(X)
isNat#(n__length(V1)) → isNatList#(activate(V1))isNatList#(n__take(V1, V2)) → activate#(V2)
isNat#(n__s(V1)) → isNat#(activate(V1))isNatList#(n__cons(V1, V2)) → activate#(V1)
activate#(n__take(X1, X2)) → take#(X1, X2)isNatIList#(n__cons(V1, V2)) → isNat#(activate(V1))
take#(s(M), cons(N, IL)) → activate#(IL)take#(0, IL) → isNatIList#(IL)
U11#(tt, L) → length#(activate(L))activate#(n__and(X1, X2)) → and#(X1, X2)
isNatIList#(n__cons(V1, V2)) → activate#(V1)take#(s(M), cons(N, IL)) → isNat#(M)
length#(cons(N, L)) → U11#(and(isNatList(activate(L)), n__isNat(N)), activate(L))length#(cons(N, L)) → activate#(L)

Rewrite Rules

zeros → cons(0, n__zeros)U11(tt, L) → s(length(activate(L)))
U21(tt) → nilU31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zerostake(X1, X2) → n__take(X1, X2)
0 → n__0length(X) → n__length(X)
s(X) → n__s(X)cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)nil → n__nil
isNatList(X) → n__isNatList(X)isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)activate(n__0) → 0
activate(n__length(X)) → length(X)activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nilactivate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Original Signature

Termination of terms over the following signature is verified: n__isNatIList, activate, isNat, n__s, n__0, length, n__nil, n__zeros, U21, cons, isNatIList, n__length, n__take, n__isNat, and, n__isNatList, n__and, n__cons, 0, s, isNatList, tt, zeros, take, U11, U31, nil

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

take#(s(M), cons(N, IL)) → isNatIList#(activate(IL))and#(tt, X) → activate#(X)
isNatList#(n__cons(V1, V2)) → and#(isNat(activate(V1)), n__isNatList(activate(V2)))zeros# → 0#
isNatList#(n__take(V1, V2)) → activate#(V1)isNatList#(n__cons(V1, V2)) → isNat#(activate(V1))
U31#(tt, IL, M, N) → activate#(N)U31#(tt, IL, M, N) → activate#(M)
U31#(tt, IL, M, N) → activate#(IL)activate#(n__isNat(X)) → isNat#(X)
isNat#(n__s(V1)) → activate#(V1)isNat#(n__length(V1)) → activate#(V1)
length#(nil) → 0#isNatList#(n__take(V1, V2)) → and#(isNat(activate(V1)), n__isNatIList(activate(V2)))
U31#(tt, IL, M, N) → cons#(activate(N), n__take(activate(M), activate(IL)))isNatList#(n__cons(V1, V2)) → activate#(V2)
isNatList#(n__take(V1, V2)) → isNat#(activate(V1))U11#(tt, L) → activate#(L)
take#(s(M), cons(N, IL)) → U31#(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)length#(cons(N, L)) → isNatList#(activate(L))
isNatList#(n__take(V1, V2)) → activate#(V2)isNat#(n__length(V1)) → isNatList#(activate(V1))
activate#(n__0) → 0#take#(s(M), cons(N, IL)) → activate#(IL)
isNatIList#(n__cons(V1, V2)) → isNat#(activate(V1))U11#(tt, L) → length#(activate(L))
length#(cons(N, L)) → U11#(and(isNatList(activate(L)), n__isNat(N)), activate(L))isNatIList#(n__cons(V1, V2)) → activate#(V2)
activate#(n__zeros) → zeros#activate#(n__isNatIList(X)) → isNatIList#(X)
isNatIList#(n__cons(V1, V2)) → and#(isNat(activate(V1)), n__isNatIList(activate(V2)))length#(cons(N, L)) → and#(isNatList(activate(L)), n__isNat(N))
take#(s(M), cons(N, IL)) → and#(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))take#(0, IL) → U21#(isNatIList(IL))
U21#(tt) → nil#U11#(tt, L) → s#(length(activate(L)))
isNatIList#(V) → activate#(V)activate#(n__cons(X1, X2)) → cons#(X1, X2)
isNatIList#(V) → isNatList#(activate(V))zeros# → cons#(0, n__zeros)
activate#(n__isNatList(X)) → isNatList#(X)activate#(n__s(X)) → s#(X)
activate#(n__length(X)) → length#(X)activate#(n__nil) → nil#
isNat#(n__s(V1)) → isNat#(activate(V1))isNatList#(n__cons(V1, V2)) → activate#(V1)
activate#(n__take(X1, X2)) → take#(X1, X2)take#(0, IL) → isNatIList#(IL)
isNatIList#(n__cons(V1, V2)) → activate#(V1)activate#(n__and(X1, X2)) → and#(X1, X2)
take#(s(M), cons(N, IL)) → isNat#(M)length#(cons(N, L)) → activate#(L)

Rewrite Rules

zeros → cons(0, n__zeros)U11(tt, L) → s(length(activate(L)))
U21(tt) → nilU31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zerostake(X1, X2) → n__take(X1, X2)
0 → n__0length(X) → n__length(X)
s(X) → n__s(X)cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)nil → n__nil
isNatList(X) → n__isNatList(X)isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)activate(n__0) → 0
activate(n__length(X)) → length(X)activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nilactivate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Original Signature

Termination of terms over the following signature is verified: n__isNatIList, n__s, isNat, activate, n__0, length, n__nil, n__zeros, U21, cons, isNatIList, n__length, n__take, n__isNat, and, n__isNatList, n__and, n__cons, 0, s, isNatList, tt, zeros, take, U11, U31, nil

Strategy

The following SCCs where found

take#(s(M), cons(N, IL)) → isNatIList#(activate(IL))and#(tt, X) → activate#(X)
isNatIList#(n__cons(V1, V2)) → activate#(V2)activate#(n__isNatIList(X)) → isNatIList#(X)
isNatList#(n__cons(V1, V2)) → and#(isNat(activate(V1)), n__isNatList(activate(V2)))isNatIList#(n__cons(V1, V2)) → and#(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList#(n__take(V1, V2)) → activate#(V1)length#(cons(N, L)) → and#(isNatList(activate(L)), n__isNat(N))
isNatList#(n__cons(V1, V2)) → isNat#(activate(V1))take#(s(M), cons(N, IL)) → and#(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
U31#(tt, IL, M, N) → activate#(N)U31#(tt, IL, M, N) → activate#(M)
U31#(tt, IL, M, N) → activate#(IL)activate#(n__isNat(X)) → isNat#(X)
isNatIList#(V) → activate#(V)isNat#(n__s(V1)) → activate#(V1)
isNat#(n__length(V1)) → activate#(V1)isNatIList#(V) → isNatList#(activate(V))
isNatList#(n__take(V1, V2)) → and#(isNat(activate(V1)), n__isNatIList(activate(V2)))isNatList#(n__cons(V1, V2)) → activate#(V2)
isNatList#(n__take(V1, V2)) → isNat#(activate(V1))U11#(tt, L) → activate#(L)
take#(s(M), cons(N, IL)) → U31#(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)activate#(n__isNatList(X)) → isNatList#(X)
activate#(n__length(X)) → length#(X)length#(cons(N, L)) → isNatList#(activate(L))
isNatList#(n__take(V1, V2)) → activate#(V2)isNat#(n__length(V1)) → isNatList#(activate(V1))
isNat#(n__s(V1)) → isNat#(activate(V1))isNatList#(n__cons(V1, V2)) → activate#(V1)
activate#(n__take(X1, X2)) → take#(X1, X2)take#(s(M), cons(N, IL)) → activate#(IL)
isNatIList#(n__cons(V1, V2)) → isNat#(activate(V1))take#(0, IL) → isNatIList#(IL)
isNatIList#(n__cons(V1, V2)) → activate#(V1)activate#(n__and(X1, X2)) → and#(X1, X2)
U11#(tt, L) → length#(activate(L))length#(cons(N, L)) → U11#(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take#(s(M), cons(N, IL)) → isNat#(M)length#(cons(N, L)) → activate#(L)