TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60014 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (589ms).
 | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (305ms), PolynomialLinearRange4iUR (10000ms), DependencyGraph (239ms), PolynomialLinearRange8NegiUR (timeout), DependencyGraph (234ms), ReductionPairSAT (4698ms), DependencyGraph (236ms), SizeChangePrinciple (timeout)].

The following open problems remain:



Open Dependency Pair Problem 2

Dependency Pairs

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)))
activate#(n__length(X))activate#(X)activate#(n__s(X))activate#(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))
isNatList#(n__cons(V1, V2))isNat#(activate(V1))activate#(n__cons(X1, X2))activate#(X1)
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__cons(V1, V2))activate#(V2)
U11#(tt, L)activate#(L)activate#(n__length(X))length#(activate(X))
activate#(n__isNatList(X))isNatList#(X)length#(cons(N, L))isNatList#(activate(L))
isNat#(n__length(V1))isNatList#(activate(V1))isNat#(n__s(V1))isNat#(activate(V1))
isNatList#(n__cons(V1, V2))activate#(V1)isNatIList#(n__cons(V1, V2))isNat#(activate(V1))
U11#(tt, L)length#(activate(L))isNatIList#(n__cons(V1, V2))activate#(V1)
length#(cons(N, L))U11#(and(isNatList(activate(L)), n__isNat(N)), activate(L))length#(cons(N, L))activate#(L)

Rewrite Rules

zeroscons(0, n__zeros)U11(tt, L)s(length(activate(L)))
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)))length(nil)0
length(cons(N, L))U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))zerosn__zeros
0n__0length(X)n__length(X)
s(X)n__s(X)cons(X1, X2)n__cons(X1, X2)
isNatIList(X)n__isNatIList(X)niln__nil
isNatList(X)n__isNatList(X)isNat(X)n__isNat(X)
activate(n__zeros)zerosactivate(n__0)0
activate(n__length(X))length(activate(X))activate(n__s(X))s(activate(X))
activate(n__cons(X1, X2))cons(activate(X1), X2)activate(n__isNatIList(X))isNatIList(X)
activate(n__nil)nilactivate(n__isNatList(X))isNatList(X)
activate(n__isNat(X))isNat(X)activate(X)X

Original Signature

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


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

zeroscons(0, n__zeros)U11(tt, L)s(length(activate(L)))
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)))length(nil)0
length(cons(N, L))U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))zerosn__zeros
0n__0length(X)n__length(X)
s(X)n__s(X)cons(X1, X2)n__cons(X1, X2)
isNatIList(X)n__isNatIList(X)niln__nil
isNatList(X)n__isNatList(X)isNat(X)n__isNat(X)
activate(n__zeros)zerosactivate(n__0)0
activate(n__length(X))length(activate(X))activate(n__s(X))s(activate(X))
activate(n__cons(X1, X2))cons(activate(X1), X2)activate(n__isNatIList(X))isNatIList(X)
activate(n__nil)nilactivate(n__isNatList(X))isNatList(X)
activate(n__isNat(X))isNat(X)activate(X)X

Original Signature

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

Strategy


The following SCCs where found

isNatIList#(n__cons(V1, V2)) → activate#(V2)and#(tt, X) → activate#(X)
activate#(n__isNatIList(X)) → isNatIList#(X)activate#(n__length(X)) → activate#(X)
isNatList#(n__cons(V1, V2)) → and#(isNat(activate(V1)), n__isNatList(activate(V2)))activate#(n__s(X)) → activate#(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))
isNatList#(n__cons(V1, V2)) → isNat#(activate(V1))activate#(n__cons(X1, X2)) → activate#(X1)
activate#(n__isNat(X)) → isNat#(X)isNatIList#(V) → activate#(V)
isNat#(n__length(V1)) → activate#(V1)isNat#(n__s(V1)) → activate#(V1)
isNatIList#(V) → isNatList#(activate(V))isNatList#(n__cons(V1, V2)) → activate#(V2)
U11#(tt, L) → activate#(L)activate#(n__length(X)) → length#(activate(X))
activate#(n__isNatList(X)) → isNatList#(X)length#(cons(N, L)) → isNatList#(activate(L))
isNat#(n__length(V1)) → isNatList#(activate(V1))isNat#(n__s(V1)) → isNat#(activate(V1))
isNatList#(n__cons(V1, V2)) → activate#(V1)isNatIList#(n__cons(V1, V2)) → isNat#(activate(V1))
isNatIList#(n__cons(V1, V2)) → activate#(V1)U11#(tt, L) → length#(activate(L))
length#(cons(N, L)) → U11#(and(isNatList(activate(L)), n__isNat(N)), activate(L))length#(cons(N, L)) → activate#(L)