Open Dependency Pair Problem 2

Dependency Pairs

mark^#(U11(X1, X2))	→	mark^#(X1)	mark^#(U31(X1, X2, X3, X4))	→	mark^#(X1)
mark^#(isNat(X))	→	a__isNat^#(X)	mark^#(take(X1, X2))	→	mark^#(X1)
mark^#(isNatIList(X))	→	a__isNatIList^#(X)	a__isNatList^#(take(V1, V2))	→	a__and^#(a__isNat(V1), isNatIList(V2))
a__length^#(cons(N, L))	→	a__U11^#(a__and(a__isNatList(L), isNat(N)), L)	a__isNatIList^#(cons(V1, V2))	→	a__and^#(a__isNat(V1), isNatIList(V2))
a__isNat^#(s(V1))	→	a__isNat^#(V1)	mark^#(U21(X))	→	mark^#(X)
a__isNatList^#(cons(V1, V2))	→	a__and^#(a__isNat(V1), isNatList(V2))	mark^#(length(X))	→	a__length^#(mark(X))
a__take^#(s(M), cons(N, IL))	→	a__isNatIList^#(IL)	a__isNatIList^#(V)	→	a__isNatList^#(V)
a__U11^#(tt, L)	→	mark^#(L)	a__and^#(tt, X)	→	mark^#(X)
mark^#(U31(X1, X2, X3, X4))	→	a__U31^#(mark(X1), X2, X3, X4)	mark^#(length(X))	→	mark^#(X)
mark^#(s(X))	→	mark^#(X)	a__length^#(cons(N, L))	→	a__isNatList^#(L)
a__take^#(0, IL)	→	a__isNatIList^#(IL)	a__isNatList^#(cons(V1, V2))	→	a__isNat^#(V1)
a__take^#(s(M), cons(N, IL))	→	a__U31^#(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)	a__U11^#(tt, L)	→	a__length^#(mark(L))
a__isNatList^#(take(V1, V2))	→	a__isNat^#(V1)	mark^#(cons(X1, X2))	→	mark^#(X1)
a__length^#(cons(N, L))	→	a__and^#(a__isNatList(L), isNat(N))	a__isNat^#(length(V1))	→	a__isNatList^#(V1)
a__take^#(s(M), cons(N, IL))	→	a__and^#(a__isNatIList(IL), and(isNat(M), isNat(N)))	mark^#(isNatList(X))	→	a__isNatList^#(X)
mark^#(and(X1, X2))	→	mark^#(X1)	a__isNatIList^#(cons(V1, V2))	→	a__isNat^#(V1)
mark^#(and(X1, X2))	→	a__and^#(mark(X1), X2)	mark^#(take(X1, X2))	→	a__take^#(mark(X1), mark(X2))
mark^#(U11(X1, X2))	→	a__U11^#(mark(X1), X2)	mark^#(take(X1, X2))	→	mark^#(X2)
a__U31^#(tt, IL, M, N)	→	mark^#(N)

Rewrite Rules

a__zeros	→	cons(0, zeros)	a__U11(tt, L)	→	s(a__length(mark(L)))
a__U21(tt)	→	nil	a__U31(tt, IL, M, N)	→	cons(mark(N), take(M, IL))
a__and(tt, X)	→	mark(X)	a__isNat(0)	→	tt
a__isNat(length(V1))	→	a__isNatList(V1)	a__isNat(s(V1))	→	a__isNat(V1)
a__isNatIList(V)	→	a__isNatList(V)	a__isNatIList(zeros)	→	tt
a__isNatIList(cons(V1, V2))	→	a__and(a__isNat(V1), isNatIList(V2))	a__isNatList(nil)	→	tt
a__isNatList(cons(V1, V2))	→	a__and(a__isNat(V1), isNatList(V2))	a__isNatList(take(V1, V2))	→	a__and(a__isNat(V1), isNatIList(V2))
a__length(nil)	→	0	a__length(cons(N, L))	→	a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL)	→	a__U21(a__isNatIList(IL))	a__take(s(M), cons(N, IL))	→	a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros)	→	a__zeros	mark(U11(X1, X2))	→	a__U11(mark(X1), X2)
mark(length(X))	→	a__length(mark(X))	mark(U21(X))	→	a__U21(mark(X))
mark(U31(X1, X2, X3, X4))	→	a__U31(mark(X1), X2, X3, X4)	mark(take(X1, X2))	→	a__take(mark(X1), mark(X2))
mark(and(X1, X2))	→	a__and(mark(X1), X2)	mark(isNat(X))	→	a__isNat(X)
mark(isNatList(X))	→	a__isNatList(X)	mark(isNatIList(X))	→	a__isNatIList(X)
mark(cons(X1, X2))	→	cons(mark(X1), X2)	mark(0)	→	0
mark(tt)	→	tt	mark(s(X))	→	s(mark(X))
mark(nil)	→	nil	a__zeros	→	zeros
a__U11(X1, X2)	→	U11(X1, X2)	a__length(X)	→	length(X)
a__U21(X)	→	U21(X)	a__U31(X1, X2, X3, X4)	→	U31(X1, X2, X3, X4)
a__take(X1, X2)	→	take(X1, X2)	a__and(X1, X2)	→	and(X1, X2)
a__isNat(X)	→	isNat(X)	a__isNatList(X)	→	isNatList(X)
a__isNatIList(X)	→	isNatIList(X)

Original Signature

Termination of terms over the following signature is verified: a__take, a__and, isNat, a__isNatList, length, a__U21, U21, cons, a__zeros, isNatIList, a__length, mark, and, 0, s, a__isNatIList, isNatList, tt, zeros, take, a__isNat, a__U31, U11, U31, a__U11, nil

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark^#(U11(X1, X2))	→	mark^#(X1)	mark^#(U31(X1, X2, X3, X4))	→	mark^#(X1)
mark^#(take(X1, X2))	→	mark^#(X1)	mark^#(isNat(X))	→	a__isNat^#(X)
mark^#(isNatIList(X))	→	a__isNatIList^#(X)	mark^#(zeros)	→	a__zeros^#
a__length^#(cons(N, L))	→	a__U11^#(a__and(a__isNatList(L), isNat(N)), L)	a__isNatList^#(take(V1, V2))	→	a__and^#(a__isNat(V1), isNatIList(V2))
a__isNatIList^#(cons(V1, V2))	→	a__and^#(a__isNat(V1), isNatIList(V2))	a__isNat^#(s(V1))	→	a__isNat^#(V1)
mark^#(U21(X))	→	mark^#(X)	mark^#(length(X))	→	a__length^#(mark(X))
a__isNatList^#(cons(V1, V2))	→	a__and^#(a__isNat(V1), isNatList(V2))	a__take^#(s(M), cons(N, IL))	→	a__isNatIList^#(IL)
a__isNatIList^#(V)	→	a__isNatList^#(V)	a__U11^#(tt, L)	→	mark^#(L)
a__and^#(tt, X)	→	mark^#(X)	mark^#(U21(X))	→	a__U21^#(mark(X))
mark^#(U31(X1, X2, X3, X4))	→	a__U31^#(mark(X1), X2, X3, X4)	mark^#(length(X))	→	mark^#(X)
mark^#(s(X))	→	mark^#(X)	a__length^#(cons(N, L))	→	a__isNatList^#(L)
a__take^#(0, IL)	→	a__isNatIList^#(IL)	a__take^#(0, IL)	→	a__U21^#(a__isNatIList(IL))
a__isNatList^#(cons(V1, V2))	→	a__isNat^#(V1)	a__take^#(s(M), cons(N, IL))	→	a__U31^#(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
a__U11^#(tt, L)	→	a__length^#(mark(L))	a__isNatList^#(take(V1, V2))	→	a__isNat^#(V1)
mark^#(cons(X1, X2))	→	mark^#(X1)	a__length^#(cons(N, L))	→	a__and^#(a__isNatList(L), isNat(N))
a__take^#(s(M), cons(N, IL))	→	a__and^#(a__isNatIList(IL), and(isNat(M), isNat(N)))	a__isNat^#(length(V1))	→	a__isNatList^#(V1)
mark^#(isNatList(X))	→	a__isNatList^#(X)	mark^#(and(X1, X2))	→	mark^#(X1)
a__isNatIList^#(cons(V1, V2))	→	a__isNat^#(V1)	mark^#(and(X1, X2))	→	a__and^#(mark(X1), X2)
mark^#(take(X1, X2))	→	a__take^#(mark(X1), mark(X2))	mark^#(U11(X1, X2))	→	a__U11^#(mark(X1), X2)
mark^#(take(X1, X2))	→	mark^#(X2)	a__U31^#(tt, IL, M, N)	→	mark^#(N)

Rewrite Rules

a__zeros	→	cons(0, zeros)	a__U11(tt, L)	→	s(a__length(mark(L)))
a__U21(tt)	→	nil	a__U31(tt, IL, M, N)	→	cons(mark(N), take(M, IL))
a__and(tt, X)	→	mark(X)	a__isNat(0)	→	tt
a__isNat(length(V1))	→	a__isNatList(V1)	a__isNat(s(V1))	→	a__isNat(V1)
a__isNatIList(V)	→	a__isNatList(V)	a__isNatIList(zeros)	→	tt
a__isNatIList(cons(V1, V2))	→	a__and(a__isNat(V1), isNatIList(V2))	a__isNatList(nil)	→	tt
a__isNatList(cons(V1, V2))	→	a__and(a__isNat(V1), isNatList(V2))	a__isNatList(take(V1, V2))	→	a__and(a__isNat(V1), isNatIList(V2))
a__length(nil)	→	0	a__length(cons(N, L))	→	a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL)	→	a__U21(a__isNatIList(IL))	a__take(s(M), cons(N, IL))	→	a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros)	→	a__zeros	mark(U11(X1, X2))	→	a__U11(mark(X1), X2)
mark(length(X))	→	a__length(mark(X))	mark(U21(X))	→	a__U21(mark(X))
mark(U31(X1, X2, X3, X4))	→	a__U31(mark(X1), X2, X3, X4)	mark(take(X1, X2))	→	a__take(mark(X1), mark(X2))
mark(and(X1, X2))	→	a__and(mark(X1), X2)	mark(isNat(X))	→	a__isNat(X)
mark(isNatList(X))	→	a__isNatList(X)	mark(isNatIList(X))	→	a__isNatIList(X)
mark(cons(X1, X2))	→	cons(mark(X1), X2)	mark(0)	→	0
mark(tt)	→	tt	mark(s(X))	→	s(mark(X))
mark(nil)	→	nil	a__zeros	→	zeros
a__U11(X1, X2)	→	U11(X1, X2)	a__length(X)	→	length(X)
a__U21(X)	→	U21(X)	a__U31(X1, X2, X3, X4)	→	U31(X1, X2, X3, X4)
a__take(X1, X2)	→	take(X1, X2)	a__and(X1, X2)	→	and(X1, X2)
a__isNat(X)	→	isNat(X)	a__isNatList(X)	→	isNatList(X)
a__isNatIList(X)	→	isNatIList(X)

Original Signature

Termination of terms over the following signature is verified: a__take, a__and, isNat, a__isNatList, length, U21, a__U21, cons, a__zeros, isNatIList, a__length, mark, and, 0, s, a__isNatIList, isNatList, tt, zeros, take, a__isNat, a__U31, U11, U31, a__U11, nil

Strategy

The following SCCs where found

mark^#(U11(X1, X2)) → mark^#(X1)	mark^#(take(X1, X2)) → mark^#(X1)
mark^#(isNat(X)) → a__isNat^#(X)	mark^#(U31(X1, X2, X3, X4)) → mark^#(X1)
mark^#(isNatIList(X)) → a__isNatIList^#(X)	a__length^#(cons(N, L)) → a__U11^#(a__and(a__isNatList(L), isNat(N)), L)
a__isNatList^#(take(V1, V2)) → a__and^#(a__isNat(V1), isNatIList(V2))	a__isNatIList^#(cons(V1, V2)) → a__and^#(a__isNat(V1), isNatIList(V2))
a__isNat^#(s(V1)) → a__isNat^#(V1)	mark^#(U21(X)) → mark^#(X)
mark^#(length(X)) → a__length^#(mark(X))	a__isNatList^#(cons(V1, V2)) → a__and^#(a__isNat(V1), isNatList(V2))
a__take^#(s(M), cons(N, IL)) → a__isNatIList^#(IL)	a__isNatIList^#(V) → a__isNatList^#(V)
a__U11^#(tt, L) → mark^#(L)	a__and^#(tt, X) → mark^#(X)
mark^#(U31(X1, X2, X3, X4)) → a__U31^#(mark(X1), X2, X3, X4)	mark^#(length(X)) → mark^#(X)
mark^#(s(X)) → mark^#(X)	a__length^#(cons(N, L)) → a__isNatList^#(L)
a__take^#(0, IL) → a__isNatIList^#(IL)	a__isNatList^#(cons(V1, V2)) → a__isNat^#(V1)
a__take^#(s(M), cons(N, IL)) → a__U31^#(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)	a__U11^#(tt, L) → a__length^#(mark(L))
a__isNatList^#(take(V1, V2)) → a__isNat^#(V1)	mark^#(cons(X1, X2)) → mark^#(X1)
a__length^#(cons(N, L)) → a__and^#(a__isNatList(L), isNat(N))	mark^#(isNatList(X)) → a__isNatList^#(X)
a__take^#(s(M), cons(N, IL)) → a__and^#(a__isNatIList(IL), and(isNat(M), isNat(N)))	a__isNat^#(length(V1)) → a__isNatList^#(V1)
mark^#(and(X1, X2)) → mark^#(X1)	a__isNatIList^#(cons(V1, V2)) → a__isNat^#(V1)
mark^#(and(X1, X2)) → a__and^#(mark(X1), X2)	mark^#(take(X1, X2)) → a__take^#(mark(X1), mark(X2))
mark^#(take(X1, X2)) → mark^#(X2)	mark^#(U11(X1, X2)) → a__U11^#(mark(X1), X2)
a__U31^#(tt, IL, M, N) → mark^#(N)

The following DP Processors were used

The following open problems remain:

Open Dependency Pair Problem 2

Dependency Pairs

Rewrite Rules

Original Signature

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

The following SCCs where found