a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
↳ QTRS
↳ DependencyPairsProof
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
A__U11(tt, L) → MARK(L)
MARK(zeros) → A__ZEROS
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__U11(tt, L) → A__LENGTH(mark(L))
A__ISNATILIST(V) → A__ISNATLIST(V)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
A__U11(tt, L) → MARK(L)
MARK(zeros) → A__ZEROS
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__U11(tt, L) → A__LENGTH(mark(L))
A__ISNATILIST(V) → A__ISNATLIST(V)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__U11(tt, L) → MARK(L)
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(length(X)) → A__LENGTH(mark(X))
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
A__U11(tt, L) → A__LENGTH(mark(L))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATILIST(V) → A__ISNATLIST(V)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(V) → A__ISNATLIST(V)
Used ordering: Polynomial interpretation [25,35]:
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__U11(tt, L) → MARK(L)
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(length(X)) → A__LENGTH(mark(X))
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
MARK(isNat(X)) → A__ISNAT(X)
A__U11(tt, L) → A__LENGTH(mark(L))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
The value of delta used in the strict ordering is 1.
POL(A__ISNAT(x1)) = 0
POL(a__zeros) = 0
POL(a__U11(x1, x2)) = (4)x_1 + (2)x_2
POL(mark(x1)) = x_1
POL(a__isNatIList(x1)) = 1
POL(and(x1, x2)) = x_1 + x_2
POL(A__ISNATLIST(x1)) = 0
POL(A__AND(x1, x2)) = x_2
POL(a__length(x1)) = (2)x_1
POL(tt) = 0
POL(isNatList(x1)) = 0
POL(zeros) = 0
POL(isNatIList(x1)) = 1
POL(a__isNat(x1)) = 0
POL(s(x1)) = x_1
POL(isNat(x1)) = 0
POL(a__isNatList(x1)) = 0
POL(nil) = 1
POL(A__LENGTH(x1)) = (2)x_1
POL(A__U11(x1, x2)) = (2)x_2
POL(a__and(x1, x2)) = x_1 + x_2
POL(U11(x1, x2)) = (4)x_1 + (2)x_2
POL(0) = 0
POL(cons(x1, x2)) = (2)x_1 + x_2
POL(MARK(x1)) = x_1
POL(A__ISNATILIST(x1)) = 1
POL(length(x1)) = (2)x_1
a__U11(tt, L) → s(a__length(mark(L)))
a__zeros → cons(0, zeros)
a__isNat(0) → tt
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
a__length(nil) → 0
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__zeros → zeros
mark(nil) → nil
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNat(X) → isNat(X)
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__U11(tt, L) → MARK(L)
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(length(X)) → A__LENGTH(mark(X))
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__U11(tt, L) → A__LENGTH(mark(L))
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__U11(tt, L) → MARK(L)
MARK(U11(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
Used ordering: Polynomial interpretation [25,35]:
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__AND(tt, X) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__U11(tt, L) → A__LENGTH(mark(L))
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
The value of delta used in the strict ordering is 1.
POL(A__ISNAT(x1)) = 0
POL(a__U11(x1, x2)) = 1 + (2)x_1 + (4)x_2
POL(a__zeros) = 1
POL(mark(x1)) = 1 + (4)x_1
POL(a__isNatIList(x1)) = 1
POL(and(x1, x2)) = x_1 + (4)x_2
POL(A__ISNATLIST(x1)) = 0
POL(A__AND(x1, x2)) = (4)x_2
POL(a__length(x1)) = 2 + x_1
POL(tt) = 1
POL(isNatList(x1)) = 0
POL(zeros) = 0
POL(isNatIList(x1)) = 0
POL(a__isNat(x1)) = 1
POL(s(x1)) = x_1
POL(isNat(x1)) = 0
POL(a__isNatList(x1)) = 1
POL(nil) = 2
POL(A__LENGTH(x1)) = x_1
POL(A__U11(x1, x2)) = 1 + (4)x_2
POL(a__and(x1, x2)) = x_1 + (4)x_2
POL(U11(x1, x2)) = 1 + (2)x_1 + x_2
POL(0) = 0
POL(cons(x1, x2)) = 1 + (4)x_1 + (4)x_2
POL(MARK(x1)) = (4)x_1
POL(A__ISNATILIST(x1)) = 0
POL(length(x1)) = 1 + x_1
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
a__length(nil) → 0
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__zeros → zeros
mark(nil) → nil
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNat(X) → isNat(X)
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatIList(X) → isNatIList(X)
a__zeros → cons(0, zeros)
a__isNatList(X) → isNatList(X)
a__isNat(0) → tt
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__AND(tt, X) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__U11(tt, L) → A__LENGTH(mark(L))
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, L) → A__LENGTH(mark(L))
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__U11(tt, L) → A__LENGTH(mark(L))
Used ordering: Polynomial interpretation [25,35]:
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
The value of delta used in the strict ordering is 4.
POL(A__LENGTH(x1)) = (4)x_1
POL(A__U11(x1, x2)) = 4 + (4)x_1 + (4)x_2
POL(a__zeros) = 4
POL(a__and(x1, x2)) = x_1 + x_2
POL(a__U11(x1, x2)) = 4 + x_2
POL(mark(x1)) = 4 + x_1
POL(a__isNatIList(x1)) = 4 + (4)x_1
POL(U11(x1, x2)) = x_2
POL(and(x1, x2)) = x_1 + x_2
POL(0) = 0
POL(cons(x1, x2)) = 4 + x_1 + (2)x_2
POL(a__length(x1)) = x_1
POL(tt) = 4
POL(isNatList(x1)) = x_1
POL(zeros) = 0
POL(isNatIList(x1)) = (4)x_1
POL(a__isNat(x1)) = 4 + x_1
POL(s(x1)) = x_1
POL(isNat(x1)) = 1 + x_1
POL(length(x1)) = x_1
POL(a__isNatList(x1)) = 2 + x_1
POL(nil) = 4
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
a__length(nil) → 0
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__zeros → zeros
mark(nil) → nil
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNat(X) → isNat(X)
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatIList(X) → isNatIList(X)
a__zeros → cons(0, zeros)
a__isNatList(X) → isNatList(X)
a__isNat(0) → tt
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__AND(tt, X) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(isNat(X)) → A__ISNAT(X)
Used ordering: Polynomial interpretation [25,35]:
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__AND(tt, X) → MARK(X)
The value of delta used in the strict ordering is 4.
POL(A__ISNAT(x1)) = 0
POL(a__zeros) = 4
POL(a__U11(x1, x2)) = 4 + (2)x_2
POL(mark(x1)) = 2
POL(a__isNatIList(x1)) = 2 + (3)x_1
POL(and(x1, x2)) = 4 + (2)x_1 + x_2
POL(A__ISNATLIST(x1)) = 0
POL(A__AND(x1, x2)) = (4)x_2
POL(a__length(x1)) = 4
POL(tt) = 0
POL(isNatList(x1)) = 0
POL(zeros) = 2
POL(isNatIList(x1)) = 2 + x_1
POL(a__isNat(x1)) = 3
POL(s(x1)) = 4 + (4)x_1
POL(isNat(x1)) = 2 + (2)x_1
POL(a__isNatList(x1)) = 2
POL(nil) = 2
POL(a__and(x1, x2)) = (4)x_1 + x_2
POL(U11(x1, x2)) = 2 + (3)x_1 + (2)x_2
POL(0) = 1
POL(cons(x1, x2)) = 4 + (3)x_1 + (4)x_2
POL(MARK(x1)) = (4)x_1
POL(A__ISNATILIST(x1)) = 4 + x_1
POL(length(x1)) = 2 + (2)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__AND(tt, X) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__AND(tt, X) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
Used ordering: Polynomial interpretation [25,35]:
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__AND(tt, X) → MARK(X)
The value of delta used in the strict ordering is 4.
POL(A__ISNAT(x1)) = 4 + (4)x_1
POL(a__zeros) = 2
POL(a__and(x1, x2)) = (2)x_1 + (3)x_2
POL(a__U11(x1, x2)) = 4 + (2)x_2
POL(mark(x1)) = 4 + (3)x_1
POL(a__isNatIList(x1)) = 1 + (4)x_1
POL(U11(x1, x2)) = 2
POL(and(x1, x2)) = 2 + x_1 + (3)x_2
POL(0) = 1
POL(A__ISNATLIST(x1)) = (4)x_1
POL(cons(x1, x2)) = 1 + x_1 + (2)x_2
POL(A__AND(x1, x2)) = (4)x_2
POL(MARK(x1)) = (4)x_1
POL(a__length(x1)) = 2
POL(tt) = 0
POL(isNatList(x1)) = x_1
POL(zeros) = 0
POL(isNatIList(x1)) = 4 + (4)x_1
POL(a__isNat(x1)) = 4 + (4)x_1
POL(s(x1)) = (4)x_1
POL(length(x1)) = x_1
POL(isNat(x1)) = 4 + (3)x_1
POL(a__isNatList(x1)) = (3)x_1
POL(nil) = 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__AND(tt, X) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
A__ISNAT(s(V1)) → A__ISNAT(V1)
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__ISNAT(s(V1)) → A__ISNAT(V1)
The value of delta used in the strict ordering is 4.
POL(A__ISNAT(x1)) = (4)x_1
POL(s(x1)) = 1 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
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__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)