U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U211(tt, V1) → U221(isNat(activate(V1)))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U311(tt, N) → ACTIVATE(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
ACTIVATE(n__0) → 01
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U121(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U111(tt, V1, V2) → ISNAT(activate(V1))
U411(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → S(activate(X))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
U121(tt, V2) → U131(isNat(activate(V2)))
U211(tt, V1) → ACTIVATE(V1)
U411(tt, M, N) → S(plus(activate(N), activate(M)))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U211(tt, V1) → U221(isNat(activate(V1)))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U311(tt, N) → ACTIVATE(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
ACTIVATE(n__0) → 01
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U121(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U111(tt, V1, V2) → ISNAT(activate(V1))
U411(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → S(activate(X))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
U121(tt, V2) → U131(isNat(activate(V2)))
U211(tt, V1) → ACTIVATE(V1)
U411(tt, M, N) → S(plus(activate(N), activate(M)))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
U311(tt, N) → ACTIVATE(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U411(tt, M, N) → ACTIVATE(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U111(tt, V1, V2) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
U311(tt, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
Used ordering: Polynomial interpretation [25,35]:
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
U411(tt, M, N) → PLUS(activate(N), activate(M))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U411(tt, M, N) → ACTIVATE(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U111(tt, V1, V2) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
The value of delta used in the strict ordering is 1/4.
POL(n__plus(x1, x2)) = (4)x_1 + (4)x_2
POL(U121(x1, x2)) = (2)x_2
POL(n__isNat(x1)) = (2)x_1
POL(U41(x1, x2, x3)) = (4)x_2 + (4)x_3
POL(U211(x1, x2)) = (2)x_2
POL(activate(x1)) = x_1
POL(U12(x1, x2)) = (4)x_2
POL(and(x1, x2)) = (5/4)x_1 + x_2
POL(n__s(x1)) = x_1
POL(U21(x1, x2)) = (7/4)x_1 + (1/4)x_2
POL(PLUS(x1, x2)) = (4)x_1 + (9/4)x_2
POL(tt) = 0
POL(AND(x1, x2)) = x_2
POL(U311(x1, x2)) = 1/4 + (4)x_2
POL(s(x1)) = x_1
POL(isNat(x1)) = (2)x_1
POL(ACTIVATE(x1)) = x_1
POL(plus(x1, x2)) = (4)x_1 + (4)x_2
POL(U31(x1, x2)) = 1/4 + (4)x_2
POL(U111(x1, x2, x3)) = (4)x_2 + (2)x_3
POL(ISNATKIND(x1)) = x_1
POL(n__isNatKind(x1)) = x_1
POL(U11(x1, x2, x3)) = (4)x_3
POL(isNatKind(x1)) = x_1
POL(U411(x1, x2, x3)) = (9/4)x_2 + (4)x_3
POL(0) = 1/4
POL(ISNAT(x1)) = (2)x_1
POL(n__and(x1, x2)) = (5/4)x_1 + x_2
POL(U22(x1)) = 0
POL(n__0) = 1/4
POL(U13(x1)) = (3/4)x_1
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__isNatKind(X)) → isNatKind(X)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
U21(tt, V1) → U22(isNat(activate(V1)))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U22(tt) → tt
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__0) → tt
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U121(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U411(tt, M, N) → ACTIVATE(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
U111(tt, V1, V2) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
U411(tt, M, N) → ACTIVATE(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
Used ordering: Polynomial interpretation [25,35]:
U411(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
The value of delta used in the strict ordering is 1/4.
POL(n__plus(x1, x2)) = 2 + (3)x_1 + (4)x_2
POL(U121(x1, x2)) = 3/4 + (1/4)x_2
POL(n__isNat(x1)) = 1 + (2)x_1
POL(U41(x1, x2, x3)) = 2 + (4)x_2 + (3)x_3
POL(U211(x1, x2)) = 1/2 + (1/4)x_2
POL(activate(x1)) = x_1
POL(U12(x1, x2)) = 0
POL(and(x1, x2)) = x_1 + x_2
POL(n__s(x1)) = x_1
POL(U21(x1, x2)) = 1 + (2)x_2
POL(PLUS(x1, x2)) = 3/4 + (3/4)x_1 + (1/4)x_2
POL(tt) = 0
POL(AND(x1, x2)) = 1/4 + (1/4)x_2
POL(s(x1)) = x_1
POL(isNat(x1)) = 1 + (2)x_1
POL(ACTIVATE(x1)) = 1/4 + (1/4)x_1
POL(plus(x1, x2)) = 2 + (3)x_1 + (4)x_2
POL(U31(x1, x2)) = 2 + x_2
POL(U111(x1, x2, x3)) = 1 + (1/4)x_2 + (1/4)x_3
POL(ISNATKIND(x1)) = 1/4 + (1/4)x_1
POL(n__isNatKind(x1)) = x_1
POL(U11(x1, x2, x3)) = 1/4 + x_2
POL(isNatKind(x1)) = x_1
POL(U411(x1, x2, x3)) = 3/4 + (1/4)x_2 + (3/4)x_3
POL(0) = 0
POL(ISNAT(x1)) = 1/2 + (1/4)x_1
POL(n__and(x1, x2)) = x_1 + x_2
POL(U22(x1)) = (3/4)x_1
POL(n__0) = 0
POL(U13(x1)) = 0
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__isNatKind(X)) → isNatKind(X)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
U21(tt, V1) → U22(isNat(activate(V1)))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U22(tt) → tt
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__0) → tt
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U411(tt, M, N) → PLUS(activate(N), activate(M))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
The value of delta used in the strict ordering is 1/4.
POL(n__plus(x1, x2)) = (4)x_1 + (2)x_2
POL(plus(x1, x2)) = (4)x_1 + (2)x_2
POL(U31(x1, x2)) = (3/2)x_2
POL(n__isNat(x1)) = 4
POL(U41(x1, x2, x3)) = 9/4 + (2)x_2 + (4)x_3
POL(n__isNatKind(x1)) = 0
POL(U11(x1, x2, x3)) = 4 + (13/4)x_1
POL(U211(x1, x2)) = 2 + (4)x_2
POL(U12(x1, x2)) = (1/2)x_1
POL(activate(x1)) = x_1
POL(n__s(x1)) = 2 + x_1
POL(isNatKind(x1)) = 0
POL(and(x1, x2)) = (4)x_1 + (4)x_2
POL(0) = 0
POL(U21(x1, x2)) = 2 + (7/2)x_1
POL(ISNAT(x1)) = 7/4 + (4)x_1
POL(n__and(x1, x2)) = (4)x_1 + (4)x_2
POL(U22(x1)) = 1/4
POL(tt) = 0
POL(n__0) = 0
POL(U13(x1)) = 0
POL(s(x1)) = 2 + x_1
POL(isNat(x1)) = 4
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__isNatKind(X)) → isNatKind(X)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
U21(tt, V1) → U22(isNat(activate(V1)))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U22(tt) → tt
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__0) → tt
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
Used ordering: Polynomial interpretation [25,35]:
U411(tt, M, N) → PLUS(activate(N), activate(M))
The value of delta used in the strict ordering is 1/8.
POL(n__plus(x1, x2)) = (2)x_1 + x_2
POL(plus(x1, x2)) = (2)x_1 + x_2
POL(U31(x1, x2)) = (1/4)x_1 + x_2
POL(n__isNat(x1)) = 0
POL(U41(x1, x2, x3)) = 1/2 + x_2 + (2)x_3
POL(n__isNatKind(x1)) = (4)x_1
POL(U11(x1, x2, x3)) = 0
POL(U12(x1, x2)) = 0
POL(activate(x1)) = x_1
POL(n__s(x1)) = 1/2 + x_1
POL(isNatKind(x1)) = (4)x_1
POL(and(x1, x2)) = x_2
POL(0) = 0
POL(U411(x1, x2, x3)) = (1/4)x_2
POL(U21(x1, x2)) = 0
POL(n__and(x1, x2)) = x_2
POL(PLUS(x1, x2)) = (1/4)x_2
POL(U22(x1)) = x_1
POL(tt) = 0
POL(n__0) = 0
POL(U13(x1)) = (13/4)x_1
POL(s(x1)) = 1/2 + x_1
POL(isNat(x1)) = 0
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__isNatKind(X)) → isNatKind(X)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
U21(tt, V1) → U22(isNat(activate(V1)))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U22(tt) → tt
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__0) → tt
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
U411(tt, M, N) → PLUS(activate(N), activate(M))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
Used ordering: Polynomial interpretation [25,35]:
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
The value of delta used in the strict ordering is 1/8.
POL(n__plus(x1, x2)) = (4)x_1 + (3/2)x_2
POL(plus(x1, x2)) = (4)x_1 + (3/2)x_2
POL(U31(x1, x2)) = (4)x_2
POL(n__isNat(x1)) = 5/4 + (7/4)x_1
POL(U41(x1, x2, x3)) = 1/4 + (3/2)x_2 + (4)x_3
POL(ISNATKIND(x1)) = (1/2)x_1
POL(n__isNatKind(x1)) = x_1
POL(U11(x1, x2, x3)) = 5/4 + (5/4)x_3
POL(U12(x1, x2)) = 5/4
POL(activate(x1)) = x_1
POL(isNatKind(x1)) = x_1
POL(n__s(x1)) = 1/4 + x_1
POL(and(x1, x2)) = (2)x_1 + (3/2)x_2
POL(0) = 0
POL(U21(x1, x2)) = 3/4 + (7/4)x_1
POL(n__and(x1, x2)) = (2)x_1 + (3/2)x_2
POL(U22(x1)) = 3/4
POL(tt) = 0
POL(AND(x1, x2)) = (3/4)x_2
POL(n__0) = 0
POL(U13(x1)) = 5/4
POL(s(x1)) = 1/4 + x_1
POL(isNat(x1)) = 5/4 + (7/4)x_1
POL(ACTIVATE(x1)) = (1/2)x_1
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__isNatKind(X)) → isNatKind(X)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
U21(tt, V1) → U22(isNat(activate(V1)))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U22(tt) → tt
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__0) → tt
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
Used ordering: Polynomial interpretation [25,35]:
AND(tt, X) → ACTIVATE(X)
The value of delta used in the strict ordering is 1/8.
POL(n__plus(x1, x2)) = 1/4 + (1/2)x_1 + (9/4)x_2
POL(plus(x1, x2)) = 4 + (4)x_1 + (13/4)x_2
POL(U31(x1, x2)) = 4 + (3/2)x_1 + (4)x_2
POL(n__isNat(x1)) = 7/2 + (11/4)x_1
POL(U41(x1, x2, x3)) = 11/4 + (13/4)x_2 + (1/4)x_3
POL(n__isNatKind(x1)) = 13/4 + (7/4)x_1
POL(U11(x1, x2, x3)) = 5/4 + (5/2)x_1 + (2)x_2 + (3/2)x_3
POL(U12(x1, x2)) = 3/2 + (3/4)x_1 + (1/2)x_2
POL(activate(x1)) = 4
POL(n__s(x1)) = 11/4
POL(and(x1, x2)) = 1/4 + (7/4)x_1 + (4)x_2
POL(isNatKind(x1)) = 7/4
POL(0) = 3/2
POL(U21(x1, x2)) = 3 + (15/4)x_1 + (3/2)x_2
POL(n__and(x1, x2)) = 1/4 + x_1 + (4)x_2
POL(U22(x1)) = 4 + (1/4)x_1
POL(tt) = 1/4
POL(AND(x1, x2)) = (1/2)x_2
POL(n__0) = 1
POL(U13(x1)) = 13/4 + (1/4)x_1
POL(s(x1)) = 9/4 + (2)x_1
POL(isNat(x1)) = 3/2 + (2)x_1
POL(ACTIVATE(x1)) = (1/2)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
AND(tt, X) → ACTIVATE(X)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X