YES
0 QTRS
↳1 QTRSToCSRProof (⇔, 0 ms)
↳2 CSR
↳3 CSDependencyPairsProof (⇔, 0 ms)
↳4 QCSDP
↳5 QCSDependencyGraphProof (⇔, 0 ms)
↳6 AND
↳7 QCSDP
↳8 QCSDPReductionPairProof (⇔, 135 ms)
↳9 QCSDP
↳10 QCSDependencyGraphProof (⇔, 0 ms)
↳11 TRUE
↳12 QCSDP
↳13 QCSDPSubtermProof (⇔, 0 ms)
↳14 QCSDP
↳15 QCSDependencyGraphProof (⇔, 0 ms)
↳16 TRUE
↳17 QCSDP
↳18 QCSDPSubtermProof (⇔, 0 ms)
↳19 QCSDP
↳20 QCSDependencyGraphProof (⇔, 0 ms)
↳21 TRUE
active(U11(tt, N)) → mark(N)
active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)
active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U31(X)) → U31(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2) → mark(U11(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U31(X)) → U31(proper(X))
proper(0) → ok(0)
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(U11(tt, N)) → mark(N)
active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)
active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U31(X)) → U31(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2) → mark(U11(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U31(X)) → U31(proper(X))
proper(0) → ok(0)
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
U11: {1}
tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
U31: {1}
0: empty set
U41: {1}
x: {1, 2}
and: {1}
isNat: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)
U11: {1}
tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
U31: {1}
0: empty set
U41: {1}
x: {1, 2}
and: {1}
isNat: empty set
U21'(tt, M, N) → PLUS(N, M)
U41'(tt, M, N) → PLUS(x(N, M), N)
U41'(tt, M, N) → X(N, M)
ISNAT(plus(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(x(V1, V2)) → ISNAT(V1)
PLUS(N, 0) → U11'(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U21'(and(isNat(M), isNat(N)), M, N)
PLUS(N, s(M)) → AND(isNat(M), isNat(N))
PLUS(N, s(M)) → ISNAT(M)
X(N, 0) → U31'(isNat(N))
X(N, 0) → ISNAT(N)
X(N, s(M)) → U41'(and(isNat(M), isNat(N)), M, N)
X(N, s(M)) → AND(isNat(M), isNat(N))
X(N, s(M)) → ISNAT(M)
U11'(tt, N) → N
U21'(tt, M, N) → N
U21'(tt, M, N) → M
U41'(tt, M, N) → N
U41'(tt, M, N) → M
AND(tt, X) → X
isNat(x0)
U11'(tt, N) → U(N)
U21'(tt, M, N) → U(N)
U21'(tt, M, N) → U(M)
U41'(tt, M, N) → U(N)
U41'(tt, M, N) → U(M)
AND(tt, X) → U(X)
U(isNat(x0)) → ISNAT(x0)
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)
AND(tt, X) → U(X)
U(isNat(x0)) → ISNAT(x0)
ISNAT(plus(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(x(V1, V2)) → ISNAT(V1)
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)
[tt, 0] > [x2, U413] > [isNat1, ISNAT1, plus2, U213] > and1 > s1
tt: multiset
isNat1: [1]
ISNAT1: [1]
plus2: [1,2]
s1: multiset
x2: [2,1]
0: multiset
and1: [1]
U213: [3,2,1]
U413: [2,3,1]
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
ISNAT(plus(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(x(V1, V2)) → ISNAT(V1)
AND(tt, X) → U(X)
U(isNat(x0)) → ISNAT(x0)
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)
PLUS(N, s(M)) → U21'(and(isNat(M), isNat(N)), M, N)
U21'(tt, M, N) → PLUS(N, M)
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PLUS(N, s(M)) → U21'(and(isNat(M), isNat(N)), M, N)
Used ordering: Combined order from the following AFS and order.
U21'(tt, M, N) → PLUS(N, M)
U21'(tt, M, N) → PLUS(N, M)
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)
U41'(tt, M, N) → X(N, M)
X(N, s(M)) → U41'(and(isNat(M), isNat(N)), M, N)
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
X(N, s(M)) → U41'(and(isNat(M), isNat(N)), M, N)
Used ordering: Combined order from the following AFS and order.
U41'(tt, M, N) → X(N, M)
U41'(tt, M, N) → X(N, M)
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)