0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
↳9 DependencyPairsProof (⇔)
↳10 QDP
↳11 DependencyGraphProof (⇔)
↳12 QDP
↳13 MRRProof (⇔)
↳14 QDP
↳15 DependencyGraphProof (⇔)
↳16 QDP
↳17 MRRProof (⇔)
↳18 QDP
↳19 MRRProof (⇔)
↳20 QDP
↳21 DependencyGraphProof (⇔)
↳22 QDP
↳23 QDPOrderProof (⇔)
↳24 QDP
↳25 QDPOrderProof (⇔)
↳26 QDP
↳27 DependencyGraphProof (⇔)
↳28 QDP
↳29 UsableRulesProof (⇔)
↳30 QDP
↳31 QDPSizeChangeProof (⇔)
↳32 TRUE
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(nil, XS) → nil
a__zip(X, nil) → nil
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__tail(cons(X, XS)) → mark(XS)
a__repItems(nil) → nil
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(a__incr(x1)) = x1
POL(a__oddNs) = 0
POL(a__pairNs) = 0
POL(a__repItems(x1)) = 2·x1
POL(a__tail(x1)) = x1
POL(a__take(x1, x2)) = 2·x1 + x2
POL(a__zip(x1, x2)) = 2 + x1 + x2
POL(cons(x1, x2)) = 2·x1 + x2
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(oddNs) = 0
POL(pair(x1, x2)) = x1 + x2
POL(pairNs) = 0
POL(repItems(x1)) = 2·x1
POL(s(x1)) = x1
POL(tail(x1)) = x1
POL(take(x1, x2)) = 2·x1 + x2
POL(zip(x1, x2)) = 2 + x1 + x2
a__zip(nil, XS) → nil
a__zip(X, nil) → nil
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__tail(cons(X, XS)) → mark(XS)
a__repItems(nil) → nil
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(a__incr(x1)) = 2·x1
POL(a__oddNs) = 0
POL(a__pairNs) = 0
POL(a__repItems(x1)) = 2·x1
POL(a__tail(x1)) = x1
POL(a__take(x1, x2)) = 2 + x1 + x2
POL(a__zip(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = 2·x1 + x2
POL(incr(x1)) = 2·x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(oddNs) = 0
POL(pair(x1, x2)) = x1 + x2
POL(pairNs) = 0
POL(repItems(x1)) = 2·x1
POL(s(x1)) = x1
POL(tail(x1)) = x1
POL(take(x1, x2)) = 2 + x1 + x2
POL(zip(x1, x2)) = x1 + x2
a__take(0, XS) → nil
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__tail(cons(X, XS)) → mark(XS)
a__repItems(nil) → nil
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(a__incr(x1)) = x1
POL(a__oddNs) = 0
POL(a__pairNs) = 0
POL(a__repItems(x1)) = 2·x1
POL(a__tail(x1)) = 2·x1
POL(a__take(x1, x2)) = x1 + x2
POL(a__zip(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 2
POL(oddNs) = 0
POL(pair(x1, x2)) = x1 + x2
POL(pairNs) = 0
POL(repItems(x1)) = 2·x1
POL(s(x1)) = x1
POL(tail(x1)) = 2·x1
POL(take(x1, x2)) = x1 + x2
POL(zip(x1, x2)) = x1 + x2
a__repItems(nil) → nil
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__tail(cons(X, XS)) → mark(XS)
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(a__incr(x1)) = 2·x1
POL(a__oddNs) = 0
POL(a__pairNs) = 0
POL(a__repItems(x1)) = 2·x1
POL(a__tail(x1)) = 1 + x1
POL(a__take(x1, x2)) = x1 + x2
POL(a__zip(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = 2·x1 + x2
POL(incr(x1)) = 2·x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(oddNs) = 0
POL(pair(x1, x2)) = x1 + x2
POL(pairNs) = 0
POL(repItems(x1)) = 2·x1
POL(s(x1)) = 2·x1
POL(tail(x1)) = 1 + x1
POL(take(x1, x2)) = x1 + x2
POL(zip(x1, x2)) = x1 + x2
a__tail(cons(X, XS)) → mark(XS)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
A__ODDNS → A__INCR(a__pairNs)
A__ODDNS → A__PAIRNS
A__INCR(cons(X, XS)) → MARK(X)
A__TAKE(s(N), cons(X, XS)) → MARK(X)
A__ZIP(cons(X, XS), cons(Y, YS)) → MARK(X)
A__ZIP(cons(X, XS), cons(Y, YS)) → MARK(Y)
A__REPITEMS(cons(X, XS)) → MARK(X)
MARK(pairNs) → A__PAIRNS
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(oddNs) → A__ODDNS
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(zip(X1, X2)) → A__ZIP(mark(X1), mark(X2))
MARK(zip(X1, X2)) → MARK(X1)
MARK(zip(X1, X2)) → MARK(X2)
MARK(tail(X)) → A__TAIL(mark(X))
MARK(tail(X)) → MARK(X)
MARK(repItems(X)) → A__REPITEMS(mark(X))
MARK(repItems(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
A__INCR(cons(X, XS)) → MARK(X)
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(oddNs) → A__ODDNS
A__ODDNS → A__INCR(a__pairNs)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(N), cons(X, XS)) → MARK(X)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(zip(X1, X2)) → A__ZIP(mark(X1), mark(X2))
A__ZIP(cons(X, XS), cons(Y, YS)) → MARK(X)
MARK(zip(X1, X2)) → MARK(X1)
MARK(zip(X1, X2)) → MARK(X2)
MARK(tail(X)) → MARK(X)
MARK(repItems(X)) → A__REPITEMS(mark(X))
A__REPITEMS(cons(X, XS)) → MARK(X)
MARK(repItems(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
A__ZIP(cons(X, XS), cons(Y, YS)) → MARK(Y)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(N), cons(X, XS)) → MARK(X)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(zip(X1, X2)) → A__ZIP(mark(X1), mark(X2))
MARK(zip(X1, X2)) → MARK(X1)
MARK(zip(X1, X2)) → MARK(X2)
MARK(tail(X)) → MARK(X)
POL(0) = 0
POL(A__INCR(x1)) = x1
POL(A__ODDNS) = 0
POL(A__REPITEMS(x1)) = x1
POL(A__TAKE(x1, x2)) = 2 + 2·x1 + 2·x2
POL(A__ZIP(x1, x2)) = x1 + x2
POL(MARK(x1)) = 2·x1
POL(a__incr(x1)) = 2·x1
POL(a__oddNs) = 0
POL(a__pairNs) = 0
POL(a__repItems(x1)) = 2·x1
POL(a__tail(x1)) = 1 + x1
POL(a__take(x1, x2)) = 2 + 2·x1 + x2
POL(a__zip(x1, x2)) = 2 + x1 + 2·x2
POL(cons(x1, x2)) = 2·x1 + x2
POL(incr(x1)) = 2·x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(oddNs) = 0
POL(pair(x1, x2)) = x1 + 2·x2
POL(pairNs) = 0
POL(repItems(x1)) = 2·x1
POL(s(x1)) = x1
POL(tail(x1)) = 1 + x1
POL(take(x1, x2)) = 2 + 2·x1 + x2
POL(zip(x1, x2)) = 2 + x1 + 2·x2
A__INCR(cons(X, XS)) → MARK(X)
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(oddNs) → A__ODDNS
A__ODDNS → A__INCR(a__pairNs)
A__ZIP(cons(X, XS), cons(Y, YS)) → MARK(X)
MARK(repItems(X)) → A__REPITEMS(mark(X))
A__REPITEMS(cons(X, XS)) → MARK(X)
MARK(repItems(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
A__ZIP(cons(X, XS), cons(Y, YS)) → MARK(Y)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
MARK(incr(X)) → A__INCR(mark(X))
A__INCR(cons(X, XS)) → MARK(X)
MARK(incr(X)) → MARK(X)
MARK(oddNs) → A__ODDNS
A__ODDNS → A__INCR(a__pairNs)
MARK(repItems(X)) → A__REPITEMS(mark(X))
A__REPITEMS(cons(X, XS)) → MARK(X)
MARK(repItems(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
MARK(repItems(X)) → A__REPITEMS(mark(X))
A__REPITEMS(cons(X, XS)) → MARK(X)
MARK(repItems(X)) → MARK(X)
POL(0) = 0
POL(A__INCR(x1)) = x1
POL(A__ODDNS) = 0
POL(A__REPITEMS(x1)) = 1 + 2·x1
POL(MARK(x1)) = x1
POL(a__incr(x1)) = 2·x1
POL(a__oddNs) = 0
POL(a__pairNs) = 0
POL(a__repItems(x1)) = 2 + 2·x1
POL(a__tail(x1)) = 2·x1
POL(a__take(x1, x2)) = 2·x1 + 2·x2
POL(a__zip(x1, x2)) = x1 + 2·x2
POL(cons(x1, x2)) = x1 + x2
POL(incr(x1)) = 2·x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(oddNs) = 0
POL(pair(x1, x2)) = x1 + x2
POL(pairNs) = 0
POL(repItems(x1)) = 2 + 2·x1
POL(s(x1)) = 2·x1
POL(tail(x1)) = 2·x1
POL(take(x1, x2)) = 2·x1 + 2·x2
POL(zip(x1, x2)) = x1 + 2·x2
MARK(incr(X)) → A__INCR(mark(X))
A__INCR(cons(X, XS)) → MARK(X)
MARK(incr(X)) → MARK(X)
MARK(oddNs) → A__ODDNS
A__ODDNS → A__INCR(a__pairNs)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
A__ODDNS → A__INCR(a__pairNs)
POL(0) = 0
POL(A__INCR(x1)) = x1
POL(A__ODDNS) = 2
POL(MARK(x1)) = 2·x1
POL(a__incr(x1)) = x1
POL(a__oddNs) = 1
POL(a__pairNs) = 1
POL(a__repItems(x1)) = 2·x1
POL(a__tail(x1)) = x1
POL(a__take(x1, x2)) = 2·x1 + x2
POL(a__zip(x1, x2)) = x1 + 2·x2
POL(cons(x1, x2)) = 2·x1 + x2
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(oddNs) = 1
POL(pair(x1, x2)) = x1 + 2·x2
POL(pairNs) = 1
POL(repItems(x1)) = 2·x1
POL(s(x1)) = x1
POL(tail(x1)) = x1
POL(take(x1, x2)) = 2·x1 + x2
POL(zip(x1, x2)) = x1 + 2·x2
MARK(incr(X)) → A__INCR(mark(X))
A__INCR(cons(X, XS)) → MARK(X)
MARK(incr(X)) → MARK(X)
MARK(oddNs) → A__ODDNS
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
A__INCR(cons(X, XS)) → MARK(X)
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
POL(0) = 0
POL(A__INCR(x1)) = x1
POL(MARK(x1)) = x1
POL(a__incr(x1)) = x1
POL(a__oddNs) = 1
POL(a__pairNs) = 1
POL(a__repItems(x1)) = 1 + x1
POL(a__tail(x1)) = 0
POL(a__take(x1, x2)) = x1 + x2
POL(a__zip(x1, x2)) = 1 + x1 + x2
POL(cons(x1, x2)) = x1
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(oddNs) = 1
POL(pair(x1, x2)) = 1 + x1 + x2
POL(pairNs) = 1
POL(repItems(x1)) = 1 + x1
POL(s(x1)) = x1
POL(tail(x1)) = 0
POL(take(x1, x2)) = x1 + x2
POL(zip(x1, x2)) = 1 + x1 + x2
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(oddNs) → a__oddNs
mark(incr(X)) → a__incr(mark(X))
mark(pairNs) → a__pairNs
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__oddNs → a__incr(a__pairNs)
a__pairNs → cons(0, incr(oddNs))
A__INCR(cons(X, XS)) → MARK(X)
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(s(X)) → MARK(X)
POL(0) = 0
POL(A__INCR(x1)) = x1
POL(MARK(x1)) = x1
POL(a__incr(x1)) = 1 + x1
POL(a__oddNs) = 1
POL(a__pairNs) = 0
POL(a__repItems(x1)) = x1
POL(a__tail(x1)) = x1
POL(a__take(x1, x2)) = x1 + x2
POL(a__zip(x1, x2)) = 1 + x2
POL(cons(x1, x2)) = x1
POL(incr(x1)) = 1 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(oddNs) = 1
POL(pair(x1, x2)) = 0
POL(pairNs) = 0
POL(repItems(x1)) = x1
POL(s(x1)) = 1 + x1
POL(tail(x1)) = x1
POL(take(x1, x2)) = x1 + x2
POL(zip(x1, x2)) = 1 + x2
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(oddNs) → a__oddNs
mark(incr(X)) → a__incr(mark(X))
mark(pairNs) → a__pairNs
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__oddNs → a__incr(a__pairNs)
a__pairNs → cons(0, incr(oddNs))
A__INCR(cons(X, XS)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
MARK(cons(X1, X2)) → MARK(X1)
a__pairNs → cons(0, incr(oddNs))
a__oddNs → a__incr(a__pairNs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__zip(cons(X, XS), cons(Y, YS)) → cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__repItems(cons(X, XS)) → cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) → a__pairNs
mark(incr(X)) → a__incr(mark(X))
mark(oddNs) → a__oddNs
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) → a__zip(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(repItems(X)) → a__repItems(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
a__pairNs → pairNs
a__incr(X) → incr(X)
a__oddNs → oddNs
a__take(X1, X2) → take(X1, X2)
a__zip(X1, X2) → zip(X1, X2)
a__tail(X) → tail(X)
a__repItems(X) → repItems(X)
MARK(cons(X1, X2)) → MARK(X1)
From the DPs we obtained the following set of size-change graphs: