0 QTRS
↳1 QTRSToCSRProof (⇔)
↳2 CSR
↳3 PoloCSRProof (⇔)
↳4 CSR
↳5 PoloCSRProof (⇔)
↳6 CSR
↳7 PoloCSRProof (⇔)
↳8 CSR
↳9 PoloCSRProof (⇔)
↳10 CSR
↳11 PoloCSRProof (⇔)
↳12 CSR
↳13 RisEmptyProof (⇔)
↳14 TRUE
active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}
nil: empty set
zip: {1, 2}
pair: {1, 2}
tail: {1}
repItems: {1}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
pairNs → cons(0, incr(oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), incr(XS))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), zip(XS, YS))
tail(cons(X, XS)) → XS
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, cons(X, repItems(XS)))
pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}
nil: empty set
zip: {1, 2}
pair: {1, 2}
tail: {1}
repItems: {1}
Used ordering:
take(0, XS) → nil
zip(X, nil) → nil
tail(cons(X, XS)) → XS
repItems(nil) → nil
POL(0) = 0
POL(cons(x1, x2)) = x1 + x2
POL(incr(x1)) = 2·x1
POL(nil) = 1
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 + 2·x1
POL(take(x1, x2)) = 2 + x1 + 2·x2
POL(zip(x1, x2)) = x1 + 2·x2
pairNs → cons(0, incr(oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), incr(XS))
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
zip(nil, XS) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), zip(XS, YS))
repItems(cons(X, XS)) → cons(X, cons(X, repItems(XS)))
pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}
nil: empty set
zip: {1, 2}
pair: {1, 2}
repItems: {1}
Used ordering:
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), zip(XS, YS))
repItems(cons(X, XS)) → cons(X, cons(X, repItems(XS)))
POL(0) = 0
POL(cons(x1, x2)) = 1 + 2·x1
POL(incr(x1)) = x1
POL(nil) = 0
POL(oddNs) = 1
POL(pair(x1, x2)) = x1 + x2
POL(pairNs) = 1
POL(repItems(x1)) = 2 + 2·x1
POL(s(x1)) = x1
POL(take(x1, x2)) = x1 + x2
POL(zip(x1, x2)) = x1 + x2
pairNs → cons(0, incr(oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), incr(XS))
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
zip(nil, XS) → nil
pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}
nil: empty set
zip: {1, 2}
Used ordering:
oddNs → incr(pairNs)
zip(nil, XS) → nil
POL(0) = 0
POL(cons(x1, x2)) = x1
POL(incr(x1)) = 2·x1
POL(nil) = 2
POL(oddNs) = 2
POL(pairNs) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x1 + 2·x2
POL(zip(x1, x2)) = 2 + x1 + x2
pairNs → cons(0, incr(oddNs))
incr(cons(X, XS)) → cons(s(X), incr(XS))
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}
Used ordering:
incr(cons(X, XS)) → cons(s(X), incr(XS))
POL(0) = 0
POL(cons(x1, x2)) = 2 + x1
POL(incr(x1)) = 2·x1
POL(oddNs) = 0
POL(pairNs) = 2
POL(s(x1)) = 2·x1
POL(take(x1, x2)) = 2·x1 + x2
pairNs → cons(0, incr(oddNs))
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}
Used ordering:
pairNs → cons(0, incr(oddNs))
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
POL(0) = 0
POL(cons(x1, x2)) = 1 + x1 + x2
POL(incr(x1)) = x1
POL(oddNs) = 0
POL(pairNs) = 2
POL(s(x1)) = 2 + 2·x1
POL(take(x1, x2)) = 2 + 2·x1 + 2·x2