0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
↳7 QDP
↳8 UsableRulesProof (⇔)
↳9 QDP
↳10 QReductionProof (⇔)
↳11 QDP
↳12 QDPSizeChangeProof (⇔)
↳13 TRUE
↳14 QDP
↳15 UsableRulesProof (⇔)
↳16 QDP
↳17 QReductionProof (⇔)
↳18 QDP
↳19 QDPOrderProof (⇔)
↳20 QDP
↳21 PisEmptyProof (⇔)
↳22 TRUE
↳23 QDP
↳24 UsableRulesProof (⇔)
↳25 QDP
↳26 QReductionProof (⇔)
↳27 QDP
↳28 QDPOrderProof (⇔)
↳29 QDP
↳30 PisEmptyProof (⇔)
↳31 TRUE
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0)) → 0
log(s(s(x))) → s(log(s(quot(x, s(s(0))))))
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0)) → 0
log(s(s(x))) → s(log(s(quot(x, s(s(0))))))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
log(s(0))
log(s(s(x0)))
MINUS(x, s(y)) → PRED(minus(x, y))
MINUS(x, s(y)) → MINUS(x, y)
QUOT(s(x), s(y)) → QUOT(minus(x, y), s(y))
QUOT(s(x), s(y)) → MINUS(x, y)
LOG(s(s(x))) → LOG(s(quot(x, s(s(0)))))
LOG(s(s(x))) → QUOT(x, s(s(0)))
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0)) → 0
log(s(s(x))) → s(log(s(quot(x, s(s(0))))))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
log(s(0))
log(s(s(x0)))
MINUS(x, s(y)) → MINUS(x, y)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0)) → 0
log(s(s(x))) → s(log(s(quot(x, s(s(0))))))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
log(s(0))
log(s(s(x0)))
MINUS(x, s(y)) → MINUS(x, y)
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
log(s(0))
log(s(s(x0)))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
log(s(0))
log(s(s(x0)))
MINUS(x, s(y)) → MINUS(x, y)
From the DPs we obtained the following set of size-change graphs:
QUOT(s(x), s(y)) → QUOT(minus(x, y), s(y))
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0)) → 0
log(s(s(x))) → s(log(s(quot(x, s(s(0))))))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
log(s(0))
log(s(s(x0)))
QUOT(s(x), s(y)) → QUOT(minus(x, y), s(y))
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
log(s(0))
log(s(s(x0)))
quot(0, s(x0))
quot(s(x0), s(x1))
log(s(0))
log(s(s(x0)))
QUOT(s(x), s(y)) → QUOT(minus(x, y), s(y))
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
QUOT(s(x), s(y)) → QUOT(minus(x, y), s(y))
POL(0) = 0
POL(QUOT(x1, x2)) = x1
POL(minus(x1, x2)) = x1
POL(pred(x1)) = x1
POL(s(x1)) = 1 + x1
pred(s(x)) → x
minus(x, s(y)) → pred(minus(x, y))
minus(x, 0) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
LOG(s(s(x))) → LOG(s(quot(x, s(s(0)))))
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0)) → 0
log(s(s(x))) → s(log(s(quot(x, s(s(0))))))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
log(s(0))
log(s(s(x0)))
LOG(s(s(x))) → LOG(s(quot(x, s(s(0)))))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
log(s(0))
log(s(s(x0)))
log(s(0))
log(s(s(x0)))
LOG(s(s(x))) → LOG(s(quot(x, s(s(0)))))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
LOG(s(s(x))) → LOG(s(quot(x, s(s(0)))))
POL(0) = 0
POL(LOG(x1)) = x1
POL(minus(x1, x2)) = x1
POL(pred(x1)) = x1
POL(quot(x1, x2)) = x1
POL(s(x1)) = 1 + x1
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
minus(x, 0) → x
quot(0, s(y)) → 0
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))