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 QDPSizeChangeProof (⇔)
↳20 TRUE
↳21 QDP
↳22 UsableRulesProof (⇔)
↳23 QDP
↳24 QReductionProof (⇔)
↳25 QDP
↳26 Narrowing (⇔)
↳27 QDP
↳28 DependencyGraphProof (⇔)
↳29 QDP
↳30 Narrowing (⇔)
↳31 QDP
↳32 DependencyGraphProof (⇔)
↳33 QDP
↳34 Narrowing (⇔)
↳35 QDP
↳36 Instantiation (⇔)
↳37 QDP
↳38 DependencyGraphProof (⇔)
↳39 QDP
↳40 ForwardInstantiation (⇔)
↳41 QDP
↳42 DependencyGraphProof (⇔)
↳43 QDP
↳44 QDPOrderProof (⇔)
↳45 QDP
↳46 DependencyGraphProof (⇔)
↳47 TRUE
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, y) → if_quot(minus(x, y), y, le(y, 0), le(y, x))
if_quot(x, y, true, z) → divByZeroError
if_quot(x, y, false, true) → s(quot(x, y))
if_quot(x, y, false, false) → 0
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, y) → if_quot(minus(x, y), y, le(y, 0), le(y, x))
if_quot(x, y, true, z) → divByZeroError
if_quot(x, y, false, true) → s(quot(x, y))
if_quot(x, y, false, false) → 0
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
MINUS(s(x), s(y)) → MINUS(x, y)
LE(s(x), s(y)) → LE(x, y)
QUOT(x, y) → IF_QUOT(minus(x, y), y, le(y, 0), le(y, x))
QUOT(x, y) → MINUS(x, y)
QUOT(x, y) → LE(y, 0)
QUOT(x, y) → LE(y, x)
IF_QUOT(x, y, false, true) → QUOT(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, y) → if_quot(minus(x, y), y, le(y, 0), le(y, x))
if_quot(x, y, true, z) → divByZeroError
if_quot(x, y, false, true) → s(quot(x, y))
if_quot(x, y, false, false) → 0
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
LE(s(x), s(y)) → LE(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, y) → if_quot(minus(x, y), y, le(y, 0), le(y, x))
if_quot(x, y, true, z) → divByZeroError
if_quot(x, y, false, true) → s(quot(x, y))
if_quot(x, y, false, false) → 0
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
LE(s(x), s(y)) → LE(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
LE(s(x), s(y)) → LE(x, y)
From the DPs we obtained the following set of size-change graphs:
MINUS(s(x), s(y)) → MINUS(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, y) → if_quot(minus(x, y), y, le(y, 0), le(y, x))
if_quot(x, y, true, z) → divByZeroError
if_quot(x, y, false, true) → s(quot(x, y))
if_quot(x, y, false, false) → 0
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
MINUS(s(x), s(y)) → MINUS(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
MINUS(s(x), s(y)) → MINUS(x, y)
From the DPs we obtained the following set of size-change graphs:
QUOT(x, y) → IF_QUOT(minus(x, y), y, le(y, 0), le(y, x))
IF_QUOT(x, y, false, true) → QUOT(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, y) → if_quot(minus(x, y), y, le(y, 0), le(y, x))
if_quot(x, y, true, z) → divByZeroError
if_quot(x, y, false, true) → s(quot(x, y))
if_quot(x, y, false, false) → 0
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
QUOT(x, y) → IF_QUOT(minus(x, y), y, le(y, 0), le(y, x))
IF_QUOT(x, y, false, true) → QUOT(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
quot(x0, x1)
if_quot(x0, x1, true, x2)
if_quot(x0, x1, false, true)
if_quot(x0, x1, false, false)
QUOT(x, y) → IF_QUOT(minus(x, y), y, le(y, 0), le(y, x))
IF_QUOT(x, y, false, true) → QUOT(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
QUOT(y0, 0) → IF_QUOT(minus(y0, 0), 0, true, le(0, y0))
QUOT(y0, s(x0)) → IF_QUOT(minus(y0, s(x0)), s(x0), false, le(s(x0), y0))
IF_QUOT(x, y, false, true) → QUOT(x, y)
QUOT(y0, 0) → IF_QUOT(minus(y0, 0), 0, true, le(0, y0))
QUOT(y0, s(x0)) → IF_QUOT(minus(y0, s(x0)), s(x0), false, le(s(x0), y0))
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
QUOT(y0, s(x0)) → IF_QUOT(minus(y0, s(x0)), s(x0), false, le(s(x0), y0))
IF_QUOT(x, y, false, true) → QUOT(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
QUOT(0, s(x0)) → IF_QUOT(minus(0, s(x0)), s(x0), false, false)
QUOT(s(x1), s(x0)) → IF_QUOT(minus(s(x1), s(x0)), s(x0), false, le(x0, x1))
IF_QUOT(x, y, false, true) → QUOT(x, y)
QUOT(0, s(x0)) → IF_QUOT(minus(0, s(x0)), s(x0), false, false)
QUOT(s(x1), s(x0)) → IF_QUOT(minus(s(x1), s(x0)), s(x0), false, le(x0, x1))
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
QUOT(s(x1), s(x0)) → IF_QUOT(minus(s(x1), s(x0)), s(x0), false, le(x0, x1))
IF_QUOT(x, y, false, true) → QUOT(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
QUOT(s(y0), s(y0)) → IF_QUOT(0, s(y0), false, le(y0, y0))
QUOT(s(x0), s(x1)) → IF_QUOT(minus(x0, x1), s(x1), false, le(x1, x0))
IF_QUOT(x, y, false, true) → QUOT(x, y)
QUOT(s(y0), s(y0)) → IF_QUOT(0, s(y0), false, le(y0, y0))
QUOT(s(x0), s(x1)) → IF_QUOT(minus(x0, x1), s(x1), false, le(x1, x0))
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
IF_QUOT(0, s(z0), false, true) → QUOT(0, s(z0))
IF_QUOT(y_0, s(z1), false, true) → QUOT(y_0, s(z1))
QUOT(s(y0), s(y0)) → IF_QUOT(0, s(y0), false, le(y0, y0))
QUOT(s(x0), s(x1)) → IF_QUOT(minus(x0, x1), s(x1), false, le(x1, x0))
IF_QUOT(0, s(z0), false, true) → QUOT(0, s(z0))
IF_QUOT(y_0, s(z1), false, true) → QUOT(y_0, s(z1))
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
IF_QUOT(y_0, s(z1), false, true) → QUOT(y_0, s(z1))
QUOT(s(y0), s(y0)) → IF_QUOT(0, s(y0), false, le(y0, y0))
QUOT(s(x0), s(x1)) → IF_QUOT(minus(x0, x1), s(x1), false, le(x1, x0))
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
IF_QUOT(s(y_0), s(x1), false, true) → QUOT(s(y_0), s(x1))
QUOT(s(y0), s(y0)) → IF_QUOT(0, s(y0), false, le(y0, y0))
QUOT(s(x0), s(x1)) → IF_QUOT(minus(x0, x1), s(x1), false, le(x1, x0))
IF_QUOT(s(y_0), s(x1), false, true) → QUOT(s(y_0), s(x1))
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
QUOT(s(x0), s(x1)) → IF_QUOT(minus(x0, x1), s(x1), false, le(x1, x0))
IF_QUOT(s(y_0), s(x1), false, true) → QUOT(s(y_0), s(x1))
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
IF_QUOT(s(y_0), s(x1), false, true) → QUOT(s(y_0), s(x1))
POL(0) = 0
POL(IF_QUOT(x1, x2, x3, x4)) = x1 + x2 + x3
POL(QUOT(x1, x2)) = x1 + x2
POL(false) = 1
POL(le(x1, x2)) = 0
POL(minus(x1, x2)) = x1
POL(s(x1)) = 1 + x1
POL(true) = 0
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
QUOT(s(x0), s(x1)) → IF_QUOT(minus(x0, x1), s(x1), false, le(x1, x0))
minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, x0)
minus(0, x0)
minus(x0, 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))