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 QDPSizeChangeProof (⇔)
↳27 TRUE
↳28 QDP
↳29 UsableRulesProof (⇔)
↳30 QDP
↳31 QReductionProof (⇔)
↳32 QDP
↳33 QDPSizeChangeProof (⇔)
↳34 TRUE
↳35 QDP
↳36 UsableRulesProof (⇔)
↳37 QDP
↳38 QReductionProof (⇔)
↳39 QDP
↳40 QDPOrderProof (⇔)
↳41 QDP
↳42 PisEmptyProof (⇔)
↳43 TRUE
↳44 QDP
↳45 UsableRulesProof (⇔)
↳46 QDP
↳47 QReductionProof (⇔)
↳48 QDP
↳49 QDPSizeChangeProof (⇔)
↳50 TRUE
↳51 QDP
↳52 UsableRulesProof (⇔)
↳53 QDP
↳54 QReductionProof (⇔)
↳55 QDP
↳56 QDPOrderProof (⇔)
↳57 QDP
↳58 PisEmptyProof (⇔)
↳59 TRUE
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
quicksort(nil) → nil
quicksort(add(n, x)) → app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
quicksort(nil) → nil
quicksort(add(n, x)) → app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
MINUS(s(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)
LE(s(x), s(y)) → LE(x, y)
APP(add(n, x), y) → APP(x, y)
LOW(n, add(m, x)) → IF_LOW(le(m, n), n, add(m, x))
LOW(n, add(m, x)) → LE(m, n)
IF_LOW(true, n, add(m, x)) → LOW(n, x)
IF_LOW(false, n, add(m, x)) → LOW(n, x)
HIGH(n, add(m, x)) → IF_HIGH(le(m, n), n, add(m, x))
HIGH(n, add(m, x)) → LE(m, n)
IF_HIGH(true, n, add(m, x)) → HIGH(n, x)
IF_HIGH(false, n, add(m, x)) → HIGH(n, x)
QUICKSORT(add(n, x)) → APP(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
QUICKSORT(add(n, x)) → QUICKSORT(low(n, x))
QUICKSORT(add(n, x)) → LOW(n, x)
QUICKSORT(add(n, x)) → QUICKSORT(high(n, x))
QUICKSORT(add(n, x)) → HIGH(n, x)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
quicksort(nil) → nil
quicksort(add(n, x)) → app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
APP(add(n, x), y) → APP(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
quicksort(nil) → nil
quicksort(add(n, x)) → app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
APP(add(n, x), y) → APP(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
APP(add(n, x), y) → APP(x, y)
From the DPs we obtained the following set of size-change graphs:
LE(s(x), s(y)) → LE(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
quicksort(nil) → nil
quicksort(add(n, x)) → app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
LE(s(x), s(y)) → LE(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
LE(s(x), s(y)) → LE(x, y)
From the DPs we obtained the following set of size-change graphs:
HIGH(n, add(m, x)) → IF_HIGH(le(m, n), n, add(m, x))
IF_HIGH(true, n, add(m, x)) → HIGH(n, x)
IF_HIGH(false, n, add(m, x)) → HIGH(n, x)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
quicksort(nil) → nil
quicksort(add(n, x)) → app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
HIGH(n, add(m, x)) → IF_HIGH(le(m, n), n, add(m, x))
IF_HIGH(true, n, add(m, x)) → HIGH(n, x)
IF_HIGH(false, n, add(m, x)) → HIGH(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
HIGH(n, add(m, x)) → IF_HIGH(le(m, n), n, add(m, x))
IF_HIGH(true, n, add(m, x)) → HIGH(n, x)
IF_HIGH(false, n, add(m, x)) → HIGH(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
From the DPs we obtained the following set of size-change graphs:
LOW(n, add(m, x)) → IF_LOW(le(m, n), n, add(m, x))
IF_LOW(true, n, add(m, x)) → LOW(n, x)
IF_LOW(false, n, add(m, x)) → LOW(n, x)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
quicksort(nil) → nil
quicksort(add(n, x)) → app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
LOW(n, add(m, x)) → IF_LOW(le(m, n), n, add(m, x))
IF_LOW(true, n, add(m, x)) → LOW(n, x)
IF_LOW(false, n, add(m, x)) → LOW(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
LOW(n, add(m, x)) → IF_LOW(le(m, n), n, add(m, x))
IF_LOW(true, n, add(m, x)) → LOW(n, x)
IF_LOW(false, n, add(m, x)) → LOW(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
From the DPs we obtained the following set of size-change graphs:
QUICKSORT(add(n, x)) → QUICKSORT(high(n, x))
QUICKSORT(add(n, x)) → QUICKSORT(low(n, x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
quicksort(nil) → nil
quicksort(add(n, x)) → app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
QUICKSORT(add(n, x)) → QUICKSORT(high(n, x))
QUICKSORT(add(n, x)) → QUICKSORT(low(n, x))
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
if_low(true, n, add(m, x)) → add(m, low(n, x))
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(false, n, add(m, x)) → add(m, high(n, x))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
quicksort(nil)
quicksort(add(x0, x1))
QUICKSORT(add(n, x)) → QUICKSORT(high(n, x))
QUICKSORT(add(n, x)) → QUICKSORT(low(n, x))
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
if_low(true, n, add(m, x)) → add(m, low(n, x))
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(false, n, add(m, x)) → add(m, high(n, x))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
QUICKSORT(add(n, x)) → QUICKSORT(high(n, x))
QUICKSORT(add(n, x)) → QUICKSORT(low(n, x))
POL(0) = 0
POL(QUICKSORT(x1)) = x1
POL(add(x1, x2)) = 1 + x2
POL(false) = 0
POL(high(x1, x2)) = x2
POL(if_high(x1, x2, x3)) = x3
POL(if_low(x1, x2, x3)) = x3
POL(le(x1, x2)) = 0
POL(low(x1, x2)) = x2
POL(nil) = 0
POL(s(x1)) = 0
POL(true) = 0
high(n, nil) → nil
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
low(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(false, n, add(m, x)) → add(m, high(n, x))
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
if_low(true, n, add(m, x)) → add(m, low(n, x))
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(false, n, add(m, x)) → add(m, high(n, x))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
MINUS(s(x), s(y)) → MINUS(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
quicksort(nil) → nil
quicksort(add(n, x)) → app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
MINUS(s(x), s(y)) → MINUS(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
MINUS(s(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))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
quicksort(nil) → nil
quicksort(add(n, x)) → app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
QUOT(s(x), s(y)) → QUOT(minus(x, y), s(y))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
QUOT(s(x), s(y)) → QUOT(minus(x, y), s(y))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
minus(x0, 0)
minus(s(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(s(x1)) = 1 + x1
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
minus(x0, 0)
minus(s(x0), s(x1))