0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
↳7 QDP
↳8 QDPOrderProof (⇔)
↳9 QDP
↳10 PisEmptyProof (⇔)
↳11 TRUE
↳12 QDP
↳13 QDPOrderProof (⇔)
↳14 QDP
↳15 PisEmptyProof (⇔)
↳16 TRUE
↳17 QDP
↳18 QDP
↳19 QDPOrderProof (⇔)
↳20 QDP
↳21 PisEmptyProof (⇔)
↳22 TRUE
↳23 QDP
↳24 QDPOrderProof (⇔)
↳25 QDP
↳26 PisEmptyProof (⇔)
↳27 TRUE
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)
APP(x, y) → HELPA(0, plus(length(x), length(y)), x, y)
APP(x, y) → PLUS(length(x), length(y))
APP(x, y) → LENGTH(x)
APP(x, y) → LENGTH(y)
PLUS(x, s(y)) → PLUS(x, y)
LENGTH(cons(x, y)) → LENGTH(y)
HELPA(c, l, ys, zs) → IF(ge(c, l), c, l, ys, zs)
HELPA(c, l, ys, zs) → GE(c, l)
GE(s(x), s(y)) → GE(x, y)
IF(false, c, l, ys, zs) → HELPB(c, l, ys, zs)
TAKE(s(c), cons(x, xs), ys) → TAKE(c, xs, ys)
TAKE(s(c), nil, cons(y, ys)) → TAKE(c, nil, ys)
HELPB(c, l, ys, zs) → TAKE(c, ys, zs)
HELPB(c, l, ys, zs) → HELPA(s(c), l, ys, zs)
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)
TAKE(s(c), nil, cons(y, ys)) → TAKE(c, nil, ys)
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TAKE(s(c), nil, cons(y, ys)) → TAKE(c, nil, ys)
trivial
trivial
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)
GE(s(x), s(y)) → GE(x, y)
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
GE(s(x), s(y)) → GE(x, y)
trivial
trivial
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)
IF(false, c, l, ys, zs) → HELPB(c, l, ys, zs)
HELPB(c, l, ys, zs) → HELPA(s(c), l, ys, zs)
HELPA(c, l, ys, zs) → IF(ge(c, l), c, l, ys, zs)
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)
LENGTH(cons(x, y)) → LENGTH(y)
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
LENGTH(cons(x, y)) → LENGTH(y)
trivial
trivial
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)
PLUS(x, s(y)) → PLUS(x, y)
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PLUS(x, s(y)) → PLUS(x, y)
trivial
trivial
app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, ys, zs)
take(0, cons(x, xs), ys) → x
take(0, nil, cons(y, ys)) → y
take(s(c), cons(x, xs), ys) → take(c, xs, ys)
take(s(c), nil, cons(y, ys)) → take(c, nil, ys)
helpb(c, l, ys, zs) → cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
app(x0, x1)
plus(x0, 0)
plus(x0, s(x1))
length(nil)
length(cons(x0, x1))
helpa(x0, x1, x2, x3)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
if(true, x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
take(0, cons(x0, xs), x1)
take(0, nil, cons(x0, x1))
take(s(x0), cons(x1, xs), x2)
take(s(x0), nil, cons(x1, x2))
helpb(x0, x1, x2, x3)