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))
↳ QTRS
↳ DependencyPairsProof
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))
LENGTH(cons(x, y)) → LENGTH(y)
APP(x, y) → PLUS(length(x), length(y))
HELPA(c, l, ys, zs) → GE(c, l)
PLUS(x, s(y)) → PLUS(x, y)
HELPB(c, l, ys, zs) → TAKE(c, ys, zs)
TAKE(s(c), nil, cons(y, ys)) → TAKE(c, nil, ys)
HELPB(c, l, ys, zs) → HELPA(s(c), l, ys, zs)
APP(x, y) → LENGTH(x)
APP(x, y) → HELPA(0, plus(length(x), length(y)), x, y)
TAKE(s(c), cons(x, xs), ys) → TAKE(c, xs, ys)
HELPA(c, l, ys, zs) → IF(ge(c, l), c, l, ys, zs)
IF(false, c, l, ys, zs) → HELPB(c, l, ys, zs)
APP(x, y) → LENGTH(y)
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))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
LENGTH(cons(x, y)) → LENGTH(y)
APP(x, y) → PLUS(length(x), length(y))
HELPA(c, l, ys, zs) → GE(c, l)
PLUS(x, s(y)) → PLUS(x, y)
HELPB(c, l, ys, zs) → TAKE(c, ys, zs)
TAKE(s(c), nil, cons(y, ys)) → TAKE(c, nil, ys)
HELPB(c, l, ys, zs) → HELPA(s(c), l, ys, zs)
APP(x, y) → LENGTH(x)
APP(x, y) → HELPA(0, plus(length(x), length(y)), x, y)
TAKE(s(c), cons(x, xs), ys) → TAKE(c, xs, ys)
HELPA(c, l, ys, zs) → IF(ge(c, l), c, l, ys, zs)
IF(false, c, l, ys, zs) → HELPB(c, l, ys, zs)
APP(x, y) → LENGTH(y)
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))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
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))
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)
The value of delta used in the strict ordering is 1/8.
POL(cons(x1, x2)) = 1/2 + (4)x_1 + (4)x_2
POL(TAKE(x1, x2, x3)) = (1/4)x_3
POL(s(x1)) = 0
POL(nil) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
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))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
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))
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)
The value of delta used in the strict ordering is 1.
POL(GE(x1, x2)) = x_2
POL(s(x1)) = 1 + x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
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))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
HELPA(c, l, ys, zs) → IF(ge(c, l), c, l, ys, zs)
IF(false, c, l, ys, zs) → HELPB(c, l, 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))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
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))
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)
The value of delta used in the strict ordering is 1.
POL(LENGTH(x1)) = (4)x_1
POL(cons(x1, x2)) = 1/4 + (4)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
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))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
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))
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)
The value of delta used in the strict ordering is 8.
POL(PLUS(x1, x2)) = (2)x_2
POL(s(x1)) = 4 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
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))