min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
↳ QTRS
↳ DependencyPairsProof
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
IF(true, c, xs, ys, z) → MIN(len(xs), len(ys))
IF(true, c, xs, ys, z) → SUM(take(c, xs), take(c, ys))
ADDLIST(x, y) → IF(le(0, min(len(x), len(y))), 0, x, y, nil)
ADDLIST(x, y) → LEN(x)
MIN(s(x), s(y)) → MIN(x, y)
LEN(cons(x, xs)) → LEN(xs)
TAKE(s(x), cons(y, ys)) → TAKE(x, ys)
ADDLIST(x, y) → MIN(len(x), len(y))
ADDLIST(x, y) → LEN(y)
LE(s(x), s(y)) → LE(x, y)
IF(true, c, xs, ys, z) → TAKE(c, ys)
IF(true, c, xs, ys, z) → IF(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
SUM(x, s(y)) → SUM(x, y)
IF(true, c, xs, ys, z) → LEN(xs)
IF(true, c, xs, ys, z) → TAKE(c, xs)
IF(true, c, xs, ys, z) → LEN(ys)
ADDLIST(x, y) → LE(0, min(len(x), len(y)))
IF(true, c, xs, ys, z) → LE(s(c), min(len(xs), len(ys)))
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
IF(true, c, xs, ys, z) → MIN(len(xs), len(ys))
IF(true, c, xs, ys, z) → SUM(take(c, xs), take(c, ys))
ADDLIST(x, y) → IF(le(0, min(len(x), len(y))), 0, x, y, nil)
ADDLIST(x, y) → LEN(x)
MIN(s(x), s(y)) → MIN(x, y)
LEN(cons(x, xs)) → LEN(xs)
TAKE(s(x), cons(y, ys)) → TAKE(x, ys)
ADDLIST(x, y) → MIN(len(x), len(y))
ADDLIST(x, y) → LEN(y)
LE(s(x), s(y)) → LE(x, y)
IF(true, c, xs, ys, z) → TAKE(c, ys)
IF(true, c, xs, ys, z) → IF(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
SUM(x, s(y)) → SUM(x, y)
IF(true, c, xs, ys, z) → LEN(xs)
IF(true, c, xs, ys, z) → TAKE(c, xs)
IF(true, c, xs, ys, z) → LEN(ys)
ADDLIST(x, y) → LE(0, min(len(x), len(y)))
IF(true, c, xs, ys, z) → LE(s(c), min(len(xs), len(ys)))
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
TAKE(s(x), cons(y, ys)) → TAKE(x, ys)
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TAKE(s(x), cons(y, ys)) → TAKE(x, ys)
The value of delta used in the strict ordering is 16.
POL(TAKE(x1, x2)) = (4)x_2
POL(cons(x1, x2)) = 4 + (2)x_1 + (4)x_2
POL(s(x1)) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
LE(s(x), s(y)) → LE(x, y)
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
LE(s(x), s(y)) → LE(x, y)
The value of delta used in the strict ordering is 1.
POL(s(x1)) = 1 + x_1
POL(LE(x1, x2)) = x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
SUM(x, s(y)) → SUM(x, y)
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SUM(x, s(y)) → SUM(x, y)
The value of delta used in the strict ordering is 1/16.
POL(SUM(x1, x2)) = (1/4)x_2
POL(s(x1)) = 1/4 + (2)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
LEN(cons(x, xs)) → LEN(xs)
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
LEN(cons(x, xs)) → LEN(xs)
The value of delta used in the strict ordering is 1.
POL(cons(x1, x2)) = 1/4 + (4)x_2
POL(LEN(x1)) = (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
MIN(s(x), s(y)) → MIN(x, y)
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MIN(s(x), s(y)) → MIN(x, y)
The value of delta used in the strict ordering is 1.
POL(MIN(x1, x2)) = x_2
POL(s(x1)) = 1 + x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
IF(true, c, xs, ys, z) → IF(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))