car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0) → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0) → 0
shorter(nil, y) → true
shorter(cons(x, l), 0) → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0), shorter(l, s(0)), l)
if(true, b, l) → s(0)
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))
↳ QTRS
↳ DependencyPairsProof
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0) → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0) → 0
shorter(nil, y) → true
shorter(cons(x, l), 0) → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0), shorter(l, s(0)), l)
if(true, b, l) → s(0)
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))
IF2(true, l) → CAR(l)
IF2(false, l) → PROD(cons(times(car(l), cadr(l)), cddr(l)))
PLUS(x, y) → IFPLUS(isZero(x), x, y)
IF(false, b, l) → IF2(b, l)
IF2(false, l) → CAR(l)
TIMES(x, y) → ISZERO(x)
IFTIMES(false, x, y) → PLUS(y, times(p(x), y))
PLUS(x, y) → ISZERO(x)
TIMES(x, y) → IFTIMES(isZero(x), x, y)
PROD(l) → SHORTER(l, s(0))
IF2(false, l) → TIMES(car(l), cadr(l))
PROD(l) → SHORTER(l, 0)
IFPLUS(false, x, y) → PLUS(p(x), y)
SHORTER(cons(x, l), s(y)) → SHORTER(l, y)
PROD(l) → IF(shorter(l, 0), shorter(l, s(0)), l)
IF2(false, l) → CADR(l)
IFPLUS(false, x, y) → P(x)
IFTIMES(false, x, y) → P(x)
IFTIMES(false, x, y) → TIMES(p(x), y)
IF2(false, l) → CDDR(l)
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0) → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0) → 0
shorter(nil, y) → true
shorter(cons(x, l), 0) → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0), shorter(l, s(0)), l)
if(true, b, l) → s(0)
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
IF2(true, l) → CAR(l)
IF2(false, l) → PROD(cons(times(car(l), cadr(l)), cddr(l)))
PLUS(x, y) → IFPLUS(isZero(x), x, y)
IF(false, b, l) → IF2(b, l)
IF2(false, l) → CAR(l)
TIMES(x, y) → ISZERO(x)
IFTIMES(false, x, y) → PLUS(y, times(p(x), y))
PLUS(x, y) → ISZERO(x)
TIMES(x, y) → IFTIMES(isZero(x), x, y)
PROD(l) → SHORTER(l, s(0))
IF2(false, l) → TIMES(car(l), cadr(l))
PROD(l) → SHORTER(l, 0)
IFPLUS(false, x, y) → PLUS(p(x), y)
SHORTER(cons(x, l), s(y)) → SHORTER(l, y)
PROD(l) → IF(shorter(l, 0), shorter(l, s(0)), l)
IF2(false, l) → CADR(l)
IFPLUS(false, x, y) → P(x)
IFTIMES(false, x, y) → P(x)
IFTIMES(false, x, y) → TIMES(p(x), y)
IF2(false, l) → CDDR(l)
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0) → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0) → 0
shorter(nil, y) → true
shorter(cons(x, l), 0) → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0), shorter(l, s(0)), l)
if(true, b, l) → s(0)
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
SHORTER(cons(x, l), s(y)) → SHORTER(l, y)
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0) → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0) → 0
shorter(nil, y) → true
shorter(cons(x, l), 0) → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0), shorter(l, s(0)), l)
if(true, b, l) → s(0)
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SHORTER(cons(x, l), s(y)) → SHORTER(l, y)
The value of delta used in the strict ordering is 4.
POL(cons(x1, x2)) = 1 + (4)x_2
POL(s(x1)) = (4)x_1
POL(SHORTER(x1, x2)) = (4)x_1 + (3)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0) → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0) → 0
shorter(nil, y) → true
shorter(cons(x, l), 0) → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0), shorter(l, s(0)), l)
if(true, b, l) → s(0)
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
PLUS(x, y) → IFPLUS(isZero(x), x, y)
IFPLUS(false, x, y) → PLUS(p(x), y)
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0) → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0) → 0
shorter(nil, y) → true
shorter(cons(x, l), 0) → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0), shorter(l, s(0)), l)
if(true, b, l) → s(0)
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
TIMES(x, y) → IFTIMES(isZero(x), x, y)
IFTIMES(false, x, y) → TIMES(p(x), y)
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0) → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0) → 0
shorter(nil, y) → true
shorter(cons(x, l), 0) → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0), shorter(l, s(0)), l)
if(true, b, l) → s(0)
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
IF2(false, l) → PROD(cons(times(car(l), cadr(l)), cddr(l)))
IF(false, b, l) → IF2(b, l)
PROD(l) → IF(shorter(l, 0), shorter(l, s(0)), l)
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0) → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0) → 0
shorter(nil, y) → true
shorter(cons(x, l), 0) → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0), shorter(l, s(0)), l)
if(true, b, l) → s(0)
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))