cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x
↳ QTRS
↳ DependencyPairsProof
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x
DIV2(s(s(x))) → DIV2(x)
EVEN(s(s(x))) → EVEN(x)
COND1(true, x) → COND2(even(x), x)
NEQ(s(x), s(y)) → NEQ(x, y)
COND2(false, x) → COND1(neq(x, 0), p(x))
COND2(true, x) → COND1(neq(x, 0), div2(x))
COND2(false, x) → NEQ(x, 0)
COND1(true, x) → EVEN(x)
COND2(true, x) → DIV2(x)
COND2(false, x) → P(x)
COND2(true, x) → NEQ(x, 0)
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
DIV2(s(s(x))) → DIV2(x)
EVEN(s(s(x))) → EVEN(x)
COND1(true, x) → COND2(even(x), x)
NEQ(s(x), s(y)) → NEQ(x, y)
COND2(false, x) → COND1(neq(x, 0), p(x))
COND2(true, x) → COND1(neq(x, 0), div2(x))
COND2(false, x) → NEQ(x, 0)
COND1(true, x) → EVEN(x)
COND2(true, x) → DIV2(x)
COND2(false, x) → P(x)
COND2(true, x) → NEQ(x, 0)
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
DIV2(s(s(x))) → DIV2(x)
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
DIV2(s(s(x))) → DIV2(x)
The value of delta used in the strict ordering is 35/8.
POL(DIV2(x1)) = x_1
POL(s(x1)) = 5/4 + (5/2)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
EVEN(s(s(x))) → EVEN(x)
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
EVEN(s(s(x))) → EVEN(x)
The value of delta used in the strict ordering is 35/8.
POL(EVEN(x1)) = x_1
POL(s(x1)) = 5/4 + (5/2)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
COND1(true, x) → COND2(even(x), x)
COND2(false, x) → COND1(neq(x, 0), p(x))
COND2(true, x) → COND1(neq(x, 0), div2(x))
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
COND2(false, x) → COND1(neq(x, 0), p(x))
Used ordering: Polynomial interpretation [25,35]:
COND1(true, x) → COND2(even(x), x)
COND2(true, x) → COND1(neq(x, 0), div2(x))
The value of delta used in the strict ordering is 1.
POL(COND1(x1, x2)) = (1/4)x_1 + (4)x_2
POL(even(x1)) = 0
POL(div2(x1)) = 1/4 + (1/2)x_1
POL(COND2(x1, x2)) = 1 + (15/4)x_2
POL(true) = 4
POL(false) = 0
POL(p(x1)) = (1/2)x_1
POL(s(x1)) = 1 + (4)x_1
POL(0) = 0
POL(neq(x1, x2)) = (4)x_1
neq(s(x), 0) → true
div2(s(0)) → 0
div2(0) → 0
p(0) → 0
div2(s(s(x))) → s(div2(x))
p(s(x)) → x
neq(0, 0) → false
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
COND1(true, x) → COND2(even(x), x)
COND2(true, x) → COND1(neq(x, 0), div2(x))
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
COND1(true, x) → COND2(even(x), x)
Used ordering: Polynomial interpretation [25,35]:
COND2(true, x) → COND1(neq(x, 0), div2(x))
The value of delta used in the strict ordering is 1/8.
POL(COND1(x1, x2)) = 4 + (1/4)x_1 + (1/2)x_2
POL(even(x1)) = (4)x_1
POL(div2(x1)) = (1/2)x_1
POL(COND2(x1, x2)) = 4 + (1/2)x_2
POL(true) = 1/2
POL(false) = 0
POL(s(x1)) = 2 + x_1
POL(0) = 0
POL(neq(x1, x2)) = (1/4)x_1 + (1/4)x_2
neq(s(x), 0) → true
div2(s(0)) → 0
div2(0) → 0
div2(s(s(x))) → s(div2(x))
neq(0, 0) → false
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
COND2(true, x) → COND1(neq(x, 0), div2(x))
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x