cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
↳ QTRS
↳ DependencyPairsProof
cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
COND1(true, x, y, z) → GR(y, z)
COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND2(true, x, y, z) → P(y)
COND2(false, x, y, z) → P(x)
COND2(true, x, y, z) → GR(y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, x, y, z) → GR(x, z)
GR(s(x), s(y)) → GR(x, y)
cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
COND1(true, x, y, z) → GR(y, z)
COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND2(true, x, y, z) → P(y)
COND2(false, x, y, z) → P(x)
COND2(true, x, y, z) → GR(y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, x, y, z) → GR(x, z)
GR(s(x), s(y)) → GR(x, y)
cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
GR(s(x), s(y)) → GR(x, y)
cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
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.
GR(s(x), s(y)) → GR(x, y)
The value of delta used in the strict ordering is 85/16.
POL(GR(x1, x2)) = x_1 + (13/4)x_2
POL(s(x1)) = 5/4 + (15/4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
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, y, z) → COND2(gr(y, z), x, y, z)
Used ordering: Polynomial interpretation [25,35]:
COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
The value of delta used in the strict ordering is 1/8.
POL(COND2(x1, x2, x3, x4)) = (11/4)x_2 + (1/4)x_4
POL(gr(x1, x2)) = (4)x_1
POL(true) = 1/4
POL(false) = 0
POL(p(x1)) = (1/4)x_1
POL(s(x1)) = 3/2 + (4)x_1
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = (1/2)x_1 + (11/4)x_2 + (1/4)x_4
p(s(x)) → x
p(0) → 0
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
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(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
The value of delta used in the strict ordering is 27/16.
POL(COND2(x1, x2, x3, x4)) = (9/4)x_1 + (9/4)x_3
POL(gr(x1, x2)) = 5/4 + (1/4)x_1
POL(true) = 2
POL(false) = 5/4
POL(p(x1)) = (1/4)x_1
POL(s(x1)) = 4 + (4)x_1
POL(0) = 0
p(s(x)) → x
p(0) → 0
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x