0 QTRS
↳1 AAECC Innermost (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
↳7 QDP
↳8 UsableRulesProof (⇔)
↳9 QDP
↳10 QReductionProof (⇔)
↳11 QDP
↳12 QDPSizeChangeProof (⇔)
↳13 TRUE
↳14 QDP
↳15 UsableRulesProof (⇔)
↳16 QDP
↳17 QReductionProof (⇔)
↳18 QDP
↳19 QDPOrderProof (⇔)
↳20 QDP
↳21 DependencyGraphProof (⇔)
↳22 QDP
↳23 QDPOrderProof (⇔)
↳24 QDP
↳25 PisEmptyProof (⇔)
↳26 TRUE
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
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) → 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)
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, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND1(true, x, y, z) → GR(y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
COND2(true, x, y, z) → GR(y, z)
COND2(true, x, y, z) → P(y)
COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND2(false, x, y, z) → GR(x, z)
COND2(false, x, y, z) → P(x)
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
cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
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
cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
GR(s(x), s(y)) → GR(x, y)
cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
GR(s(x), s(y)) → GR(x, y)
From the DPs we obtained the following set of size-change graphs:
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)
COND1(true, x, y, z) → COND2(gr(y, z), 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)
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, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
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)
COND1(true, x, y, z) → COND2(gr(y, z), 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, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
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)
COND1(true, x, y, z) → COND2(gr(y, z), 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
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
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)
The value of delta used in the strict ordering is 3/16.
POL(COND2(x1, x2, x3, x4)) = (4)x2 + (1/2)x3 + (1/2)x4
POL(true) = 1/4
POL(gr(x1, x2)) = x1
POL(p(x1)) = (1/2)x1
POL(false) = 0
POL(COND1(x1, x2, x3, x4)) = (3/4)x1 + (4)x2 + (1/2)x3 + (1/2)x4
POL(s(x1)) = 1/4 + (2)x1
POL(0) = 0
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
gr(0, x) → false
gr(s(x), 0) → true
p(s(x)) → x
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
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
COND2(true, x, y, z) → COND2(gr(y, z), x, p(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
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
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 2.
POL(COND2(x1, x2, x3, x4)) = (2)x1 + (4)x3
POL(true) = 1
POL(gr(x1, x2)) = x1
POL(p(x1)) = (1/2)x1
POL(s(x1)) = 2 + (2)x1
POL(0) = 0
POL(false) = 0
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
gr(0, x) → false
gr(s(x), 0) → true
p(s(x)) → x
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))