times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
↳ QTRS
↳ DependencyPairsProof
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
TIMES(x, plus(y, s(z))) → TIMES(x, s(z))
TIMES(x, s(y)) → TIMES(x, y)
TIMES(x, s(y)) → PLUS(times(x, y), x)
TIMES(x, plus(y, s(z))) → TIMES(x, plus(y, times(s(z), 0)))
TIMES(x, plus(y, s(z))) → PLUS(y, times(s(z), 0))
PLUS(x, s(y)) → PLUS(x, y)
TIMES(x, plus(y, s(z))) → PLUS(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
TIMES(x, plus(y, s(z))) → TIMES(s(z), 0)
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
TIMES(x, plus(y, s(z))) → TIMES(x, s(z))
TIMES(x, s(y)) → TIMES(x, y)
TIMES(x, s(y)) → PLUS(times(x, y), x)
TIMES(x, plus(y, s(z))) → TIMES(x, plus(y, times(s(z), 0)))
TIMES(x, plus(y, s(z))) → PLUS(y, times(s(z), 0))
PLUS(x, s(y)) → PLUS(x, y)
TIMES(x, plus(y, s(z))) → PLUS(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
TIMES(x, plus(y, s(z))) → TIMES(s(z), 0)
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
PLUS(x, s(y)) → PLUS(x, y)
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PLUS(x, s(y)) → PLUS(x, y)
The value of delta used in the strict ordering is 4.
POL(PLUS(x1, x2)) = (4)x_2
POL(s(x1)) = 1 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
TIMES(x, plus(y, s(z))) → TIMES(x, s(z))
TIMES(x, s(y)) → TIMES(x, y)
TIMES(x, plus(y, s(z))) → TIMES(x, plus(y, times(s(z), 0)))
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TIMES(x, plus(y, s(z))) → TIMES(x, s(z))
TIMES(x, s(y)) → TIMES(x, y)
Used ordering: Polynomial interpretation [25,35]:
TIMES(x, plus(y, s(z))) → TIMES(x, plus(y, times(s(z), 0)))
The value of delta used in the strict ordering is 4.
POL(plus(x1, x2)) = 4 + x_1 + (4)x_2
POL(TIMES(x1, x2)) = (4)x_2
POL(times(x1, x2)) = x_2
POL(s(x1)) = 1 + x_1
POL(0) = 1
times(x, 0) → 0
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
TIMES(x, plus(y, s(z))) → TIMES(x, plus(y, times(s(z), 0)))
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TIMES(x, plus(y, s(z))) → TIMES(x, plus(y, times(s(z), 0)))
The value of delta used in the strict ordering is 1.
POL(plus(x1, x2)) = 1 + (2)x_1 + x_2
POL(TIMES(x1, x2)) = x_2
POL(times(x1, x2)) = 0
POL(s(x1)) = 1
POL(0) = 0
times(x, 0) → 0
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))