minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
f(0) → s(0)
f(s(x)) → minus(s(x), g(f(x)))
g(0) → 0
g(s(x)) → minus(s(x), f(g(x)))
↳ QTRS
↳ DependencyPairsProof
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
f(0) → s(0)
f(s(x)) → minus(s(x), g(f(x)))
g(0) → 0
g(s(x)) → minus(s(x), f(g(x)))
G(s(x)) → MINUS(s(x), f(g(x)))
F(s(x)) → F(x)
MINUS(s(x), s(y)) → MINUS(x, y)
F(s(x)) → G(f(x))
G(s(x)) → G(x)
F(s(x)) → MINUS(s(x), g(f(x)))
G(s(x)) → F(g(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
f(0) → s(0)
f(s(x)) → minus(s(x), g(f(x)))
g(0) → 0
g(s(x)) → minus(s(x), f(g(x)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
G(s(x)) → MINUS(s(x), f(g(x)))
F(s(x)) → F(x)
MINUS(s(x), s(y)) → MINUS(x, y)
F(s(x)) → G(f(x))
G(s(x)) → G(x)
F(s(x)) → MINUS(s(x), g(f(x)))
G(s(x)) → F(g(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
f(0) → s(0)
f(s(x)) → minus(s(x), g(f(x)))
g(0) → 0
g(s(x)) → minus(s(x), f(g(x)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
MINUS(s(x), s(y)) → MINUS(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
f(0) → s(0)
f(s(x)) → minus(s(x), g(f(x)))
g(0) → 0
g(s(x)) → minus(s(x), f(g(x)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MINUS(s(x), s(y)) → MINUS(x, y)
The value of delta used in the strict ordering is 12.
POL(MINUS(x1, x2)) = (3)x_2
POL(s(x1)) = 4 + (2)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
f(0) → s(0)
f(s(x)) → minus(s(x), g(f(x)))
g(0) → 0
g(s(x)) → minus(s(x), f(g(x)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
F(s(x)) → F(x)
F(s(x)) → G(f(x))
G(s(x)) → G(x)
G(s(x)) → F(g(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
f(0) → s(0)
f(s(x)) → minus(s(x), g(f(x)))
g(0) → 0
g(s(x)) → minus(s(x), f(g(x)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
F(s(x)) → F(x)
G(s(x)) → G(x)
G(s(x)) → F(g(x))
Used ordering: Polynomial interpretation [25,35]:
F(s(x)) → G(f(x))
The value of delta used in the strict ordering is 2.
POL(f(x1)) = 1 + x_1
POL(minus(x1, x2)) = x_1
POL(g(x1)) = (2)x_1
POL(s(x1)) = 1 + (2)x_1
POL(G(x1)) = 2 + (3)x_1
POL(0) = 0
POL(F(x1)) = 3 + (2)x_1
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
f(0) → s(0)
f(s(x)) → minus(s(x), g(f(x)))
g(0) → 0
g(s(x)) → minus(s(x), f(g(x)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
F(s(x)) → G(f(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
f(0) → s(0)
f(s(x)) → minus(s(x), g(f(x)))
g(0) → 0
g(s(x)) → minus(s(x), f(g(x)))