+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))
↳ QTRS
↳ DependencyPairsProof
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))
F(h(x, h(y, z))) → +1(x, y)
+1(x, +(y, z)) → +1(x, y)
F(g(h(x, y))) → F(h(s(x), y))
F(g(f(x))) → F(h(s(0), x))
+1(x, +(y, z)) → +1(+(x, y), z)
+1(x, s(y)) → +1(x, y)
F(h(x, h(y, z))) → F(h(+(x, y), z))
+1(s(x), y) → +1(x, y)
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
F(h(x, h(y, z))) → +1(x, y)
+1(x, +(y, z)) → +1(x, y)
F(g(h(x, y))) → F(h(s(x), y))
F(g(f(x))) → F(h(s(0), x))
+1(x, +(y, z)) → +1(+(x, y), z)
+1(x, s(y)) → +1(x, y)
F(h(x, h(y, z))) → F(h(+(x, y), z))
+1(s(x), y) → +1(x, y)
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
+1(x, +(y, z)) → +1(x, y)
+1(x, +(y, z)) → +1(+(x, y), z)
+1(x, s(y)) → +1(x, y)
+1(s(x), y) → +1(x, y)
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+1(x, +(y, z)) → +1(x, y)
+1(x, +(y, z)) → +1(+(x, y), z)
Used ordering: Polynomial interpretation [25,35]:
+1(x, s(y)) → +1(x, y)
+1(s(x), y) → +1(x, y)
The value of delta used in the strict ordering is 1/4.
POL(s(x1)) = x_1
POL(0) = 0
POL(+1(x1, x2)) = (1/4)x_2
POL(+(x1, x2)) = 1 + x_1 + (4)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
+1(x, s(y)) → +1(x, y)
+1(s(x), y) → +1(x, y)
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+1(s(x), y) → +1(x, y)
Used ordering: Polynomial interpretation [25,35]:
+1(x, s(y)) → +1(x, y)
The value of delta used in the strict ordering is 1/4.
POL(s(x1)) = 1/4 + x_1
POL(+1(x1, x2)) = x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
+1(x, s(y)) → +1(x, y)
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+1(x, s(y)) → +1(x, y)
The value of delta used in the strict ordering is 1/16.
POL(s(x1)) = 1/4 + (2)x_1
POL(+1(x1, x2)) = (1/4)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
F(h(x, h(y, z))) → F(h(+(x, y), z))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
F(h(x, h(y, z))) → F(h(+(x, y), z))
The value of delta used in the strict ordering is 1/4.
POL(h(x1, x2)) = 1 + x_2
POL(s(x1)) = 0
POL(0) = 0
POL(F(x1)) = (1/4)x_1
POL(+(x1, x2)) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))