:2(x, x) -> e
:2(x, e) -> x
i1(:2(x, y)) -> :2(y, x)
:2(:2(x, y), z) -> :2(x, :2(z, i1(y)))
:2(e, x) -> i1(x)
i1(i1(x)) -> x
i1(e) -> e
:2(x, :2(y, i1(x))) -> i1(y)
:2(x, :2(y, :2(i1(x), z))) -> :2(i1(z), y)
:2(i1(x), :2(y, x)) -> i1(y)
:2(i1(x), :2(y, :2(x, z))) -> :2(i1(z), y)
↳ QTRS
↳ DependencyPairsProof
:2(x, x) -> e
:2(x, e) -> x
i1(:2(x, y)) -> :2(y, x)
:2(:2(x, y), z) -> :2(x, :2(z, i1(y)))
:2(e, x) -> i1(x)
i1(i1(x)) -> x
i1(e) -> e
:2(x, :2(y, i1(x))) -> i1(y)
:2(x, :2(y, :2(i1(x), z))) -> :2(i1(z), y)
:2(i1(x), :2(y, x)) -> i1(y)
:2(i1(x), :2(y, :2(x, z))) -> :2(i1(z), y)
:12(x, :2(y, i1(x))) -> I1(y)
:12(e, x) -> I1(x)
:12(:2(x, y), z) -> I1(y)
:12(:2(x, y), z) -> :12(z, i1(y))
:12(x, :2(y, :2(i1(x), z))) -> I1(z)
:12(i1(x), :2(y, x)) -> I1(y)
:12(i1(x), :2(y, :2(x, z))) -> I1(z)
:12(:2(x, y), z) -> :12(x, :2(z, i1(y)))
I1(:2(x, y)) -> :12(y, x)
:12(x, :2(y, :2(i1(x), z))) -> :12(i1(z), y)
:12(i1(x), :2(y, :2(x, z))) -> :12(i1(z), y)
:2(x, x) -> e
:2(x, e) -> x
i1(:2(x, y)) -> :2(y, x)
:2(:2(x, y), z) -> :2(x, :2(z, i1(y)))
:2(e, x) -> i1(x)
i1(i1(x)) -> x
i1(e) -> e
:2(x, :2(y, i1(x))) -> i1(y)
:2(x, :2(y, :2(i1(x), z))) -> :2(i1(z), y)
:2(i1(x), :2(y, x)) -> i1(y)
:2(i1(x), :2(y, :2(x, z))) -> :2(i1(z), y)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
:12(x, :2(y, i1(x))) -> I1(y)
:12(e, x) -> I1(x)
:12(:2(x, y), z) -> I1(y)
:12(:2(x, y), z) -> :12(z, i1(y))
:12(x, :2(y, :2(i1(x), z))) -> I1(z)
:12(i1(x), :2(y, x)) -> I1(y)
:12(i1(x), :2(y, :2(x, z))) -> I1(z)
:12(:2(x, y), z) -> :12(x, :2(z, i1(y)))
I1(:2(x, y)) -> :12(y, x)
:12(x, :2(y, :2(i1(x), z))) -> :12(i1(z), y)
:12(i1(x), :2(y, :2(x, z))) -> :12(i1(z), y)
:2(x, x) -> e
:2(x, e) -> x
i1(:2(x, y)) -> :2(y, x)
:2(:2(x, y), z) -> :2(x, :2(z, i1(y)))
:2(e, x) -> i1(x)
i1(i1(x)) -> x
i1(e) -> e
:2(x, :2(y, i1(x))) -> i1(y)
:2(x, :2(y, :2(i1(x), z))) -> :2(i1(z), y)
:2(i1(x), :2(y, x)) -> i1(y)
:2(i1(x), :2(y, :2(x, z))) -> :2(i1(z), y)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
:12(x, :2(y, i1(x))) -> I1(y)
:12(e, x) -> I1(x)
:12(:2(x, y), z) -> I1(y)
:12(:2(x, y), z) -> :12(z, i1(y))
:12(x, :2(y, :2(i1(x), z))) -> I1(z)
:12(i1(x), :2(y, x)) -> I1(y)
:12(i1(x), :2(y, :2(x, z))) -> I1(z)
I1(:2(x, y)) -> :12(y, x)
:12(x, :2(y, :2(i1(x), z))) -> :12(i1(z), y)
:12(i1(x), :2(y, :2(x, z))) -> :12(i1(z), y)
Used ordering: Polynomial interpretation [21]:
:12(:2(x, y), z) -> :12(x, :2(z, i1(y)))
POL(:2(x1, x2)) = 2 + x1 + x2
POL(:12(x1, x2)) = 1 + 3·x1 + 3·x2
POL(I1(x1)) = 3·x1
POL(e) = 0
POL(i1(x1)) = x1
:2(x, e) -> x
i1(i1(x)) -> x
i1(e) -> e
:2(x, x) -> e
:2(e, x) -> i1(x)
:2(i1(x), :2(y, x)) -> i1(y)
:2(:2(x, y), z) -> :2(x, :2(z, i1(y)))
:2(i1(x), :2(y, :2(x, z))) -> :2(i1(z), y)
i1(:2(x, y)) -> :2(y, x)
:2(x, :2(y, :2(i1(x), z))) -> :2(i1(z), y)
:2(x, :2(y, i1(x))) -> i1(y)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
:12(:2(x, y), z) -> :12(x, :2(z, i1(y)))
:2(x, x) -> e
:2(x, e) -> x
i1(:2(x, y)) -> :2(y, x)
:2(:2(x, y), z) -> :2(x, :2(z, i1(y)))
:2(e, x) -> i1(x)
i1(i1(x)) -> x
i1(e) -> e
:2(x, :2(y, i1(x))) -> i1(y)
:2(x, :2(y, :2(i1(x), z))) -> :2(i1(z), y)
:2(i1(x), :2(y, x)) -> i1(y)
:2(i1(x), :2(y, :2(x, z))) -> :2(i1(z), y)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
:12(:2(x, y), z) -> :12(x, :2(z, i1(y)))
POL(:2(x1, x2)) = 3 + 2·x1 + 3·x2
POL(:12(x1, x2)) = 3·x1
POL(e) = 0
POL(i1(x1)) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
:2(x, x) -> e
:2(x, e) -> x
i1(:2(x, y)) -> :2(y, x)
:2(:2(x, y), z) -> :2(x, :2(z, i1(y)))
:2(e, x) -> i1(x)
i1(i1(x)) -> x
i1(e) -> e
:2(x, :2(y, i1(x))) -> i1(y)
:2(x, :2(y, :2(i1(x), z))) -> :2(i1(z), y)
:2(i1(x), :2(y, x)) -> i1(y)
:2(i1(x), :2(y, :2(x, z))) -> :2(i1(z), y)