01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
↳ QTRS
↳ DependencyPairsProof
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
+12(j1(x), j1(y)) -> +12(x, y)
*12(*2(x, y), z) -> *12(y, z)
OPP1(j1(x)) -> OPP1(x)
+12(11(x), j1(y)) -> +12(x, y)
+12(j1(x), 11(y)) -> +12(x, y)
*12(*2(x, y), z) -> *12(x, *2(y, z))
-12(x, y) -> +12(x, opp1(y))
+12(j1(x), 11(y)) -> 011(+2(x, y))
+12(11(x), j1(y)) -> 011(+2(x, y))
+12(01(x), 01(y)) -> +12(x, y)
-12(x, y) -> OPP1(y)
*12(j1(x), y) -> *12(x, y)
OPP1(01(x)) -> OPP1(x)
+12(+2(x, y), z) -> +12(y, z)
*12(11(x), y) -> 011(*2(x, y))
*12(11(x), y) -> *12(x, y)
+12(j1(x), j1(y)) -> +12(+2(x, y), j1(#))
+12(01(x), 01(y)) -> 011(+2(x, y))
*12(j1(x), y) -> 011(*2(x, y))
*12(11(x), y) -> +12(01(*2(x, y)), y)
*12(j1(x), y) -> -12(01(*2(x, y)), y)
+12(11(x), 11(y)) -> +12(x, y)
OPP1(01(x)) -> 011(opp1(x))
+12(j1(x), 01(y)) -> +12(x, y)
+12(01(x), j1(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(x, +2(y, z))
OPP1(11(x)) -> OPP1(x)
*12(01(x), y) -> 011(*2(x, y))
*12(01(x), y) -> *12(x, y)
+12(11(x), 01(y)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
+12(j1(x), j1(y)) -> +12(x, y)
*12(*2(x, y), z) -> *12(y, z)
OPP1(j1(x)) -> OPP1(x)
+12(11(x), j1(y)) -> +12(x, y)
+12(j1(x), 11(y)) -> +12(x, y)
*12(*2(x, y), z) -> *12(x, *2(y, z))
-12(x, y) -> +12(x, opp1(y))
+12(j1(x), 11(y)) -> 011(+2(x, y))
+12(11(x), j1(y)) -> 011(+2(x, y))
+12(01(x), 01(y)) -> +12(x, y)
-12(x, y) -> OPP1(y)
*12(j1(x), y) -> *12(x, y)
OPP1(01(x)) -> OPP1(x)
+12(+2(x, y), z) -> +12(y, z)
*12(11(x), y) -> 011(*2(x, y))
*12(11(x), y) -> *12(x, y)
+12(j1(x), j1(y)) -> +12(+2(x, y), j1(#))
+12(01(x), 01(y)) -> 011(+2(x, y))
*12(j1(x), y) -> 011(*2(x, y))
*12(11(x), y) -> +12(01(*2(x, y)), y)
*12(j1(x), y) -> -12(01(*2(x, y)), y)
+12(11(x), 11(y)) -> +12(x, y)
OPP1(01(x)) -> 011(opp1(x))
+12(j1(x), 01(y)) -> +12(x, y)
+12(01(x), j1(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(x, +2(y, z))
OPP1(11(x)) -> OPP1(x)
*12(01(x), y) -> 011(*2(x, y))
*12(01(x), y) -> *12(x, y)
+12(11(x), 01(y)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
OPP1(01(x)) -> OPP1(x)
OPP1(11(x)) -> OPP1(x)
OPP1(j1(x)) -> OPP1(x)
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
OPP1(j1(x)) -> OPP1(x)
Used ordering: Polynomial interpretation [21]:
OPP1(01(x)) -> OPP1(x)
OPP1(11(x)) -> OPP1(x)
POL(01(x1)) = x1
POL(11(x1)) = x1
POL(OPP1(x1)) = x1
POL(j1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
OPP1(11(x)) -> OPP1(x)
OPP1(01(x)) -> OPP1(x)
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
OPP1(11(x)) -> OPP1(x)
Used ordering: Polynomial interpretation [21]:
OPP1(01(x)) -> OPP1(x)
POL(01(x1)) = x1
POL(11(x1)) = 1 + x1
POL(OPP1(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
OPP1(01(x)) -> OPP1(x)
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
OPP1(01(x)) -> OPP1(x)
POL(01(x1)) = 1 + x1
POL(OPP1(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
+12(j1(x), j1(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(x, y)
+12(11(x), j1(y)) -> +12(x, y)
+12(j1(x), 11(y)) -> +12(x, y)
+12(01(x), j1(y)) -> +12(x, y)
+12(j1(x), 01(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(x, +2(y, z))
+12(01(x), 01(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
+12(j1(x), j1(y)) -> +12(+2(x, y), j1(#))
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 01(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+12(j1(x), j1(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(x, y)
+12(11(x), j1(y)) -> +12(x, y)
+12(j1(x), 11(y)) -> +12(x, y)
+12(01(x), j1(y)) -> +12(x, y)
+12(j1(x), 01(y)) -> +12(x, y)
+12(j1(x), j1(y)) -> +12(+2(x, y), j1(#))
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 01(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
Used ordering: Polynomial interpretation [21]:
+12(+2(x, y), z) -> +12(x, +2(y, z))
+12(01(x), 01(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
POL(#) = 0
POL(+2(x1, x2)) = x1 + x2
POL(+12(x1, x2)) = x1 + x2
POL(01(x1)) = x1
POL(11(x1)) = 1 + x1
POL(j1(x1)) = 1 + x1
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(#, x) -> x
01(#) -> #
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(x, #) -> x
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(01(x), 11(y)) -> 11(+2(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
+12(01(x), 01(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+12(01(x), 01(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))
POL(#) = 0
POL(+2(x1, x2)) = 1 + x1 + x2
POL(+12(x1, x2)) = x1
POL(01(x1)) = 1 + x1
POL(11(x1)) = 0
POL(j1(x1)) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
*12(*2(x, y), z) -> *12(y, z)
*12(j1(x), y) -> *12(x, y)
*12(11(x), y) -> *12(x, y)
*12(01(x), y) -> *12(x, y)
*12(*2(x, y), z) -> *12(x, *2(y, z))
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
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(y, z)
*12(01(x), y) -> *12(x, y)
*12(*2(x, y), z) -> *12(x, *2(y, z))
Used ordering: Polynomial interpretation [21]:
*12(j1(x), y) -> *12(x, y)
*12(11(x), y) -> *12(x, y)
POL(#) = 0
POL(*2(x1, x2)) = 1 + x1 + x2
POL(*12(x1, x2)) = x1
POL(+2(x1, x2)) = 0
POL(-2(x1, x2)) = 0
POL(01(x1)) = 1 + x1
POL(11(x1)) = x1
POL(j1(x1)) = x1
POL(opp1(x1)) = 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
*12(j1(x), y) -> *12(x, y)
*12(11(x), y) -> *12(x, y)
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
*12(j1(x), y) -> *12(x, y)
Used ordering: Polynomial interpretation [21]:
*12(11(x), y) -> *12(x, y)
POL(*12(x1, x2)) = x1
POL(11(x1)) = x1
POL(j1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
*12(11(x), y) -> *12(x, y)
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
*12(11(x), y) -> *12(x, y)
POL(*12(x1, x2)) = x1
POL(11(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
01(#) -> #
+2(#, x) -> x
+2(x, #) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(01(x), j1(y)) -> j1(+2(x, y))
+2(j1(x), 01(y)) -> j1(+2(x, y))
+2(11(x), 11(y)) -> j1(+2(+2(x, y), 11(#)))
+2(j1(x), j1(y)) -> 11(+2(+2(x, y), j1(#)))
+2(11(x), j1(y)) -> 01(+2(x, y))
+2(j1(x), 11(y)) -> 01(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
opp1(#) -> #
opp1(01(x)) -> 01(opp1(x))
opp1(11(x)) -> j1(opp1(x))
opp1(j1(x)) -> 11(opp1(x))
-2(x, y) -> +2(x, opp1(y))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(j1(x), y) -> -2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))