O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
↳ QTRS
↳ DependencyPairsProof
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
+1(I(x), O(y)) → +1(x, y)
+1(O(x), I(y)) → +1(x, y)
*1(I(x), y) → O1(*(x, y))
*1(O(x), y) → O1(*(x, y))
*1(I(x), y) → +1(O(*(x, y)), y)
+1(O(x), O(y)) → +1(x, y)
+1(I(x), I(y)) → +1(+(x, y), I(0))
*1(O(x), y) → *1(x, y)
+1(O(x), O(y)) → O1(+(x, y))
+1(I(x), I(y)) → +1(x, y)
*1(I(x), y) → *1(x, y)
+1(I(x), I(y)) → O1(+(+(x, y), I(0)))
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
+1(I(x), O(y)) → +1(x, y)
+1(O(x), I(y)) → +1(x, y)
*1(I(x), y) → O1(*(x, y))
*1(O(x), y) → O1(*(x, y))
*1(I(x), y) → +1(O(*(x, y)), y)
+1(O(x), O(y)) → +1(x, y)
+1(I(x), I(y)) → +1(+(x, y), I(0))
*1(O(x), y) → *1(x, y)
+1(O(x), O(y)) → O1(+(x, y))
+1(I(x), I(y)) → +1(x, y)
*1(I(x), y) → *1(x, y)
+1(I(x), I(y)) → O1(+(+(x, y), I(0)))
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
+1(I(x), O(y)) → +1(x, y)
+1(O(x), I(y)) → +1(x, y)
+1(O(x), O(y)) → +1(x, y)
+1(I(x), I(y)) → +1(+(x, y), I(0))
+1(I(x), I(y)) → +1(x, y)
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+1(I(x), O(y)) → +1(x, y)
+1(O(x), O(y)) → +1(x, y)
Used ordering: Polynomial interpretation [25,35]:
+1(O(x), I(y)) → +1(x, y)
+1(I(x), I(y)) → +1(+(x, y), I(0))
+1(I(x), I(y)) → +1(x, y)
The value of delta used in the strict ordering is 4.
POL(0) = 0
POL(+1(x1, x2)) = (4)x_2
POL(I(x1)) = (4)x_1
POL(O(x1)) = 1 + (4)x_1
POL(+(x1, x2)) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
+1(O(x), I(y)) → +1(x, y)
+1(I(x), I(y)) → +1(+(x, y), I(0))
+1(I(x), I(y)) → +1(x, y)
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+1(O(x), I(y)) → +1(x, y)
+1(I(x), I(y)) → +1(x, y)
Used ordering: Polynomial interpretation [25,35]:
+1(I(x), I(y)) → +1(+(x, y), I(0))
The value of delta used in the strict ordering is 2.
POL(0) = 0
POL(+1(x1, x2)) = (2)x_2
POL(I(x1)) = 1 + (3)x_1
POL(O(x1)) = 2 + (3)x_1
POL(+(x1, x2)) = (4)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
+1(I(x), I(y)) → +1(+(x, y), I(0))
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+1(I(x), I(y)) → +1(+(x, y), I(0))
The value of delta used in the strict ordering is 8.
POL(0) = 0
POL(+1(x1, x2)) = (4)x_1 + (2)x_2
POL(I(x1)) = 2 + (2)x_1
POL(+(x1, x2)) = x_1 + x_2
POL(O(x1)) = (2)x_1
+(O(x), I(y)) → I(+(x, y))
+(O(x), O(y)) → O(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(I(x), O(y)) → I(+(x, y))
O(0) → 0
+(0, x) → x
+(x, 0) → x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
*1(O(x), y) → *1(x, y)
*1(I(x), y) → *1(x, y)
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
*1(O(x), y) → *1(x, y)
*1(I(x), y) → *1(x, y)
The value of delta used in the strict ordering is 16.
POL(*1(x1, x2)) = (4)x_1
POL(I(x1)) = 4 + x_1
POL(O(x1)) = 4 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)