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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, 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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
+1(O(x), I(y)) → +1(x, y)
+1(I(x), O(y)) → +1(x, y)
*1(I(x), y) → O1(*(x, y))
*1(O(x), y) → O1(*(x, y))
-1(I(x), I(y)) → O1(-(x, y))
*1(I(x), y) → +1(O(*(x, y)), y)
+1(O(x), O(y)) → +1(x, y)
-1(O(x), I(y)) → -1(-(x, y), I(1))
-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(I(x), I(y)) → +1(x, y)
+1(I(x), I(y)) → O1(+(+(x, y), I(0)))
*1(I(x), y) → *1(x, y)
-1(I(x), I(y)) → -1(x, y)
-1(I(x), O(y)) → -1(x, y)
-1(O(x), I(y)) → -1(x, y)
+1(O(x), O(y)) → O1(+(x, y))
-1(O(x), O(y)) → O1(-(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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
+1(O(x), I(y)) → +1(x, y)
+1(I(x), O(y)) → +1(x, y)
*1(I(x), y) → O1(*(x, y))
*1(O(x), y) → O1(*(x, y))
-1(I(x), I(y)) → O1(-(x, y))
*1(I(x), y) → +1(O(*(x, y)), y)
+1(O(x), O(y)) → +1(x, y)
-1(O(x), I(y)) → -1(-(x, y), I(1))
-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(I(x), I(y)) → +1(x, y)
+1(I(x), I(y)) → O1(+(+(x, y), I(0)))
*1(I(x), y) → *1(x, y)
-1(I(x), I(y)) → -1(x, y)
-1(I(x), O(y)) → -1(x, y)
-1(O(x), I(y)) → -1(x, y)
+1(O(x), O(y)) → O1(+(x, y))
-1(O(x), O(y)) → O1(-(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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
-1(I(x), I(y)) → -1(x, y)
-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(O(x), I(y)) → -1(-(x, y), I(1))
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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, 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(I(x), I(y)) → -1(x, y)
-1(O(x), I(y)) → -1(x, y)
-1(O(x), I(y)) → -1(-(x, y), I(1))
The value of delta used in the strict ordering is 2.
POL(-(x1, x2)) = 4 + (4)x_1 + (4)x_2
POL(-1(x1, x2)) = (1/2)x_2
POL(1) = 0
POL(0) = 11/4
POL(I(x1)) = (2)x_1
POL(O(x1)) = 4 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
-1(O(x), I(y)) → -1(x, y)
-1(I(x), I(y)) → -1(x, y)
-1(O(x), I(y)) → -1(-(x, y), I(1))
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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, 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(O(x), I(y)) → -1(-(x, y), I(1))
The value of delta used in the strict ordering is 4.
POL(-(x1, x2)) = 4 + (11/4)x_2
POL(-1(x1, x2)) = (4)x_2
POL(1) = 0
POL(0) = 1/4
POL(I(x1)) = 1 + x_1
POL(O(x1)) = 5/2 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
-1(O(x), I(y)) → -1(-(x, y), I(1))
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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, 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), I(1))
The value of delta used in the strict ordering is 5/8.
POL(-(x1, x2)) = x_1 + (1/4)x_2
POL(-1(x1, x2)) = (1/2)x_1 + (1/4)x_2
POL(1) = 0
POL(0) = 1/4
POL(I(x1)) = 1 + x_1
POL(O(x1)) = 5/4 + x_1
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
O(0) → 0
-(I(x), I(y)) → O(-(x, y))
-(x, 0) → x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ 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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ 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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, 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), O(y)) → +1(x, y)
+1(O(x), O(y)) → +1(x, y)
+1(I(x), I(y)) → +1(+(x, y), I(0))
The value of delta used in the strict ordering is 1/8.
POL(0) = 0
POL(+1(x1, x2)) = (1/2)x_2
POL(I(x1)) = 1/4 + x_1
POL(O(x1)) = (4)x_1
POL(+(x1, x2)) = 1/2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
+1(I(x), O(y)) → +1(x, y)
+1(O(x), O(y)) → +1(x, y)
+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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ 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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, 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 2.
POL(0) = 0
POL(+1(x1, x2)) = (2)x_1 + (4)x_2
POL(I(x1)) = 5/4 + x_1
POL(+(x1, x2)) = 1/4 + x_1 + (2)x_2
POL(O(x1)) = 3/4 + x_1
+(0, x) → x
O(0) → 0
+(O(x), O(y)) → O(+(x, y))
+(x, 0) → x
+(I(x), O(y)) → I(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ 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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
+1(I(x), O(y)) → +1(x, y)
+1(O(x), O(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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, 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)
The value of delta used in the strict ordering is 3/16.
POL(+1(x1, x2)) = (1/2)x_1 + (1/4)x_2
POL(I(x1)) = 5/4 + (4)x_1
POL(O(x1)) = 1/4 + x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ 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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ 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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, 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 1/4.
POL(*1(x1, x2)) = (1/2)x_1
POL(I(x1)) = 1/2 + (3/2)x_1
POL(O(x1)) = 9/4 + x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ 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)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))