YES(O(1), O(n^2)) 5.17/1.79 YES(O(1), O(n^2)) 5.55/1.82 5.55/1.82 5.55/1.82
5.55/1.82 5.55/1.820 CpxTRS5.55/1.82
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))5.55/1.82
↳2 CdtProblem5.55/1.82
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))5.55/1.82
↳4 CdtProblem5.55/1.82
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))5.55/1.82
↳6 CdtProblem5.55/1.82
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))5.55/1.82
↳8 CdtProblem5.55/1.82
↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))5.55/1.82
↳10 BOUNDS(O(1), O(1))5.55/1.82
sqr(0) → 0 5.55/1.82
sqr(s(x)) → +(sqr(x), s(double(x))) 5.55/1.82
double(0) → 0 5.55/1.82
double(s(x)) → s(s(double(x))) 5.55/1.82
+(x, 0) → x 5.55/1.82
+(x, s(y)) → s(+(x, y)) 5.55/1.82
sqr(s(x)) → s(+(sqr(x), double(x)))
Tuples:
sqr(0) → 0 5.55/1.82
sqr(s(z0)) → +(sqr(z0), s(double(z0))) 5.55/1.82
sqr(s(z0)) → s(+(sqr(z0), double(z0))) 5.55/1.82
double(0) → 0 5.55/1.82
double(s(z0)) → s(s(double(z0))) 5.55/1.82
+(z0, 0) → z0 5.55/1.82
+(z0, s(z1)) → s(+(z0, z1))
S tuples:
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.82
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0)) 5.55/1.82
DOUBLE(s(z0)) → c4(DOUBLE(z0)) 5.55/1.82
+'(z0, s(z1)) → c6(+'(z0, z1))
K tuples:none
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.82
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0)) 5.55/1.82
DOUBLE(s(z0)) → c4(DOUBLE(z0)) 5.55/1.82
+'(z0, s(z1)) → c6(+'(z0, z1))
sqr, double, +
SQR, DOUBLE, +'
c1, c2, c4, c6
We considered the (Usable) Rules:
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.82
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
And the Tuples:
sqr(0) → 0 5.55/1.82
sqr(s(z0)) → +(sqr(z0), s(double(z0))) 5.55/1.82
sqr(s(z0)) → s(+(sqr(z0), double(z0))) 5.55/1.82
double(0) → 0 5.55/1.82
double(s(z0)) → s(s(double(z0))) 5.55/1.82
+(z0, 0) → z0 5.55/1.82
+(z0, s(z1)) → s(+(z0, z1))
The order we found is given by the following interpretation:
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.82
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0)) 5.55/1.82
DOUBLE(s(z0)) → c4(DOUBLE(z0)) 5.55/1.82
+'(z0, s(z1)) → c6(+'(z0, z1))
POL(+(x1, x2)) = [5] + [3]x2 5.55/1.82
POL(+'(x1, x2)) = 0 5.55/1.82
POL(0) = [1] 5.55/1.82
POL(DOUBLE(x1)) = 0 5.55/1.82
POL(SQR(x1)) = [4]x1 5.55/1.82
POL(c1(x1, x2, x3)) = x1 + x2 + x3 5.55/1.82
POL(c2(x1, x2, x3)) = x1 + x2 + x3 5.55/1.82
POL(c4(x1)) = x1 5.55/1.82
POL(c6(x1)) = x1 5.55/1.82
POL(double(x1)) = 0 5.55/1.82
POL(s(x1)) = [1] + x1 5.55/1.82
POL(sqr(x1)) = [4] + [4]x1
Tuples:
sqr(0) → 0 5.55/1.82
sqr(s(z0)) → +(sqr(z0), s(double(z0))) 5.55/1.82
sqr(s(z0)) → s(+(sqr(z0), double(z0))) 5.55/1.82
double(0) → 0 5.55/1.82
double(s(z0)) → s(s(double(z0))) 5.55/1.83
+(z0, 0) → z0 5.55/1.83
+(z0, s(z1)) → s(+(z0, z1))
S tuples:
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.83
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0)) 5.55/1.83
DOUBLE(s(z0)) → c4(DOUBLE(z0)) 5.55/1.83
+'(z0, s(z1)) → c6(+'(z0, z1))
K tuples:
DOUBLE(s(z0)) → c4(DOUBLE(z0)) 5.55/1.83
+'(z0, s(z1)) → c6(+'(z0, z1))
Defined Rule Symbols:
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.83
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0))
sqr, double, +
SQR, DOUBLE, +'
c1, c2, c4, c6
We considered the (Usable) Rules:
DOUBLE(s(z0)) → c4(DOUBLE(z0))
And the Tuples:
sqr(0) → 0 5.55/1.83
sqr(s(z0)) → +(sqr(z0), s(double(z0))) 5.55/1.83
sqr(s(z0)) → s(+(sqr(z0), double(z0))) 5.55/1.83
double(0) → 0 5.55/1.83
double(s(z0)) → s(s(double(z0))) 5.55/1.83
+(z0, 0) → z0 5.55/1.83
+(z0, s(z1)) → s(+(z0, z1))
The order we found is given by the following interpretation:
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.83
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0)) 5.55/1.83
DOUBLE(s(z0)) → c4(DOUBLE(z0)) 5.55/1.83
+'(z0, s(z1)) → c6(+'(z0, z1))
POL(+(x1, x2)) = [3] 5.55/1.83
POL(+'(x1, x2)) = 0 5.55/1.83
POL(0) = 0 5.55/1.83
POL(DOUBLE(x1)) = x1 5.55/1.83
POL(SQR(x1)) = [2]x12 5.55/1.83
POL(c1(x1, x2, x3)) = x1 + x2 + x3 5.55/1.83
POL(c2(x1, x2, x3)) = x1 + x2 + x3 5.55/1.83
POL(c4(x1)) = x1 5.55/1.83
POL(c6(x1)) = x1 5.55/1.83
POL(double(x1)) = [2] + x1 + x12 5.55/1.83
POL(s(x1)) = [3] + x1 5.55/1.83
POL(sqr(x1)) = 0
Tuples:
sqr(0) → 0 5.55/1.83
sqr(s(z0)) → +(sqr(z0), s(double(z0))) 5.55/1.83
sqr(s(z0)) → s(+(sqr(z0), double(z0))) 5.55/1.83
double(0) → 0 5.55/1.83
double(s(z0)) → s(s(double(z0))) 5.55/1.83
+(z0, 0) → z0 5.55/1.83
+(z0, s(z1)) → s(+(z0, z1))
S tuples:
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.83
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0)) 5.55/1.83
DOUBLE(s(z0)) → c4(DOUBLE(z0)) 5.55/1.83
+'(z0, s(z1)) → c6(+'(z0, z1))
K tuples:
+'(z0, s(z1)) → c6(+'(z0, z1))
Defined Rule Symbols:
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.83
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0)) 5.55/1.83
DOUBLE(s(z0)) → c4(DOUBLE(z0))
sqr, double, +
SQR, DOUBLE, +'
c1, c2, c4, c6
We considered the (Usable) Rules:
+'(z0, s(z1)) → c6(+'(z0, z1))
And the Tuples:
sqr(0) → 0 5.55/1.83
sqr(s(z0)) → +(sqr(z0), s(double(z0))) 5.55/1.83
sqr(s(z0)) → s(+(sqr(z0), double(z0))) 5.55/1.83
double(0) → 0 5.55/1.83
double(s(z0)) → s(s(double(z0))) 5.55/1.83
+(z0, 0) → z0 5.55/1.83
+(z0, s(z1)) → s(+(z0, z1))
The order we found is given by the following interpretation:
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.83
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0)) 5.55/1.83
DOUBLE(s(z0)) → c4(DOUBLE(z0)) 5.55/1.83
+'(z0, s(z1)) → c6(+'(z0, z1))
POL(+(x1, x2)) = [3] 5.55/1.83
POL(+'(x1, x2)) = x2 5.55/1.83
POL(0) = 0 5.55/1.83
POL(DOUBLE(x1)) = 0 5.55/1.83
POL(SQR(x1)) = x12 5.55/1.83
POL(c1(x1, x2, x3)) = x1 + x2 + x3 5.55/1.83
POL(c2(x1, x2, x3)) = x1 + x2 + x3 5.55/1.83
POL(c4(x1)) = x1 5.55/1.83
POL(c6(x1)) = x1 5.55/1.83
POL(double(x1)) = [2]x1 5.55/1.83
POL(s(x1)) = [1] + x1 5.55/1.83
POL(sqr(x1)) = 0
Tuples:
sqr(0) → 0 5.55/1.83
sqr(s(z0)) → +(sqr(z0), s(double(z0))) 5.55/1.83
sqr(s(z0)) → s(+(sqr(z0), double(z0))) 5.55/1.83
double(0) → 0 5.55/1.83
double(s(z0)) → s(s(double(z0))) 5.55/1.83
+(z0, 0) → z0 5.55/1.83
+(z0, s(z1)) → s(+(z0, z1))
S tuples:none
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.83
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0)) 5.55/1.83
DOUBLE(s(z0)) → c4(DOUBLE(z0)) 5.55/1.83
+'(z0, s(z1)) → c6(+'(z0, z1))
Defined Rule Symbols:
SQR(s(z0)) → c1(+'(sqr(z0), s(double(z0))), SQR(z0), DOUBLE(z0)) 5.55/1.83
SQR(s(z0)) → c2(+'(sqr(z0), double(z0)), SQR(z0), DOUBLE(z0)) 5.55/1.83
DOUBLE(s(z0)) → c4(DOUBLE(z0)) 5.55/1.83
+'(z0, s(z1)) → c6(+'(z0, z1))
sqr, double, +
SQR, DOUBLE, +'
c1, c2, c4, c6