YES(O(1), O(n^2)) 2.36/1.09 YES(O(1), O(n^2)) 2.77/1.13 2.77/1.13 2.77/1.13 2.77/1.13 2.77/1.13 2.77/1.13 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 2.77/1.13 2.77/1.13 2.77/1.13
2.77/1.13
2.77/1.13
2.77/1.13
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
2.77/1.13 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
2.77/1.13 
↳2 CdtProblem
2.77/1.13 
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
2.77/1.13 
↳4 CdtProblem
2.77/1.13 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.77/1.13 
↳6 CdtProblem
2.77/1.13 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.77/1.13 
↳8 CdtProblem
2.77/1.13 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.77/1.13 
↳10 CdtProblem
2.77/1.13 
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
2.77/1.13 
↳12 BOUNDS(O(1), O(1))
2.77/1.13
2.77/1.13 2.77/1.13
2.77/1.13
2.77/1.13

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

dx(X) → one 2.77/1.13
dx(a) → zero 2.77/1.13
dx(plus(ALPHA, BETA)) → plus(dx(ALPHA), dx(BETA)) 2.77/1.13
dx(times(ALPHA, BETA)) → plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 2.77/1.13
dx(minus(ALPHA, BETA)) → minus(dx(ALPHA), dx(BETA)) 2.77/1.13
dx(neg(ALPHA)) → neg(dx(ALPHA)) 2.77/1.13
dx(div(ALPHA, BETA)) → minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 2.77/1.13
dx(ln(ALPHA)) → div(dx(ALPHA), ALPHA) 2.77/1.13
dx(exp(ALPHA, BETA)) → plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Rewrite Strategy: INNERMOST
2.77/1.13
2.77/1.13

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
2.77/1.13
2.77/1.13

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

dx(z0) → one 2.77/1.13
dx(a) → zero 2.77/1.13
dx(plus(z0, z1)) → plus(dx(z0), dx(z1)) 2.77/1.13
dx(times(z0, z1)) → plus(times(z1, dx(z0)), times(z0, dx(z1))) 2.77/1.13
dx(minus(z0, z1)) → minus(dx(z0), dx(z1)) 2.77/1.13
dx(neg(z0)) → neg(dx(z0)) 2.77/1.13
dx(div(z0, z1)) → minus(div(dx(z0), z1), times(z0, div(dx(z1), exp(z1, two)))) 2.77/1.13
dx(ln(z0)) → div(dx(z0), z0) 2.77/1.13
dx(exp(z0, z1)) → plus(times(z1, times(exp(z0, minus(z1, one)), dx(z0))), times(exp(z0, z1), times(ln(z0), dx(z1))))
Tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.13
DX(neg(z0)) → c5(DX(z0)) 2.77/1.13
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.13
DX(ln(z0)) → c7(DX(z0)) 2.77/1.13
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
S tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.13
DX(neg(z0)) → c5(DX(z0)) 2.77/1.13
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.13
DX(ln(z0)) → c7(DX(z0)) 2.77/1.13
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
K tuples:none
Defined Rule Symbols:

dx

Defined Pair Symbols:

DX

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8

2.77/1.13
2.77/1.13

(3) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1))
We considered the (Usable) Rules:none
And the Tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.13
DX(neg(z0)) → c5(DX(z0)) 2.77/1.13
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.13
DX(ln(z0)) → c7(DX(z0)) 2.77/1.13
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.77/1.13

POL(DX(x1)) = [2]x1 + [2]x12    2.77/1.13
POL(c2(x1, x2)) = x1 + x2    2.77/1.13
POL(c3(x1, x2)) = x1 + x2    2.77/1.13
POL(c4(x1, x2)) = x1 + x2    2.77/1.13
POL(c5(x1)) = x1    2.77/1.13
POL(c6(x1, x2)) = x1 + x2    2.77/1.13
POL(c7(x1)) = x1    2.77/1.13
POL(c8(x1, x2)) = x1 + x2    2.77/1.13
POL(div(x1, x2)) = x1 + x2    2.77/1.13
POL(exp(x1, x2)) = x1 + x2    2.77/1.13
POL(ln(x1)) = x1    2.77/1.13
POL(minus(x1, x2)) = [2] + x1 + x2    2.77/1.13
POL(neg(x1)) = x1    2.77/1.13
POL(plus(x1, x2)) = [1] + x1 + x2    2.77/1.13
POL(times(x1, x2)) = x1 + x2   
2.77/1.13
2.77/1.13

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

dx(z0) → one 2.77/1.13
dx(a) → zero 2.77/1.13
dx(plus(z0, z1)) → plus(dx(z0), dx(z1)) 2.77/1.13
dx(times(z0, z1)) → plus(times(z1, dx(z0)), times(z0, dx(z1))) 2.77/1.13
dx(minus(z0, z1)) → minus(dx(z0), dx(z1)) 2.77/1.13
dx(neg(z0)) → neg(dx(z0)) 2.77/1.13
dx(div(z0, z1)) → minus(div(dx(z0), z1), times(z0, div(dx(z1), exp(z1, two)))) 2.77/1.13
dx(ln(z0)) → div(dx(z0), z0) 2.77/1.13
dx(exp(z0, z1)) → plus(times(z1, times(exp(z0, minus(z1, one)), dx(z0))), times(exp(z0, z1), times(ln(z0), dx(z1))))
Tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.13
DX(neg(z0)) → c5(DX(z0)) 2.77/1.13
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.13
DX(ln(z0)) → c7(DX(z0)) 2.77/1.13
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
S tuples:

DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(neg(z0)) → c5(DX(z0)) 2.77/1.13
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.13
DX(ln(z0)) → c7(DX(z0)) 2.77/1.13
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
K tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1))
Defined Rule Symbols:

dx

Defined Pair Symbols:

DX

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8

2.77/1.13
2.77/1.13

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.13
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
We considered the (Usable) Rules:none
And the Tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.13
DX(neg(z0)) → c5(DX(z0)) 2.77/1.13
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.13
DX(ln(z0)) → c7(DX(z0)) 2.77/1.13
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.77/1.13

POL(DX(x1)) = [4] + [2]x1    2.77/1.13
POL(c2(x1, x2)) = x1 + x2    2.77/1.13
POL(c3(x1, x2)) = x1 + x2    2.77/1.13
POL(c4(x1, x2)) = x1 + x2    2.77/1.13
POL(c5(x1)) = x1    2.77/1.13
POL(c6(x1, x2)) = x1 + x2    2.77/1.13
POL(c7(x1)) = x1    2.77/1.13
POL(c8(x1, x2)) = x1 + x2    2.77/1.13
POL(div(x1, x2)) = [5] + x1 + x2    2.77/1.13
POL(exp(x1, x2)) = [5] + x1 + x2    2.77/1.13
POL(ln(x1)) = x1    2.77/1.13
POL(minus(x1, x2)) = [4] + x1 + x2    2.77/1.13
POL(neg(x1)) = x1    2.77/1.13
POL(plus(x1, x2)) = [5] + x1 + x2    2.77/1.13
POL(times(x1, x2)) = [5] + x1 + x2   
2.77/1.13
2.77/1.13

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

dx(z0) → one 2.77/1.13
dx(a) → zero 2.77/1.13
dx(plus(z0, z1)) → plus(dx(z0), dx(z1)) 2.77/1.13
dx(times(z0, z1)) → plus(times(z1, dx(z0)), times(z0, dx(z1))) 2.77/1.13
dx(minus(z0, z1)) → minus(dx(z0), dx(z1)) 2.77/1.13
dx(neg(z0)) → neg(dx(z0)) 2.77/1.13
dx(div(z0, z1)) → minus(div(dx(z0), z1), times(z0, div(dx(z1), exp(z1, two)))) 2.77/1.13
dx(ln(z0)) → div(dx(z0), z0) 2.77/1.13
dx(exp(z0, z1)) → plus(times(z1, times(exp(z0, minus(z1, one)), dx(z0))), times(exp(z0, z1), times(ln(z0), dx(z1))))
Tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.13
DX(neg(z0)) → c5(DX(z0)) 2.77/1.13
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.13
DX(ln(z0)) → c7(DX(z0)) 2.77/1.13
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
S tuples:

DX(neg(z0)) → c5(DX(z0)) 2.77/1.13
DX(ln(z0)) → c7(DX(z0))
K tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.13
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.13
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
Defined Rule Symbols:

dx

Defined Pair Symbols:

DX

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8

2.77/1.13
2.77/1.13

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

DX(ln(z0)) → c7(DX(z0))
We considered the (Usable) Rules:none
And the Tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.13
DX(neg(z0)) → c5(DX(z0)) 2.77/1.13
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.13
DX(ln(z0)) → c7(DX(z0)) 2.77/1.13
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.77/1.13

POL(DX(x1)) = [2]x1    2.77/1.13
POL(c2(x1, x2)) = x1 + x2    2.77/1.13
POL(c3(x1, x2)) = x1 + x2    2.77/1.13
POL(c4(x1, x2)) = x1 + x2    2.77/1.13
POL(c5(x1)) = x1    2.77/1.13
POL(c6(x1, x2)) = x1 + x2    2.77/1.13
POL(c7(x1)) = x1    2.77/1.13
POL(c8(x1, x2)) = x1 + x2    2.77/1.13
POL(div(x1, x2)) = x1 + x2    2.77/1.13
POL(exp(x1, x2)) = x1 + x2    2.77/1.13
POL(ln(x1)) = [2] + x1    2.77/1.13
POL(minus(x1, x2)) = x1 + x2    2.77/1.13
POL(neg(x1)) = x1    2.77/1.13
POL(plus(x1, x2)) = [3] + x1 + x2    2.77/1.13
POL(times(x1, x2)) = [3] + x1 + x2   
2.77/1.13
2.77/1.13

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

dx(z0) → one 2.77/1.13
dx(a) → zero 2.77/1.13
dx(plus(z0, z1)) → plus(dx(z0), dx(z1)) 2.77/1.13
dx(times(z0, z1)) → plus(times(z1, dx(z0)), times(z0, dx(z1))) 2.77/1.13
dx(minus(z0, z1)) → minus(dx(z0), dx(z1)) 2.77/1.13
dx(neg(z0)) → neg(dx(z0)) 2.77/1.13
dx(div(z0, z1)) → minus(div(dx(z0), z1), times(z0, div(dx(z1), exp(z1, two)))) 2.77/1.13
dx(ln(z0)) → div(dx(z0), z0) 2.77/1.13
dx(exp(z0, z1)) → plus(times(z1, times(exp(z0, minus(z1, one)), dx(z0))), times(exp(z0, z1), times(ln(z0), dx(z1))))
Tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.13
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.13
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.14
DX(neg(z0)) → c5(DX(z0)) 2.77/1.14
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.14
DX(ln(z0)) → c7(DX(z0)) 2.77/1.14
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
S tuples:

DX(neg(z0)) → c5(DX(z0))
K tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.14
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.14
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.14
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.14
DX(exp(z0, z1)) → c8(DX(z0), DX(z1)) 2.77/1.14
DX(ln(z0)) → c7(DX(z0))
Defined Rule Symbols:

dx

Defined Pair Symbols:

DX

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8

2.77/1.14
2.77/1.14

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

DX(neg(z0)) → c5(DX(z0))
We considered the (Usable) Rules:none
And the Tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.14
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.14
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.14
DX(neg(z0)) → c5(DX(z0)) 2.77/1.14
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.14
DX(ln(z0)) → c7(DX(z0)) 2.77/1.14
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.77/1.14

POL(DX(x1)) = [3] + [4]x1    2.77/1.14
POL(c2(x1, x2)) = x1 + x2    2.77/1.14
POL(c3(x1, x2)) = x1 + x2    2.77/1.14
POL(c4(x1, x2)) = x1 + x2    2.77/1.14
POL(c5(x1)) = x1    2.77/1.14
POL(c6(x1, x2)) = x1 + x2    2.77/1.14
POL(c7(x1)) = x1    2.77/1.14
POL(c8(x1, x2)) = x1 + x2    2.77/1.14
POL(div(x1, x2)) = [1] + x1 + x2    2.77/1.14
POL(exp(x1, x2)) = [1] + x1 + x2    2.77/1.14
POL(ln(x1)) = x1    2.77/1.14
POL(minus(x1, x2)) = [1] + x1 + x2    2.77/1.14
POL(neg(x1)) = [1] + x1    2.77/1.14
POL(plus(x1, x2)) = [1] + x1 + x2    2.77/1.14
POL(times(x1, x2)) = [1] + x1 + x2   
2.77/1.14
2.77/1.14

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

dx(z0) → one 2.77/1.14
dx(a) → zero 2.77/1.14
dx(plus(z0, z1)) → plus(dx(z0), dx(z1)) 2.77/1.14
dx(times(z0, z1)) → plus(times(z1, dx(z0)), times(z0, dx(z1))) 2.77/1.14
dx(minus(z0, z1)) → minus(dx(z0), dx(z1)) 2.77/1.14
dx(neg(z0)) → neg(dx(z0)) 2.77/1.14
dx(div(z0, z1)) → minus(div(dx(z0), z1), times(z0, div(dx(z1), exp(z1, two)))) 2.77/1.14
dx(ln(z0)) → div(dx(z0), z0) 2.77/1.14
dx(exp(z0, z1)) → plus(times(z1, times(exp(z0, minus(z1, one)), dx(z0))), times(exp(z0, z1), times(ln(z0), dx(z1))))
Tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.14
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.14
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.14
DX(neg(z0)) → c5(DX(z0)) 2.77/1.14
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.14
DX(ln(z0)) → c7(DX(z0)) 2.77/1.14
DX(exp(z0, z1)) → c8(DX(z0), DX(z1))
S tuples:none
K tuples:

DX(plus(z0, z1)) → c2(DX(z0), DX(z1)) 2.77/1.14
DX(minus(z0, z1)) → c4(DX(z0), DX(z1)) 2.77/1.14
DX(times(z0, z1)) → c3(DX(z0), DX(z1)) 2.77/1.14
DX(div(z0, z1)) → c6(DX(z0), DX(z1)) 2.77/1.14
DX(exp(z0, z1)) → c8(DX(z0), DX(z1)) 2.77/1.14
DX(ln(z0)) → c7(DX(z0)) 2.77/1.14
DX(neg(z0)) → c5(DX(z0))
Defined Rule Symbols:

dx

Defined Pair Symbols:

DX

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8

2.77/1.14
2.77/1.14

(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
2.77/1.14
2.77/1.14

(12) BOUNDS(O(1), O(1))

2.77/1.14
2.77/1.14
2.95/1.20 EOF