YES(O(1), O(n^2)) 2.42/1.05 YES(O(1), O(n^2)) 2.42/1.08 2.42/1.08 2.42/1.08 2.42/1.08 2.42/1.08 2.42/1.08 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 2.42/1.08 2.42/1.08 2.42/1.08
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
2.42/1.08 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
2.42/1.08 
↳2 CdtProblem
2.42/1.08 
↳3 CdtUnreachableProof (⇔)
2.42/1.08 
↳4 CdtProblem
2.42/1.08 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.42/1.08 
↳6 CdtProblem
2.42/1.08 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
2.42/1.08 
↳8 CdtProblem
2.42/1.08 
↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))
2.42/1.08 
↳10 BOUNDS(O(1), O(1))
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2.42/1.08

(0) Obligation:

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

times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z))) 2.42/1.08
times(x, 0) → 0 2.42/1.08
times(x, s(y)) → plus(times(x, y), x) 2.42/1.08
plus(x, 0) → x 2.42/1.08
plus(x, s(y)) → s(plus(x, y))

Rewrite Strategy: INNERMOST
2.42/1.08
2.42/1.08

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

Converted CpxTRS to CDT
2.42/1.08
2.42/1.08

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, plus(z1, s(z2))) → plus(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2))) 2.42/1.08
times(z0, 0) → 0 2.42/1.08
times(z0, s(z1)) → plus(times(z0, z1), z0) 2.42/1.08
plus(z0, 0) → z0 2.42/1.08
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:

TIMES(z0, plus(z1, s(z2))) → c(PLUS(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2))), TIMES(z0, plus(z1, times(s(z2), 0))), PLUS(z1, times(s(z2), 0)), TIMES(s(z2), 0), TIMES(z0, s(z2))) 2.42/1.08
TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1)) 2.42/1.08
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
S tuples:

TIMES(z0, plus(z1, s(z2))) → c(PLUS(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2))), TIMES(z0, plus(z1, times(s(z2), 0))), PLUS(z1, times(s(z2), 0)), TIMES(s(z2), 0), TIMES(z0, s(z2))) 2.42/1.08
TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1)) 2.42/1.08
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c, c2, c4

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2.42/1.08

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

TIMES(z0, plus(z1, s(z2))) → c(PLUS(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2))), TIMES(z0, plus(z1, times(s(z2), 0))), PLUS(z1, times(s(z2), 0)), TIMES(s(z2), 0), TIMES(z0, s(z2)))
2.42/1.08
2.42/1.08

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, plus(z1, s(z2))) → plus(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2))) 2.42/1.08
times(z0, 0) → 0 2.42/1.08
times(z0, s(z1)) → plus(times(z0, z1), z0) 2.42/1.08
plus(z0, 0) → z0 2.42/1.08
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1)) 2.42/1.08
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
S tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1)) 2.42/1.08
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c2, c4

2.42/1.08
2.42/1.08

(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.

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
We considered the (Usable) Rules:

times(z0, 0) → 0 2.42/1.08
times(z0, s(z1)) → plus(times(z0, z1), z0) 2.42/1.08
plus(z0, 0) → z0 2.42/1.08
plus(z0, s(z1)) → s(plus(z0, z1))
And the Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1)) 2.42/1.08
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.42/1.08

POL(0) = [3]    2.42/1.08
POL(PLUS(x1, x2)) = [1]    2.42/1.08
POL(TIMES(x1, x2)) = [4]x2    2.42/1.08
POL(c2(x1, x2)) = x1 + x2    2.42/1.08
POL(c4(x1)) = x1    2.42/1.08
POL(plus(x1, x2)) = [5]    2.42/1.08
POL(s(x1)) = [1] + x1    2.42/1.08
POL(times(x1, x2)) = 0   
2.42/1.08
2.42/1.08

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, plus(z1, s(z2))) → plus(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2))) 2.42/1.08
times(z0, 0) → 0 2.42/1.08
times(z0, s(z1)) → plus(times(z0, z1), z0) 2.42/1.08
plus(z0, 0) → z0 2.42/1.08
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1)) 2.42/1.08
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
S tuples:

PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
K tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c2, c4

2.42/1.08
2.42/1.08

(7) 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.

PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
We considered the (Usable) Rules:

times(z0, 0) → 0 2.42/1.08
times(z0, s(z1)) → plus(times(z0, z1), z0) 2.42/1.08
plus(z0, 0) → z0 2.42/1.08
plus(z0, s(z1)) → s(plus(z0, z1))
And the Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1)) 2.42/1.08
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.42/1.08

POL(0) = [1]    2.42/1.08
POL(PLUS(x1, x2)) = [2]x2    2.42/1.08
POL(TIMES(x1, x2)) = x22 + x1·x2    2.42/1.08
POL(c2(x1, x2)) = x1 + x2    2.42/1.08
POL(c4(x1)) = x1    2.42/1.08
POL(plus(x1, x2)) = [1] + x1 + [3]x2    2.42/1.08
POL(s(x1)) = [3] + x1    2.42/1.08
POL(times(x1, x2)) = [1] + [2]x1 + x2 + x22 + [2]x1·x2   
2.42/1.08
2.42/1.08

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, plus(z1, s(z2))) → plus(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2))) 2.42/1.08
times(z0, 0) → 0 2.42/1.08
times(z0, s(z1)) → plus(times(z0, z1), z0) 2.42/1.08
plus(z0, 0) → z0 2.42/1.08
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1)) 2.42/1.08
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
S tuples:none
K tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1)) 2.42/1.08
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c2, c4

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2.42/1.08

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

The set S is empty
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2.42/1.08

(10) BOUNDS(O(1), O(1))

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2.76/1.12 EOF