YES(O(1), O(n^1)) 0.00/0.82 YES(O(1), O(n^1)) 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.84 0.00/0.84 0.00/0.84
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
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↳2 CdtProblem
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↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳4 CdtProblem
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↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
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↳6 CdtProblem
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↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳8 CdtProblem
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↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳10 BOUNDS(O(1), O(1))
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(0) Obligation:

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

f(x, nil) → g(nil, x) 0.00/0.85
f(x, g(y, z)) → g(f(x, y), z) 0.00/0.85
++(x, nil) → x 0.00/0.85
++(x, g(y, z)) → g(++(x, y), z) 0.00/0.85
null(nil) → true 0.00/0.85
null(g(x, y)) → false 0.00/0.85
mem(nil, y) → false 0.00/0.85
mem(g(x, y), z) → or(=(y, z), mem(x, z)) 0.00/0.85
mem(x, max(x)) → not(null(x)) 0.00/0.85
max(g(g(nil, x), y)) → max'(x, y) 0.00/0.85
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u)

Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, nil) → g(nil, z0) 0.00/0.85
f(z0, g(z1, z2)) → g(f(z0, z1), z2) 0.00/0.85
++(z0, nil) → z0 0.00/0.85
++(z0, g(z1, z2)) → g(++(z0, z1), z2) 0.00/0.85
null(nil) → true 0.00/0.85
null(g(z0, z1)) → false 0.00/0.85
mem(nil, z0) → false 0.00/0.85
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2)) 0.00/0.85
mem(z0, max(z0)) → not(null(z0)) 0.00/0.85
max(g(g(nil, z0), z1)) → max'(z0, z1) 0.00/0.85
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MEM(z0, max(z0)) → c8(NULL(z0)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2)))
S tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MEM(z0, max(z0)) → c8(NULL(z0)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2)))
K tuples:none
Defined Rule Symbols:

f, ++, null, mem, max

Defined Pair Symbols:

F, ++', MEM, MAX

Compound Symbols:

c1, c3, c7, c8, c10

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(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, nil) → g(nil, z0) 0.00/0.85
f(z0, g(z1, z2)) → g(f(z0, z1), z2) 0.00/0.85
++(z0, nil) → z0 0.00/0.85
++(z0, g(z1, z2)) → g(++(z0, z1), z2) 0.00/0.85
null(nil) → true 0.00/0.85
null(g(z0, z1)) → false 0.00/0.85
mem(nil, z0) → false 0.00/0.85
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2)) 0.00/0.85
mem(z0, max(z0)) → not(null(z0)) 0.00/0.85
max(g(g(nil, z0), z1)) → max'(z0, z1) 0.00/0.85
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
S tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
K tuples:none
Defined Rule Symbols:

f, ++, null, mem, max

Defined Pair Symbols:

F, ++', MEM, MAX

Compound Symbols:

c1, c3, c7, c10, c8

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(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

MEM(z0, max(z0)) → c8
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, nil) → g(nil, z0) 0.00/0.85
f(z0, g(z1, z2)) → g(f(z0, z1), z2) 0.00/0.85
++(z0, nil) → z0 0.00/0.85
++(z0, g(z1, z2)) → g(++(z0, z1), z2) 0.00/0.85
null(nil) → true 0.00/0.85
null(g(z0, z1)) → false 0.00/0.85
mem(nil, z0) → false 0.00/0.85
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2)) 0.00/0.85
mem(z0, max(z0)) → not(null(z0)) 0.00/0.85
max(g(g(nil, z0), z1)) → max'(z0, z1) 0.00/0.85
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
S tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
K tuples:none
Defined Rule Symbols:

f, ++, null, mem, max

Defined Pair Symbols:

F, ++', MEM, MAX

Compound Symbols:

c1, c3, c7, c10, c8

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(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.

F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
We considered the (Usable) Rules:none
And the Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.85

POL(++'(x1, x2)) = [3]x2    0.00/0.85
POL(F(x1, x2)) = [3]x2    0.00/0.85
POL(MAX(x1)) = [5]x1    0.00/0.85
POL(MEM(x1, x2)) = [3] + [3]x1 + [3]x2    0.00/0.85
POL(c1(x1)) = x1    0.00/0.85
POL(c10(x1)) = x1    0.00/0.85
POL(c3(x1)) = x1    0.00/0.85
POL(c7(x1)) = x1    0.00/0.85
POL(c8) = 0    0.00/0.85
POL(g(x1, x2)) = [3] + x1 + x2    0.00/0.85
POL(max(x1)) = [3]    0.00/0.85
POL(u) = [3]   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, nil) → g(nil, z0) 0.00/0.85
f(z0, g(z1, z2)) → g(f(z0, z1), z2) 0.00/0.85
++(z0, nil) → z0 0.00/0.85
++(z0, g(z1, z2)) → g(++(z0, z1), z2) 0.00/0.85
null(nil) → true 0.00/0.85
null(g(z0, z1)) → false 0.00/0.85
mem(nil, z0) → false 0.00/0.85
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2)) 0.00/0.85
mem(z0, max(z0)) → not(null(z0)) 0.00/0.85
max(g(g(nil, z0), z1)) → max'(z0, z1) 0.00/0.85
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
S tuples:none
K tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1)) 0.00/0.85
++'(z0, g(z1, z2)) → c3(++'(z0, z1)) 0.00/0.85
MEM(g(z0, z1), z2) → c7(MEM(z0, z2)) 0.00/0.85
MAX(g(g(g(z0, z1), z2), u)) → c10(MAX(g(g(z0, z1), z2))) 0.00/0.85
MEM(z0, max(z0)) → c8
Defined Rule Symbols:

f, ++, null, mem, max

Defined Pair Symbols:

F, ++', MEM, MAX

Compound Symbols:

c1, c3, c7, c10, c8

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

The set S is empty
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(10) BOUNDS(O(1), O(1))

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0.00/0.90 EOF