MAYBE 11.65/6.70 MAYBE 11.65/6.72 11.65/6.72 11.65/6.72 11.65/6.72 11.65/6.73 11.65/6.73 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 11.65/6.73 11.65/6.73 11.65/6.73
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), INFINITE).

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 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
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↳6 CdtProblem
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↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳8 CdtProblem
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↳9 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
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↳10 CdtProblem
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↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳12 CdtProblem
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↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
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↳14 CdtProblem
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↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
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↳16 CdtProblem
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(0) Obligation:

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

is_empty(nil) → true 11.65/6.73
is_empty(cons(x, l)) → false 11.65/6.73
hd(cons(x, l)) → x 11.65/6.73
tl(cons(x, l)) → cons(x, l) 11.65/6.73
append(l1, l2) → ifappend(l1, l2, is_empty(l1)) 11.65/6.73
ifappend(l1, l2, true) → l2 11.65/6.73
ifappend(l1, l2, false) → cons(hd(l1), append(tl(l1), l2))

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:

is_empty(nil) → true 11.65/6.73
is_empty(cons(z0, z1)) → false 11.65/6.73
hd(cons(z0, z1)) → z0 11.65/6.73
tl(cons(z0, z1)) → cons(z0, z1) 11.65/6.73
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 11.65/6.73
ifappend(z0, z1, true) → z1 11.65/6.73
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0)) 11.65/6.73
IFAPPEND(z0, z1, false) → c6(HD(z0), APPEND(tl(z0), z1), TL(z0))
S tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0)) 11.65/6.73
IFAPPEND(z0, z1, false) → c6(HD(z0), APPEND(tl(z0), z1), TL(z0))
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c6

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

Removed 3 trailing tuple parts
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 11.65/6.73
is_empty(cons(z0, z1)) → false 11.65/6.73
hd(cons(z0, z1)) → z0 11.65/6.73
tl(cons(z0, z1)) → cons(z0, z1) 11.65/6.73
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 11.65/6.73
ifappend(z0, z1, true) → z1 11.65/6.73
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, is_empty(z0))) 11.65/6.73
IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1))
S tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, is_empty(z0))) 11.65/6.73
IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1))
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c6

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

Use narrowing to replace APPEND(z0, z1) → c4(IFAPPEND(z0, z1, is_empty(z0))) by

APPEND(nil, x1) → c4(IFAPPEND(nil, x1, true)) 11.65/6.73
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false))
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 11.65/6.73
is_empty(cons(z0, z1)) → false 11.65/6.73
hd(cons(z0, z1)) → z0 11.65/6.73
tl(cons(z0, z1)) → cons(z0, z1) 11.65/6.73
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 11.65/6.73
ifappend(z0, z1, true) → z1 11.65/6.73
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 11.65/6.73
APPEND(nil, x1) → c4(IFAPPEND(nil, x1, true)) 11.65/6.73
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false))
S tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 11.65/6.73
APPEND(nil, x1) → c4(IFAPPEND(nil, x1, true)) 11.65/6.73
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false))
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

IFAPPEND, APPEND

Compound Symbols:

c6, c4

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

Removed 1 trailing tuple parts
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 11.65/6.73
is_empty(cons(z0, z1)) → false 11.65/6.73
hd(cons(z0, z1)) → z0 11.65/6.73
tl(cons(z0, z1)) → cons(z0, z1) 11.65/6.73
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 11.65/6.73
ifappend(z0, z1, true) → z1 11.65/6.73
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 11.65/6.73
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 11.65/6.73
APPEND(nil, x1) → c4
S tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 11.65/6.73
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 11.65/6.73
APPEND(nil, x1) → c4
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

IFAPPEND, APPEND

Compound Symbols:

c6, c4, c4

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

Removed 1 trailing nodes:

APPEND(nil, x1) → c4
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(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 11.65/6.73
is_empty(cons(z0, z1)) → false 11.65/6.73
hd(cons(z0, z1)) → z0 11.65/6.73
tl(cons(z0, z1)) → cons(z0, z1) 11.65/6.73
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 11.65/6.73
ifappend(z0, z1, true) → z1 11.65/6.73
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 11.65/6.73
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 11.65/6.73
APPEND(nil, x1) → c4
S tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 11.65/6.73
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 11.65/6.73
APPEND(nil, x1) → c4
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

IFAPPEND, APPEND

Compound Symbols:

c6, c4, c4

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(11) 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.

APPEND(nil, x1) → c4
We considered the (Usable) Rules:

tl(cons(z0, z1)) → cons(z0, z1)
And the Tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 11.65/6.73
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 11.65/6.73
APPEND(nil, x1) → c4
The order we found is given by the following interpretation:
Polynomial interpretation : 11.65/6.73

POL(APPEND(x1, x2)) = [1]    11.65/6.73
POL(IFAPPEND(x1, x2, x3)) = [1]    11.65/6.73
POL(c4) = 0    11.65/6.73
POL(c4(x1)) = x1    11.65/6.73
POL(c6(x1)) = x1    11.65/6.73
POL(cons(x1, x2)) = [2]    11.65/6.73
POL(false) = [1]    11.65/6.73
POL(nil) = [1]    11.65/6.73
POL(tl(x1)) = [4]   
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(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 11.65/6.73
is_empty(cons(z0, z1)) → false 11.65/6.73
hd(cons(z0, z1)) → z0 11.65/6.73
tl(cons(z0, z1)) → cons(z0, z1) 11.65/6.73
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 11.65/6.73
ifappend(z0, z1, true) → z1 11.65/6.73
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 11.65/6.73
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 11.65/6.73
APPEND(nil, x1) → c4
S tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 11.65/6.73
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false))
K tuples:

APPEND(nil, x1) → c4
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

IFAPPEND, APPEND

Compound Symbols:

c6, c4, c4

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(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) by

IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(cons(z0, z1), x1))
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(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 11.65/6.73
is_empty(cons(z0, z1)) → false 11.65/6.73
hd(cons(z0, z1)) → z0 11.65/6.73
tl(cons(z0, z1)) → cons(z0, z1) 11.65/6.73
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 11.65/6.73
ifappend(z0, z1, true) → z1 11.65/6.73
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 11.65/6.73
APPEND(nil, x1) → c4 11.65/6.73
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(cons(z0, z1), x1))
S tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 11.65/6.73
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(cons(z0, z1), x1))
K tuples:

APPEND(nil, x1) → c4
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c4, c6

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

Removed 1 trailing nodes:

APPEND(nil, x1) → c4
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(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 11.65/6.73
is_empty(cons(z0, z1)) → false 11.65/6.73
hd(cons(z0, z1)) → z0 11.65/6.73
tl(cons(z0, z1)) → cons(z0, z1) 11.65/6.73
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 11.65/6.73
ifappend(z0, z1, true) → z1 11.65/6.73
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 11.65/6.73
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(cons(z0, z1), x1))
S tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 11.65/6.73
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(cons(z0, z1), x1))
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c6

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11.65/6.78 EOF