YES(O(1), O(n^2)) 0.00/0.89 YES(O(1), O(n^2)) 0.00/0.91 0.00/0.91 0.00/0.91 0.00/0.91 0.00/0.91 0.00/0.91 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.91 0.00/0.91 0.00/0.91
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0.00/0.91

(0) Obligation:

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

selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs)) 0.00/0.91
select(Cons(x, xs)) → selects(x, Nil, xs) 0.00/0.91
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest)) 0.00/0.91
selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil) 0.00/0.91
select(Nil) → Nil 0.00/0.91
revapp(Nil, rest) → rest

Rewrite Strategy: INNERMOST
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0.00/0.91

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

Converted CpxTRS to CDT
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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

selects(z0, z1, Cons(z2, z3)) → Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3)) 0.00/0.91
selects(z0, z1, Nil) → Cons(Cons(z0, revapp(z1, Nil)), Nil) 0.00/0.91
select(Cons(z0, z1)) → selects(z0, Nil, z1) 0.00/0.91
select(Nil) → Nil 0.00/0.91
revapp(Cons(z0, z1), z2) → revapp(z1, Cons(z0, z2)) 0.00/0.91
revapp(Nil, z0) → z0
Tuples:

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.91
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil)) 0.00/0.91
SELECT(Cons(z0, z1)) → c2(SELECTS(z0, Nil, z1)) 0.00/0.92
REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
S tuples:

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.92
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil)) 0.00/0.92
SELECT(Cons(z0, z1)) → c2(SELECTS(z0, Nil, z1)) 0.00/0.92
REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
K tuples:none
Defined Rule Symbols:

selects, select, revapp

Defined Pair Symbols:

SELECTS, SELECT, REVAPP

Compound Symbols:

c, c1, c2, c4

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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

SELECT(Cons(z0, z1)) → c2(SELECTS(z0, Nil, z1))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

selects(z0, z1, Cons(z2, z3)) → Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3)) 0.00/0.92
selects(z0, z1, Nil) → Cons(Cons(z0, revapp(z1, Nil)), Nil) 0.00/0.92
select(Cons(z0, z1)) → selects(z0, Nil, z1) 0.00/0.92
select(Nil) → Nil 0.00/0.92
revapp(Cons(z0, z1), z2) → revapp(z1, Cons(z0, z2)) 0.00/0.92
revapp(Nil, z0) → z0
Tuples:

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.92
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil)) 0.00/0.92
REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
S tuples:

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.92
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil)) 0.00/0.92
REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
K tuples:none
Defined Rule Symbols:

selects, select, revapp

Defined Pair Symbols:

SELECTS, REVAPP

Compound Symbols:

c, c1, c4

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(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.

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.92
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil))
We considered the (Usable) Rules:none
And the Tuples:

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.92
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil)) 0.00/0.92
REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.92

POL(Cons(x1, x2)) = [1] + x1 + x2    0.00/0.92
POL(Nil) = [5]    0.00/0.92
POL(REVAPP(x1, x2)) = [4]    0.00/0.92
POL(SELECTS(x1, x2, x3)) = [5]x3    0.00/0.92
POL(c(x1, x2)) = x1 + x2    0.00/0.92
POL(c1(x1)) = x1    0.00/0.92
POL(c4(x1)) = x1   
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0.00/0.92

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

selects(z0, z1, Cons(z2, z3)) → Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3)) 0.00/0.92
selects(z0, z1, Nil) → Cons(Cons(z0, revapp(z1, Nil)), Nil) 0.00/0.92
select(Cons(z0, z1)) → selects(z0, Nil, z1) 0.00/0.92
select(Nil) → Nil 0.00/0.92
revapp(Cons(z0, z1), z2) → revapp(z1, Cons(z0, z2)) 0.00/0.92
revapp(Nil, z0) → z0
Tuples:

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.92
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil)) 0.00/0.92
REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
S tuples:

REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
K tuples:

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.92
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil))
Defined Rule Symbols:

selects, select, revapp

Defined Pair Symbols:

SELECTS, REVAPP

Compound Symbols:

c, c1, c4

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(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.

REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
We considered the (Usable) Rules:none
And the Tuples:

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.92
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil)) 0.00/0.92
REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.92

POL(Cons(x1, x2)) = [1] + x2    0.00/0.92
POL(Nil) = 0    0.00/0.92
POL(REVAPP(x1, x2)) = x1    0.00/0.92
POL(SELECTS(x1, x2, x3)) = x2 + [2]x32 + x2·x3    0.00/0.92
POL(c(x1, x2)) = x1 + x2    0.00/0.92
POL(c1(x1)) = x1    0.00/0.92
POL(c4(x1)) = x1   
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0.00/0.92

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

selects(z0, z1, Cons(z2, z3)) → Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3)) 0.00/0.92
selects(z0, z1, Nil) → Cons(Cons(z0, revapp(z1, Nil)), Nil) 0.00/0.92
select(Cons(z0, z1)) → selects(z0, Nil, z1) 0.00/0.92
select(Nil) → Nil 0.00/0.92
revapp(Cons(z0, z1), z2) → revapp(z1, Cons(z0, z2)) 0.00/0.92
revapp(Nil, z0) → z0
Tuples:

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.92
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil)) 0.00/0.92
REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
S tuples:none
K tuples:

SELECTS(z0, z1, Cons(z2, z3)) → c(REVAPP(z1, Cons(z2, z3)), SELECTS(z2, Cons(z0, z1), z3)) 0.00/0.92
SELECTS(z0, z1, Nil) → c1(REVAPP(z1, Nil)) 0.00/0.92
REVAPP(Cons(z0, z1), z2) → c4(REVAPP(z1, Cons(z0, z2)))
Defined Rule Symbols:

selects, select, revapp

Defined Pair Symbols:

SELECTS, REVAPP

Compound Symbols:

c, c1, c4

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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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0.00/0.96 EOF