YES(O(1), O(n^1)) 6.27/2.05 YES(O(1), O(n^1)) 6.71/2.12 6.71/2.12 6.71/2.12 6.71/2.12 6.71/2.12 6.71/2.12 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 6.71/2.12 6.71/2.12 6.71/2.12
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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
6.71/2.12 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
6.71/2.12 
↳2 CdtProblem
6.71/2.12 
↳3 CdtUnreachableProof (⇔)
6.71/2.12 
↳4 CdtProblem
6.71/2.12 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.71/2.12 
↳6 CdtProblem
6.71/2.12 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.71/2.12 
↳8 CdtProblem
6.71/2.12 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.71/2.12 
↳10 CdtProblem
6.71/2.12 
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
6.71/2.12 
↳12 BOUNDS(O(1), O(1))
6.71/2.12
6.71/2.12 6.71/2.12
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6.71/2.12

(0) Obligation:

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

flatten(nil) → nil 6.71/2.12
flatten(unit(x)) → flatten(x) 6.71/2.12
flatten(++(x, y)) → ++(flatten(x), flatten(y)) 6.71/2.12
flatten(++(unit(x), y)) → ++(flatten(x), flatten(y)) 6.71/2.12
flatten(flatten(x)) → flatten(x) 6.71/2.12
rev(nil) → nil 6.71/2.12
rev(unit(x)) → unit(x) 6.71/2.12
rev(++(x, y)) → ++(rev(y), rev(x)) 6.71/2.12
rev(rev(x)) → x 6.71/2.12
++(x, nil) → x 6.71/2.12
++(nil, y) → y 6.71/2.12
++(++(x, y), z) → ++(x, ++(y, z))

Rewrite Strategy: INNERMOST
6.71/2.12
6.71/2.12

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

Converted CpxTRS to CDT
6.71/2.12
6.71/2.12

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

flatten(nil) → nil 6.71/2.12
flatten(unit(z0)) → flatten(z0) 6.71/2.12
flatten(++(z0, z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.12
flatten(++(unit(z0), z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.12
flatten(flatten(z0)) → flatten(z0) 6.71/2.12
rev(nil) → nil 6.71/2.12
rev(unit(z0)) → unit(z0) 6.71/2.12
rev(++(z0, z1)) → ++(rev(z1), rev(z0)) 6.71/2.12
rev(rev(z0)) → z0 6.71/2.12
++(z0, nil) → z0 6.71/2.12
++(nil, z0) → z0 6.71/2.12
++(++(z0, z1), z2) → ++(z0, ++(z1, z2))
Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.12
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.12
REV(++(z0, z1)) → c7(++'(rev(z1), rev(z0)), REV(z1), REV(z0)) 6.71/2.12
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
S tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.12
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.12
REV(++(z0, z1)) → c7(++'(rev(z1), rev(z0)), REV(z1), REV(z0)) 6.71/2.12
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
K tuples:none
Defined Rule Symbols:

flatten, rev, ++

Defined Pair Symbols:

FLATTEN, REV, ++'

Compound Symbols:

c1, c2, c3, c4, c7, c11

6.71/2.12
6.71/2.12

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

REV(++(z0, z1)) → c7(++'(rev(z1), rev(z0)), REV(z1), REV(z0))
6.71/2.12
6.71/2.12

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

flatten(nil) → nil 6.71/2.12
flatten(unit(z0)) → flatten(z0) 6.71/2.12
flatten(++(z0, z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.12
flatten(++(unit(z0), z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.12
flatten(flatten(z0)) → flatten(z0) 6.71/2.12
rev(nil) → nil 6.71/2.12
rev(unit(z0)) → unit(z0) 6.71/2.12
rev(++(z0, z1)) → ++(rev(z1), rev(z0)) 6.71/2.12
rev(rev(z0)) → z0 6.71/2.12
++(z0, nil) → z0 6.71/2.12
++(nil, z0) → z0 6.71/2.12
++(++(z0, z1), z2) → ++(z0, ++(z1, z2))
Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.12
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.12
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
S tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.12
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.12
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
K tuples:none
Defined Rule Symbols:

flatten, rev, ++

Defined Pair Symbols:

FLATTEN, ++'

Compound Symbols:

c1, c2, c3, c4, c11

6.71/2.12
6.71/2.12

(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.

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.12
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1))
We considered the (Usable) Rules:

++(z0, nil) → z0 6.71/2.12
++(++(z0, z1), z2) → ++(z0, ++(z1, z2)) 6.71/2.12
++(nil, z0) → z0 6.71/2.12
flatten(nil) → nil 6.71/2.12
flatten(unit(z0)) → flatten(z0) 6.71/2.12
flatten(++(z0, z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.12
flatten(++(unit(z0), z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.12
flatten(flatten(z0)) → flatten(z0)
And the Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.12
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.12
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.71/2.12

POL(++(x1, x2)) = [2] + [5]x1 + [4]x2    6.71/2.12
POL(++'(x1, x2)) = 0    6.71/2.12
POL(FLATTEN(x1)) = [4]x1    6.71/2.12
POL(c1(x1)) = x1    6.71/2.12
POL(c11(x1, x2)) = x1 + x2    6.71/2.12
POL(c2(x1, x2, x3)) = x1 + x2 + x3    6.71/2.12
POL(c3(x1, x2, x3)) = x1 + x2 + x3    6.71/2.12
POL(c4(x1)) = x1    6.71/2.12
POL(flatten(x1)) = [4]x1    6.71/2.12
POL(nil) = [3]    6.71/2.12
POL(unit(x1)) = [2] + x1   
6.71/2.12
6.71/2.12

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

flatten(nil) → nil 6.71/2.12
flatten(unit(z0)) → flatten(z0) 6.71/2.12
flatten(++(z0, z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.12
flatten(++(unit(z0), z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.12
flatten(flatten(z0)) → flatten(z0) 6.71/2.12
rev(nil) → nil 6.71/2.12
rev(unit(z0)) → unit(z0) 6.71/2.12
rev(++(z0, z1)) → ++(rev(z1), rev(z0)) 6.71/2.12
rev(rev(z0)) → z0 6.71/2.12
++(z0, nil) → z0 6.71/2.12
++(nil, z0) → z0 6.71/2.12
++(++(z0, z1), z2) → ++(z0, ++(z1, z2))
Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.12
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.12
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
S tuples:

FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.12
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
K tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.12
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1))
Defined Rule Symbols:

flatten, rev, ++

Defined Pair Symbols:

FLATTEN, ++'

Compound Symbols:

c1, c2, c3, c4, c11

6.71/2.12
6.71/2.12

(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.

FLATTEN(flatten(z0)) → c4(FLATTEN(z0))
We considered the (Usable) Rules:

++(z0, nil) → z0 6.71/2.12
++(++(z0, z1), z2) → ++(z0, ++(z1, z2)) 6.71/2.12
++(nil, z0) → z0 6.71/2.12
flatten(nil) → nil 6.71/2.12
flatten(unit(z0)) → flatten(z0) 6.71/2.12
flatten(++(z0, z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.12
flatten(++(unit(z0), z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.12
flatten(flatten(z0)) → flatten(z0)
And the Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.12
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.12
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.12
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.71/2.12

POL(++(x1, x2)) = [5] + [2]x1 + x2    6.71/2.12
POL(++'(x1, x2)) = 0    6.71/2.12
POL(FLATTEN(x1)) = [5] + x1    6.71/2.12
POL(c1(x1)) = x1    6.71/2.12
POL(c11(x1, x2)) = x1 + x2    6.71/2.13
POL(c2(x1, x2, x3)) = x1 + x2 + x3    6.71/2.13
POL(c3(x1, x2, x3)) = x1 + x2 + x3    6.71/2.13
POL(c4(x1)) = x1    6.71/2.13
POL(flatten(x1)) = [4] + [4]x1    6.71/2.13
POL(nil) = [2]    6.71/2.13
POL(unit(x1)) = [1] + x1   
6.71/2.13
6.71/2.13

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

flatten(nil) → nil 6.71/2.13
flatten(unit(z0)) → flatten(z0) 6.71/2.13
flatten(++(z0, z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.13
flatten(++(unit(z0), z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.13
flatten(flatten(z0)) → flatten(z0) 6.71/2.13
rev(nil) → nil 6.71/2.13
rev(unit(z0)) → unit(z0) 6.71/2.13
rev(++(z0, z1)) → ++(rev(z1), rev(z0)) 6.71/2.13
rev(rev(z0)) → z0 6.71/2.13
++(z0, nil) → z0 6.71/2.13
++(nil, z0) → z0 6.71/2.13
++(++(z0, z1), z2) → ++(z0, ++(z1, z2))
Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.13
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.13
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.13
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.13
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
S tuples:

++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
K tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.13
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.13
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.13
FLATTEN(flatten(z0)) → c4(FLATTEN(z0))
Defined Rule Symbols:

flatten, rev, ++

Defined Pair Symbols:

FLATTEN, ++'

Compound Symbols:

c1, c2, c3, c4, c11

6.71/2.13
6.71/2.13

(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.

++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
We considered the (Usable) Rules:

++(z0, nil) → z0 6.71/2.13
++(++(z0, z1), z2) → ++(z0, ++(z1, z2)) 6.71/2.13
++(nil, z0) → z0 6.71/2.13
flatten(nil) → nil 6.71/2.13
flatten(unit(z0)) → flatten(z0) 6.71/2.13
flatten(++(z0, z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.13
flatten(++(unit(z0), z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.13
flatten(flatten(z0)) → flatten(z0)
And the Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.13
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.13
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.13
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.13
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.71/2.13

POL(++(x1, x2)) = [1] + [4]x1 + x2    6.71/2.13
POL(++'(x1, x2)) = x1    6.71/2.13
POL(FLATTEN(x1)) = [2]x1    6.71/2.13
POL(c1(x1)) = x1    6.71/2.13
POL(c11(x1, x2)) = x1 + x2    6.71/2.13
POL(c2(x1, x2, x3)) = x1 + x2 + x3    6.71/2.13
POL(c3(x1, x2, x3)) = x1 + x2 + x3    6.71/2.13
POL(c4(x1)) = x1    6.71/2.13
POL(flatten(x1)) = x1    6.71/2.13
POL(nil) = 0    6.71/2.13
POL(unit(x1)) = x1   
6.71/2.13
6.71/2.13

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

flatten(nil) → nil 6.71/2.13
flatten(unit(z0)) → flatten(z0) 6.71/2.13
flatten(++(z0, z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.13
flatten(++(unit(z0), z1)) → ++(flatten(z0), flatten(z1)) 6.71/2.13
flatten(flatten(z0)) → flatten(z0) 6.71/2.13
rev(nil) → nil 6.71/2.13
rev(unit(z0)) → unit(z0) 6.71/2.13
rev(++(z0, z1)) → ++(rev(z1), rev(z0)) 6.71/2.13
rev(rev(z0)) → z0 6.71/2.13
++(z0, nil) → z0 6.71/2.13
++(nil, z0) → z0 6.71/2.13
++(++(z0, z1), z2) → ++(z0, ++(z1, z2))
Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.13
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.13
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.13
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.13
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
S tuples:none
K tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0)) 6.71/2.13
FLATTEN(++(z0, z1)) → c2(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.13
FLATTEN(++(unit(z0), z1)) → c3(++'(flatten(z0), flatten(z1)), FLATTEN(z0), FLATTEN(z1)) 6.71/2.13
FLATTEN(flatten(z0)) → c4(FLATTEN(z0)) 6.71/2.13
++'(++(z0, z1), z2) → c11(++'(z0, ++(z1, z2)), ++'(z1, z2))
Defined Rule Symbols:

flatten, rev, ++

Defined Pair Symbols:

FLATTEN, ++'

Compound Symbols:

c1, c2, c3, c4, c11

6.71/2.13
6.71/2.13

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

The set S is empty
6.71/2.13
6.71/2.13

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

6.71/2.13
6.71/2.13
7.08/2.23 EOF