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
0.00/0.84
0.00/0.84
0.00/0.84
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
0.00/0.84 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳2 CdtProblem
0.00/0.84 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳4 CdtProblem
0.00/0.84 
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳6 CdtProblem
0.00/0.84 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.84 
↳8 CdtProblem
0.00/0.84 
↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳10 CdtProblem
0.00/0.84 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.84 
↳12 CdtProblem
0.00/0.84 
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳14 BOUNDS(O(1), O(1))
0.00/0.84
0.00/0.84 0.00/0.84
0.00/0.84
0.00/0.84

(0) Obligation:

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

a__f(X, g(X), Y) → a__f(Y, Y, Y) 0.00/0.84
a__g(b) → c 0.00/0.84
a__b → c 0.00/0.84
mark(f(X1, X2, X3)) → a__f(X1, X2, X3) 0.00/0.84
mark(g(X)) → a__g(mark(X)) 0.00/0.84
mark(b) → a__b 0.00/0.84
mark(c) → c 0.00/0.84
a__f(X1, X2, X3) → f(X1, X2, X3) 0.00/0.84
a__g(X) → g(X) 0.00/0.84
a__b → b

Rewrite Strategy: INNERMOST
0.00/0.84
0.00/0.84

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

Converted CpxTRS to CDT
0.00/0.84
0.00/0.84

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(z0, g(z0), z1) → a__f(z1, z1, z1) 0.00/0.84
a__f(z0, z1, z2) → f(z0, z1, z2) 0.00/0.84
a__g(b) → c 0.00/0.84
a__g(z0) → g(z0) 0.00/0.84
a__b → c 0.00/0.84
a__b → b 0.00/0.84
mark(f(z0, z1, z2)) → a__f(z0, z1, z2) 0.00/0.84
mark(g(z0)) → a__g(mark(z0)) 0.00/0.84
mark(b) → a__b 0.00/0.84
mark(c) → c
Tuples:

A__F(z0, g(z0), z1) → c1(A__F(z1, z1, z1)) 0.00/0.84
MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
MARK(g(z0)) → c8(A__G(mark(z0)), MARK(z0)) 0.00/0.84
MARK(b) → c9(A__B)
S tuples:

A__F(z0, g(z0), z1) → c1(A__F(z1, z1, z1)) 0.00/0.84
MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
MARK(g(z0)) → c8(A__G(mark(z0)), MARK(z0)) 0.00/0.84
MARK(b) → c9(A__B)
K tuples:none
Defined Rule Symbols:

a__f, a__g, a__b, mark

Defined Pair Symbols:

A__F, MARK

Compound Symbols:

c1, c7, c8, c9

0.00/0.84
0.00/0.84

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

Removed 3 trailing tuple parts
0.00/0.84
0.00/0.84

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(z0, g(z0), z1) → a__f(z1, z1, z1) 0.00/0.84
a__f(z0, z1, z2) → f(z0, z1, z2) 0.00/0.84
a__g(b) → c 0.00/0.84
a__g(z0) → g(z0) 0.00/0.84
a__b → c 0.00/0.84
a__b → b 0.00/0.84
mark(f(z0, z1, z2)) → a__f(z0, z1, z2) 0.00/0.84
mark(g(z0)) → a__g(mark(z0)) 0.00/0.84
mark(b) → a__b 0.00/0.84
mark(c) → c
Tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0)) 0.00/0.84
MARK(b) → c9
S tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0)) 0.00/0.84
MARK(b) → c9
K tuples:none
Defined Rule Symbols:

a__f, a__g, a__b, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c7, c1, c8, c9

0.00/0.84
0.00/0.84

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

Removed 3 trailing nodes:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
MARK(b) → c9 0.00/0.84
A__F(z0, g(z0), z1) → c1
0.00/0.84
0.00/0.84

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(z0, g(z0), z1) → a__f(z1, z1, z1) 0.00/0.84
a__f(z0, z1, z2) → f(z0, z1, z2) 0.00/0.84
a__g(b) → c 0.00/0.84
a__g(z0) → g(z0) 0.00/0.84
a__b → c 0.00/0.84
a__b → b 0.00/0.84
mark(f(z0, z1, z2)) → a__f(z0, z1, z2) 0.00/0.84
mark(g(z0)) → a__g(mark(z0)) 0.00/0.84
mark(b) → a__b 0.00/0.84
mark(c) → c
Tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0)) 0.00/0.84
MARK(b) → c9
S tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0)) 0.00/0.84
MARK(b) → c9
K tuples:none
Defined Rule Symbols:

a__f, a__g, a__b, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c7, c1, c8, c9

0.00/0.84
0.00/0.84

(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.

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
MARK(b) → c9
We considered the (Usable) Rules:none
And the Tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0)) 0.00/0.84
MARK(b) → c9
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.84

POL(A__F(x1, x2, x3)) = [4]x1 + [2]x2 + [4]x3    0.00/0.84
POL(MARK(x1)) = [1] + [4]x1    0.00/0.84
POL(b) = 0    0.00/0.84
POL(c1) = 0    0.00/0.84
POL(c7(x1)) = x1    0.00/0.84
POL(c8(x1)) = x1    0.00/0.84
POL(c9) = 0    0.00/0.84
POL(f(x1, x2, x3)) = x1 + x2 + x3    0.00/0.84
POL(g(x1)) = x1   
0.00/0.84
0.00/0.84

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(z0, g(z0), z1) → a__f(z1, z1, z1) 0.00/0.84
a__f(z0, z1, z2) → f(z0, z1, z2) 0.00/0.84
a__g(b) → c 0.00/0.84
a__g(z0) → g(z0) 0.00/0.84
a__b → c 0.00/0.84
a__b → b 0.00/0.84
mark(f(z0, z1, z2)) → a__f(z0, z1, z2) 0.00/0.84
mark(g(z0)) → a__g(mark(z0)) 0.00/0.84
mark(b) → a__b 0.00/0.84
mark(c) → c
Tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0)) 0.00/0.84
MARK(b) → c9
S tuples:

A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0))
K tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
MARK(b) → c9
Defined Rule Symbols:

a__f, a__g, a__b, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c7, c1, c8, c9

0.00/0.84
0.00/0.84

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

The following tuples could be moved from S to K by knowledge propagation:

A__F(z0, g(z0), z1) → c1
0.00/0.84
0.00/0.84

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(z0, g(z0), z1) → a__f(z1, z1, z1) 0.00/0.84
a__f(z0, z1, z2) → f(z0, z1, z2) 0.00/0.84
a__g(b) → c 0.00/0.84
a__g(z0) → g(z0) 0.00/0.84
a__b → c 0.00/0.84
a__b → b 0.00/0.84
mark(f(z0, z1, z2)) → a__f(z0, z1, z2) 0.00/0.84
mark(g(z0)) → a__g(mark(z0)) 0.00/0.84
mark(b) → a__b 0.00/0.84
mark(c) → c
Tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0)) 0.00/0.84
MARK(b) → c9
S tuples:

MARK(g(z0)) → c8(MARK(z0))
K tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
MARK(b) → c9 0.00/0.84
A__F(z0, g(z0), z1) → c1
Defined Rule Symbols:

a__f, a__g, a__b, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c7, c1, c8, c9

0.00/0.84
0.00/0.84

(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.

MARK(g(z0)) → c8(MARK(z0))
We considered the (Usable) Rules:none
And the Tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0)) 0.00/0.84
MARK(b) → c9
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.84

POL(A__F(x1, x2, x3)) = [1] + [3]x1 + x2    0.00/0.84
POL(MARK(x1)) = [3] + [3]x1    0.00/0.84
POL(b) = 0    0.00/0.84
POL(c1) = 0    0.00/0.84
POL(c7(x1)) = x1    0.00/0.84
POL(c8(x1)) = x1    0.00/0.84
POL(c9) = 0    0.00/0.84
POL(f(x1, x2, x3)) = [3] + x1 + x2 + x3    0.00/0.84
POL(g(x1)) = [1] + x1   
0.00/0.84
0.00/0.84

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(z0, g(z0), z1) → a__f(z1, z1, z1) 0.00/0.84
a__f(z0, z1, z2) → f(z0, z1, z2) 0.00/0.84
a__g(b) → c 0.00/0.84
a__g(z0) → g(z0) 0.00/0.84
a__b → c 0.00/0.84
a__b → b 0.00/0.84
mark(f(z0, z1, z2)) → a__f(z0, z1, z2) 0.00/0.84
mark(g(z0)) → a__g(mark(z0)) 0.00/0.84
mark(b) → a__b 0.00/0.84
mark(c) → c
Tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0)) 0.00/0.84
MARK(b) → c9
S tuples:none
K tuples:

MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2)) 0.00/0.84
MARK(b) → c9 0.00/0.84
A__F(z0, g(z0), z1) → c1 0.00/0.84
MARK(g(z0)) → c8(MARK(z0))
Defined Rule Symbols:

a__f, a__g, a__b, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c7, c1, c8, c9

0.00/0.84
0.00/0.84

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

The set S is empty
0.00/0.84
0.00/0.84

(14) BOUNDS(O(1), O(1))

0.00/0.84
0.00/0.84
0.00/0.88 EOF