YES(O(1), O(1)) 0.00/0.53 YES(O(1), O(1)) 0.00/0.54 0.00/0.54 0.00/0.54 0.00/0.54 0.00/0.54 0.00/0.54 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.54 0.00/0.54 0.00/0.54
0.00/0.54 0.00/0.54 0.00/0.54
0.00/0.54
0.00/0.54

(0) Obligation:

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

f(f(X)) → f(a(b(f(X)))) 0.00/0.54
f(a(g(X))) → b(X) 0.00/0.54
b(X) → a(X)

Rewrite Strategy: INNERMOST
0.00/0.54
0.00/0.54

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

Converted CpxTRS to CDT
0.00/0.54
0.00/0.54

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(a(b(f(z0)))) 0.00/0.54
f(a(g(z0))) → b(z0) 0.00/0.54
b(z0) → a(z0)
Tuples:

F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0)) 0.00/0.54
F(a(g(z0))) → c1(B(z0))
S tuples:

F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0)) 0.00/0.54
F(a(g(z0))) → c1(B(z0))
K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:

F

Compound Symbols:

c, c1

0.00/0.54
0.00/0.54

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
0.00/0.54
0.00/0.54

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(a(b(f(z0)))) 0.00/0.54
f(a(g(z0))) → b(z0) 0.00/0.54
b(z0) → a(z0)
Tuples:

F(a(g(z0))) → c1(B(z0))
S tuples:

F(a(g(z0))) → c1(B(z0))
K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:

F

Compound Symbols:

c1

0.00/0.54
0.00/0.54

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

Removed 1 trailing tuple parts
0.00/0.54
0.00/0.54

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(a(b(f(z0)))) 0.00/0.54
f(a(g(z0))) → b(z0) 0.00/0.54
b(z0) → a(z0)
Tuples:

F(a(g(z0))) → c1
S tuples:

F(a(g(z0))) → c1
K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:

F

Compound Symbols:

c1

0.00/0.54
0.00/0.54

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

Removed 1 trailing nodes:

F(a(g(z0))) → c1
0.00/0.54
0.00/0.54

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(a(b(f(z0)))) 0.00/0.54
f(a(g(z0))) → b(z0) 0.00/0.54
b(z0) → a(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:none

Compound Symbols:none

0.00/0.54
0.00/0.54

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

The set S is empty
0.00/0.54
0.00/0.54

(10) BOUNDS(O(1), O(1))

0.00/0.54
0.00/0.54
0.00/0.57 EOF