3.25/1.54	WORST_CASE(?, O(1))
3.25/1.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.25/1.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.25/1.56	
3.25/1.56	
3.25/1.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.25/1.56	
3.25/1.56	(0) CpxTRS
3.25/1.56	(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
3.25/1.56	(2) CpxTRS
3.25/1.56	(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
3.25/1.56	(4) BOUNDS(1, 1)
3.25/1.56	
3.25/1.56	
3.25/1.56	----------------------------------------
3.25/1.56	
3.25/1.56	(0)
3.25/1.56	Obligation:
3.25/1.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.25/1.56	
3.25/1.56	
3.25/1.56	The TRS R consists of the following rules:
3.25/1.56	
3.25/1.56	   f(f(X)) -> f(a(b(f(X))))
3.25/1.56	   f(a(g(X))) -> b(X)
3.25/1.56	   b(X) -> a(X)
3.25/1.56	
3.25/1.56	S is empty.
3.25/1.56	Rewrite Strategy: FULL
3.25/1.56	----------------------------------------
3.25/1.56	
3.25/1.56	(1) DependencyGraphProof (UPPER BOUND(ID))
3.25/1.56	The following rules are not reachable from basic terms in the dependency graph and can be removed:
3.25/1.56	
3.25/1.56	f(f(X)) -> f(a(b(f(X))))
3.25/1.56	
3.25/1.56	----------------------------------------
3.25/1.56	
3.25/1.56	(2)
3.25/1.56	Obligation:
3.25/1.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.25/1.56	
3.25/1.56	
3.25/1.56	The TRS R consists of the following rules:
3.25/1.56	
3.25/1.56	   f(a(g(X))) -> b(X)
3.25/1.56	   b(X) -> a(X)
3.25/1.56	
3.25/1.56	S is empty.
3.25/1.56	Rewrite Strategy: FULL
3.25/1.56	----------------------------------------
3.25/1.56	
3.25/1.56	(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
3.25/1.56	Constant runtime complexity proven by termination of constructor-based narrowing.
3.25/1.56	
3.25/1.56	The maximal most general narrowing sequences give rise to the following rewrite sequences:
3.25/1.56	
3.25/1.56	f(a(g(x0))) ->^* a(x0)
3.25/1.56	
3.25/1.56	b(x0) ->^* a(x0)
3.25/1.56	
3.25/1.56	
3.25/1.56	
3.25/1.56	
3.25/1.56	----------------------------------------
3.25/1.56	
3.25/1.56	(4)
3.25/1.56	BOUNDS(1, 1)
3.42/1.61	EOF