3.24/1.57	WORST_CASE(?, O(1))
3.24/1.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.24/1.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.24/1.59	
3.24/1.59	
3.24/1.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.24/1.59	
3.24/1.59	(0) CpxTRS
3.24/1.59	(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
3.24/1.59	(2) CpxTRS
3.24/1.59	(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
3.24/1.59	(4) BOUNDS(1, 1)
3.24/1.59	
3.24/1.59	
3.24/1.59	----------------------------------------
3.24/1.59	
3.24/1.59	(0)
3.24/1.59	Obligation:
3.24/1.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.24/1.59	
3.24/1.59	
3.24/1.59	The TRS R consists of the following rules:
3.24/1.59	
3.24/1.59	   f(a) -> g(h(a))
3.24/1.59	   h(g(x)) -> g(h(f(x)))
3.24/1.59	   k(x, h(x), a) -> h(x)
3.24/1.59	   k(f(x), y, x) -> f(x)
3.24/1.59	
3.24/1.59	S is empty.
3.24/1.59	Rewrite Strategy: FULL
3.24/1.59	----------------------------------------
3.24/1.59	
3.24/1.59	(1) DependencyGraphProof (UPPER BOUND(ID))
3.24/1.59	The following rules are not reachable from basic terms in the dependency graph and can be removed:
3.24/1.59	
3.24/1.59	k(x, h(x), a) -> h(x)
3.24/1.59	k(f(x), y, x) -> f(x)
3.24/1.59	
3.24/1.59	----------------------------------------
3.24/1.59	
3.24/1.59	(2)
3.24/1.59	Obligation:
3.24/1.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.24/1.59	
3.24/1.59	
3.24/1.59	The TRS R consists of the following rules:
3.24/1.59	
3.24/1.59	   f(a) -> g(h(a))
3.24/1.59	   h(g(x)) -> g(h(f(x)))
3.24/1.59	
3.24/1.59	S is empty.
3.24/1.59	Rewrite Strategy: FULL
3.24/1.59	----------------------------------------
3.24/1.59	
3.24/1.59	(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
3.24/1.59	Constant runtime complexity proven by termination of constructor-based narrowing.
3.24/1.59	
3.24/1.59	The maximal most general narrowing sequences give rise to the following rewrite sequences:
3.24/1.59	
3.24/1.59	h(g(a)) ->^* g(g(h(f(h(a)))))
3.24/1.59	
3.24/1.59	f(a) ->^* g(h(a))
3.24/1.59	
3.24/1.59	
3.24/1.59	
3.24/1.59	
3.24/1.59	----------------------------------------
3.24/1.59	
3.24/1.59	(4)
3.24/1.59	BOUNDS(1, 1)
3.43/1.63	EOF