3.14/1.52	WORST_CASE(?, O(1))
3.20/1.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.20/1.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.20/1.53	
3.20/1.53	
3.20/1.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.20/1.53	
3.20/1.53	(0) CpxTRS
3.20/1.53	(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
3.20/1.53	(2) CpxTRS
3.20/1.53	(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
3.20/1.53	(4) BOUNDS(1, 1)
3.20/1.53	
3.20/1.53	
3.20/1.53	----------------------------------------
3.20/1.53	
3.20/1.53	(0)
3.20/1.53	Obligation:
3.20/1.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.20/1.53	
3.20/1.53	
3.20/1.53	The TRS R consists of the following rules:
3.20/1.53	
3.20/1.53	   f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
3.20/1.53	   p(0) -> g(0)
3.20/1.53	   g(s(p(x))) -> p(x)
3.20/1.53	
3.20/1.53	S is empty.
3.20/1.53	Rewrite Strategy: FULL
3.20/1.53	----------------------------------------
3.20/1.53	
3.20/1.53	(1) DependencyGraphProof (UPPER BOUND(ID))
3.20/1.53	The following rules are not reachable from basic terms in the dependency graph and can be removed:
3.20/1.53	
3.20/1.53	f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
3.20/1.53	g(s(p(x))) -> p(x)
3.20/1.53	
3.20/1.53	----------------------------------------
3.20/1.53	
3.20/1.53	(2)
3.20/1.53	Obligation:
3.20/1.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.20/1.53	
3.20/1.53	
3.20/1.53	The TRS R consists of the following rules:
3.20/1.53	
3.20/1.53	   p(0) -> g(0)
3.20/1.53	
3.20/1.53	S is empty.
3.20/1.53	Rewrite Strategy: FULL
3.20/1.53	----------------------------------------
3.20/1.53	
3.20/1.53	(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
3.20/1.53	Constant runtime complexity proven by termination of constructor-based narrowing.
3.20/1.53	
3.20/1.53	The maximal most general narrowing sequences give rise to the following rewrite sequences:
3.20/1.53	
3.20/1.53	p(0) ->^* g(0)
3.20/1.53	
3.20/1.53	
3.20/1.53	
3.20/1.53	
3.20/1.53	----------------------------------------
3.20/1.53	
3.20/1.53	(4)
3.20/1.53	BOUNDS(1, 1)
3.20/1.57	EOF