3.18/1.58	WORST_CASE(?, O(1))
3.18/1.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.18/1.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.18/1.58	
3.18/1.58	
3.18/1.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.18/1.58	
3.18/1.58	(0) CpxTRS
3.18/1.58	(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
3.18/1.58	(2) CpxTRS
3.18/1.58	(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
3.18/1.58	(4) BOUNDS(1, 1)
3.18/1.58	
3.18/1.58	
3.18/1.58	----------------------------------------
3.18/1.58	
3.18/1.58	(0)
3.18/1.58	Obligation:
3.18/1.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.18/1.58	
3.18/1.58	
3.18/1.58	The TRS R consists of the following rules:
3.18/1.58	
3.18/1.58	   a(c(d(x))) -> c(x)
3.18/1.58	   u(b(d(d(x)))) -> b(x)
3.18/1.58	   v(a(a(x))) -> u(v(x))
3.18/1.58	   v(a(c(x))) -> u(b(d(x)))
3.18/1.58	   v(c(x)) -> b(x)
3.18/1.58	   w(a(a(x))) -> u(w(x))
3.18/1.58	   w(a(c(x))) -> u(b(d(x)))
3.18/1.58	   w(c(x)) -> b(x)
3.18/1.58	
3.18/1.58	S is empty.
3.18/1.58	Rewrite Strategy: FULL
3.18/1.58	----------------------------------------
3.18/1.58	
3.18/1.58	(1) DependencyGraphProof (UPPER BOUND(ID))
3.18/1.58	The following rules are not reachable from basic terms in the dependency graph and can be removed:
3.18/1.58	
3.18/1.58	v(a(a(x))) -> u(v(x))
3.18/1.58	v(a(c(x))) -> u(b(d(x)))
3.18/1.59	w(a(a(x))) -> u(w(x))
3.18/1.59	w(a(c(x))) -> u(b(d(x)))
3.18/1.59	
3.18/1.59	----------------------------------------
3.18/1.59	
3.18/1.59	(2)
3.18/1.59	Obligation:
3.18/1.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.18/1.59	
3.18/1.59	
3.18/1.59	The TRS R consists of the following rules:
3.18/1.59	
3.18/1.59	   a(c(d(x))) -> c(x)
3.18/1.59	   u(b(d(d(x)))) -> b(x)
3.18/1.59	   v(c(x)) -> b(x)
3.18/1.59	   w(c(x)) -> b(x)
3.18/1.59	
3.18/1.59	S is empty.
3.18/1.59	Rewrite Strategy: FULL
3.18/1.59	----------------------------------------
3.18/1.59	
3.18/1.59	(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
3.18/1.59	Constant runtime complexity proven by termination of constructor-based narrowing.
3.18/1.59	
3.18/1.59	The maximal most general narrowing sequences give rise to the following rewrite sequences:
3.18/1.59	
3.18/1.59	u(b(d(d(x0)))) ->^* b(x0)
3.18/1.59	
3.18/1.59	w(c(x0)) ->^* b(x0)
3.18/1.59	
3.18/1.59	v(c(x0)) ->^* b(x0)
3.18/1.59	
3.18/1.59	a(c(d(x0))) ->^* c(x0)
3.18/1.59	
3.18/1.59	
3.18/1.59	
3.18/1.59	
3.18/1.59	----------------------------------------
3.18/1.59	
3.18/1.59	(4)
3.18/1.59	BOUNDS(1, 1)
3.18/1.65	EOF