2.91/1.51	WORST_CASE(?, O(1))
2.91/1.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.91/1.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.91/1.52	
2.91/1.52	
2.91/1.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
2.91/1.52	
2.91/1.52	(0) CpxTRS
2.91/1.52	(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
2.91/1.52	(2) CpxTRS
2.91/1.52	(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
2.91/1.52	(4) BOUNDS(1, 1)
2.91/1.52	
2.91/1.52	
2.91/1.52	----------------------------------------
2.91/1.52	
2.91/1.52	(0)
2.91/1.52	Obligation:
2.91/1.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
2.91/1.52	
2.91/1.52	
2.91/1.52	The TRS R consists of the following rules:
2.91/1.52	
2.91/1.52	   h(c(x, y), c(s(z), z), t(w)) -> h(z, c(y, x), t(t(c(x, c(y, t(w))))))
2.91/1.52	   h(x, c(y, z), t(w)) -> h(c(s(y), x), z, t(c(t(w), w)))
2.91/1.52	   h(c(s(x), c(s(0), y)), z, t(x)) -> h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
2.91/1.52	   t(t(x)) -> t(c(t(x), x))
2.91/1.52	   t(x) -> x
2.91/1.52	   t(x) -> c(0, c(0, c(0, c(0, c(0, x)))))
2.91/1.52	
2.91/1.52	S is empty.
2.91/1.52	Rewrite Strategy: FULL
2.91/1.52	----------------------------------------
2.91/1.52	
2.91/1.52	(1) DependencyGraphProof (UPPER BOUND(ID))
2.91/1.52	The following rules are not reachable from basic terms in the dependency graph and can be removed:
2.91/1.52	
2.91/1.52	h(c(x, y), c(s(z), z), t(w)) -> h(z, c(y, x), t(t(c(x, c(y, t(w))))))
2.91/1.52	h(x, c(y, z), t(w)) -> h(c(s(y), x), z, t(c(t(w), w)))
2.91/1.52	h(c(s(x), c(s(0), y)), z, t(x)) -> h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
2.91/1.52	t(t(x)) -> t(c(t(x), x))
2.91/1.52	
2.91/1.52	----------------------------------------
2.91/1.52	
2.91/1.52	(2)
2.91/1.52	Obligation:
2.91/1.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
2.91/1.52	
2.91/1.52	
2.91/1.52	The TRS R consists of the following rules:
2.91/1.52	
2.91/1.52	   t(x) -> x
2.91/1.52	   t(x) -> c(0, c(0, c(0, c(0, c(0, x)))))
2.91/1.52	
2.91/1.52	S is empty.
2.91/1.52	Rewrite Strategy: FULL
2.91/1.52	----------------------------------------
2.91/1.52	
2.91/1.52	(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
2.91/1.52	Constant runtime complexity proven by termination of constructor-based narrowing.
2.91/1.52	
2.91/1.52	The maximal most general narrowing sequences give rise to the following rewrite sequences:
2.91/1.52	
2.91/1.52	t(x0) ->^* c(0, c(0, c(0, c(0, c(0, x0)))))
2.91/1.52	
2.91/1.52	
2.91/1.52	
2.91/1.52	
2.91/1.52	----------------------------------------
2.91/1.52	
2.91/1.52	(4)
2.91/1.52	BOUNDS(1, 1)
3.16/1.54	EOF