3.26/1.63	WORST_CASE(?, O(1))
3.26/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.26/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.26/1.64	
3.26/1.64	
3.26/1.64	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.26/1.64	
3.26/1.64	(0) CpxTRS
3.26/1.64	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
3.26/1.64	(2) CpxTRS
3.26/1.64	(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 12 ms]
3.26/1.64	(4) BOUNDS(1, 1)
3.26/1.64	
3.26/1.64	
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(0)
3.26/1.64	Obligation:
3.26/1.64	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.26/1.64	
3.26/1.64	
3.26/1.64	The TRS R consists of the following rules:
3.26/1.64	
3.26/1.64	   f(f(X)) -> c(n__f(g(n__f(X))))
3.26/1.64	   c(X) -> d(activate(X))
3.26/1.64	   h(X) -> c(n__d(X))
3.26/1.64	   f(X) -> n__f(X)
3.26/1.64	   d(X) -> n__d(X)
3.26/1.64	   activate(n__f(X)) -> f(X)
3.26/1.64	   activate(n__d(X)) -> d(X)
3.26/1.64	   activate(X) -> X
3.26/1.64	
3.26/1.64	S is empty.
3.26/1.64	Rewrite Strategy: FULL
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
3.26/1.64	The following defined symbols can occur below the 0th argument of d: d, f, activate
3.26/1.64	
3.26/1.64	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
3.26/1.64	f(f(X)) -> c(n__f(g(n__f(X))))
3.26/1.64	
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(2)
3.26/1.64	Obligation:
3.26/1.64	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.26/1.64	
3.26/1.64	
3.26/1.64	The TRS R consists of the following rules:
3.26/1.64	
3.26/1.64	   c(X) -> d(activate(X))
3.26/1.64	   h(X) -> c(n__d(X))
3.26/1.64	   f(X) -> n__f(X)
3.26/1.64	   d(X) -> n__d(X)
3.26/1.64	   activate(n__f(X)) -> f(X)
3.26/1.64	   activate(n__d(X)) -> d(X)
3.26/1.64	   activate(X) -> X
3.26/1.64	
3.26/1.64	S is empty.
3.26/1.64	Rewrite Strategy: FULL
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
3.26/1.64	Constant runtime complexity proven by termination of constructor-based narrowing.
3.26/1.64	
3.26/1.64	The maximal most general narrowing sequences give rise to the following rewrite sequences:
3.26/1.64	
3.26/1.64	activate(n__d(x0)) ->^* n__d(x0)
3.26/1.64	
3.26/1.64	activate(n__f(x0)) ->^* n__f(x0)
3.26/1.64	
3.26/1.64	h(x0) ->^* n__d(n__d(x0))
3.26/1.64	
3.26/1.64	f(x0) ->^* n__f(x0)
3.26/1.64	
3.26/1.64	d(x0) ->^* n__d(x0)
3.26/1.64	
3.26/1.64	c(x0) ->^* n__d(x0)
3.26/1.64	
3.26/1.64	c(n__d(x0)) ->^* n__d(n__d(x0))
3.26/1.64	
3.26/1.64	c(n__f(x0)) ->^* n__d(n__f(x0))
3.26/1.64	
3.26/1.64	
3.26/1.64	
3.26/1.64	
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(4)
3.26/1.64	BOUNDS(1, 1)
3.50/1.68	EOF