3.17/1.61	WORST_CASE(?, O(1))
3.17/1.62	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.17/1.62	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.17/1.62	
3.17/1.62	
3.17/1.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.17/1.62	
3.17/1.62	(0) CpxTRS
3.17/1.62	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
3.17/1.62	(2) CpxTRS
3.17/1.62	(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 28 ms]
3.17/1.62	(4) BOUNDS(1, 1)
3.17/1.62	
3.17/1.62	
3.17/1.62	----------------------------------------
3.17/1.62	
3.17/1.62	(0)
3.17/1.62	Obligation:
3.17/1.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.17/1.62	
3.17/1.62	
3.17/1.62	The TRS R consists of the following rules:
3.17/1.62	
3.17/1.62	   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
3.17/1.62	   g(b) -> c
3.17/1.62	   b -> c
3.17/1.62	   g(X) -> n__g(X)
3.17/1.62	   activate(n__g(X)) -> g(X)
3.17/1.62	   activate(X) -> X
3.17/1.62	
3.17/1.62	S is empty.
3.17/1.62	Rewrite Strategy: FULL
3.17/1.62	----------------------------------------
3.17/1.62	
3.17/1.62	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
3.17/1.62	The following defined symbols can occur below the 0th argument of f: activate, g
3.17/1.62	The following defined symbols can occur below the 1th argument of f: activate, g
3.17/1.62	The following defined symbols can occur below the 2th argument of f: activate, g
3.17/1.62	The following defined symbols can occur below the 0th argument of activate: activate, g
3.17/1.62	
3.17/1.62	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
3.17/1.62	g(b) -> c
3.17/1.62	
3.17/1.62	----------------------------------------
3.17/1.62	
3.17/1.62	(2)
3.17/1.62	Obligation:
3.17/1.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.17/1.62	
3.17/1.62	
3.17/1.62	The TRS R consists of the following rules:
3.17/1.62	
3.17/1.62	   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
3.17/1.62	   b -> c
3.17/1.62	   g(X) -> n__g(X)
3.17/1.62	   activate(n__g(X)) -> g(X)
3.17/1.62	   activate(X) -> X
3.17/1.62	
3.17/1.62	S is empty.
3.17/1.62	Rewrite Strategy: FULL
3.17/1.62	----------------------------------------
3.17/1.62	
3.17/1.62	(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
3.17/1.62	Constant runtime complexity proven by termination of constructor-based narrowing.
3.17/1.62	
3.17/1.62	The maximal most general narrowing sequences give rise to the following rewrite sequences:
3.17/1.62	
3.17/1.62	g(x0) ->^* n__g(x0)
3.17/1.62	
3.17/1.62	activate(n__g(x0)) ->^* n__g(x0)
3.17/1.62	
3.17/1.62	f(x0, n__g(x0), x1) ->^* f(x1, x1, x1)
3.17/1.62	
3.17/1.62	f(x0, n__g(x0), n__g(x1)) ->^* f(n__g(x1), n__g(x1), n__g(x1))
3.17/1.62	
3.17/1.62	b ->^* c
3.17/1.62	
3.17/1.62	
3.17/1.62	
3.17/1.62	
3.17/1.62	----------------------------------------
3.17/1.62	
3.17/1.62	(4)
3.17/1.62	BOUNDS(1, 1)
3.55/1.64	EOF