3.42/1.67	YES
3.42/1.68	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.42/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.42/1.68	
3.42/1.68	
3.42/1.68	Quasi decreasingness of the given CTRS could be proven:
3.42/1.68	
3.42/1.68	(0) CTRS
3.42/1.68	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.42/1.68	(2) QTRS
3.42/1.68	(3) QTRSRRRProof [EQUIVALENT, 47 ms]
3.42/1.68	(4) QTRS
3.42/1.68	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.42/1.68	(6) YES
3.42/1.68	
3.42/1.68	
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(0)
3.42/1.68	Obligation:
3.42/1.68	Conditional term rewrite system:
3.42/1.68	The TRS R consists of the following rules:
3.42/1.68	
3.42/1.68	   plus(0, X) -> X
3.42/1.68	   plus(s(X), Y) -> plus(X, s(Y))
3.42/1.68	   fib(0) -> pair(s(0), 0)
3.42/1.68	
3.42/1.68	The conditional TRS C consists of the following conditional rules:
3.42/1.68	
3.42/1.68	   fib(s(X)) -> pair(W, Y) <= fib(X) -> pair(Y, Z), plus(Y, Z) -> W
3.42/1.68	
3.42/1.68	
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(1) CTRSToQTRSProof (SOUND)
3.42/1.68	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(2)
3.42/1.68	Obligation:
3.42/1.68	Q restricted rewrite system:
3.42/1.68	The TRS R consists of the following rules:
3.42/1.68	
3.42/1.68	   fib(s(X)) -> U1(fib(X))
3.42/1.68	   U1(pair(Y, Z)) -> U2(plus(Y, Z), Y)
3.42/1.68	   U2(W, Y) -> pair(W, Y)
3.42/1.68	   plus(0, X) -> X
3.42/1.68	   plus(s(X), Y) -> plus(X, s(Y))
3.42/1.68	   fib(0) -> pair(s(0), 0)
3.42/1.68	
3.42/1.68	Q is empty.
3.42/1.68	
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(3) QTRSRRRProof (EQUIVALENT)
3.42/1.68	Used ordering:
3.42/1.68	Quasi precedence:
3.42/1.68	fib_1 > [U1_1, U2_2, plus_2] > s_1
3.42/1.68	fib_1 > [U1_1, U2_2, plus_2] > pair_2
3.42/1.68	fib_1 > 0
3.42/1.68	
3.42/1.68	
3.42/1.68	Status:
3.42/1.68	fib_1: multiset status
3.42/1.68	s_1: multiset status
3.42/1.68	U1_1: [1]
3.42/1.68	pair_2: multiset status
3.42/1.68	U2_2: [2,1]
3.42/1.68	plus_2: [1,2]
3.42/1.68	0: multiset status
3.42/1.68	
3.42/1.68	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.42/1.68	
3.42/1.68	   fib(s(X)) -> U1(fib(X))
3.42/1.68	   U1(pair(Y, Z)) -> U2(plus(Y, Z), Y)
3.42/1.68	   U2(W, Y) -> pair(W, Y)
3.42/1.68	   plus(0, X) -> X
3.42/1.68	   plus(s(X), Y) -> plus(X, s(Y))
3.42/1.68	   fib(0) -> pair(s(0), 0)
3.42/1.68	
3.42/1.68	
3.42/1.68	
3.42/1.68	
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(4)
3.42/1.68	Obligation:
3.42/1.68	Q restricted rewrite system:
3.42/1.68	R is empty.
3.42/1.68	Q is empty.
3.42/1.68	
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(5) RisEmptyProof (EQUIVALENT)
3.42/1.68	The TRS R is empty. Hence, termination is trivially proven.
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(6)
3.42/1.68	YES
3.42/1.70	EOF