3.58/1.71	YES
3.61/1.72	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.61/1.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.61/1.72	
3.61/1.72	
3.61/1.72	Quasi decreasingness of the given CTRS could be proven:
3.61/1.72	
3.61/1.72	(0) CTRS
3.61/1.72	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.61/1.72	(2) QTRS
3.61/1.72	(3) QTRSRRRProof [EQUIVALENT, 51 ms]
3.61/1.72	(4) QTRS
3.61/1.72	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.61/1.72	(6) QTRS
3.61/1.72	(7) AAECC Innermost [EQUIVALENT, 0 ms]
3.61/1.72	(8) QTRS
3.61/1.72	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
3.61/1.72	(10) QDP
3.61/1.72	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
3.61/1.72	(12) TRUE
3.61/1.72	
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(0)
3.61/1.72	Obligation:
3.61/1.72	Conditional term rewrite system:
3.61/1.72	The TRS R consists of the following rules:
3.61/1.72	
3.61/1.72	   b -> c
3.61/1.72	   f(b) -> f(a)
3.61/1.72	
3.61/1.72	The conditional TRS C consists of the following conditional rules:
3.61/1.72	
3.61/1.72	   a -> c <= b -> c
3.61/1.72	
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(1) CTRSToQTRSProof (SOUND)
3.61/1.72	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(2)
3.61/1.72	Obligation:
3.61/1.72	Q restricted rewrite system:
3.61/1.72	The TRS R consists of the following rules:
3.61/1.72	
3.61/1.72	   a -> U1(b)
3.61/1.72	   U1(c) -> c
3.61/1.72	   b -> c
3.61/1.72	   f(b) -> f(a)
3.61/1.72	
3.61/1.72	Q is empty.
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(3) QTRSRRRProof (EQUIVALENT)
3.61/1.72	Used ordering:
3.61/1.72	Polynomial interpretation [POLO]:
3.61/1.72	
3.61/1.72	   POL(U1(x_1)) = x_1
3.61/1.72	   POL(a) = 1
3.61/1.72	   POL(b) = 1
3.61/1.72	   POL(c) = 0
3.61/1.72	   POL(f(x_1)) = x_1
3.61/1.72	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.61/1.72	
3.61/1.72	   b -> c
3.61/1.72	
3.61/1.72	
3.61/1.72	
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(4)
3.61/1.72	Obligation:
3.61/1.72	Q restricted rewrite system:
3.61/1.72	The TRS R consists of the following rules:
3.61/1.72	
3.61/1.72	   a -> U1(b)
3.61/1.72	   U1(c) -> c
3.61/1.72	   f(b) -> f(a)
3.61/1.72	
3.61/1.72	Q is empty.
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(5) QTRSRRRProof (EQUIVALENT)
3.61/1.72	Used ordering:
3.61/1.72	Polynomial interpretation [POLO]:
3.61/1.72	
3.61/1.72	   POL(U1(x_1)) = 2*x_1
3.61/1.72	   POL(a) = 0
3.61/1.72	   POL(b) = 0
3.61/1.72	   POL(c) = 2
3.61/1.72	   POL(f(x_1)) = 2*x_1
3.61/1.72	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.61/1.72	
3.61/1.72	   U1(c) -> c
3.61/1.72	
3.61/1.72	
3.61/1.72	
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(6)
3.61/1.72	Obligation:
3.61/1.72	Q restricted rewrite system:
3.61/1.72	The TRS R consists of the following rules:
3.61/1.72	
3.61/1.72	   a -> U1(b)
3.61/1.72	   f(b) -> f(a)
3.61/1.72	
3.61/1.72	Q is empty.
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(7) AAECC Innermost (EQUIVALENT)
3.61/1.72	We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 
3.61/1.72	   a -> U1(b)
3.61/1.72	
3.61/1.72	The TRS R 2 is 
3.61/1.72	   f(b) -> f(a)
3.61/1.72	
3.61/1.72	The signature Sigma is {f_1}
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(8)
3.61/1.72	Obligation:
3.61/1.72	Q restricted rewrite system:
3.61/1.72	The TRS R consists of the following rules:
3.61/1.72	
3.61/1.72	   a -> U1(b)
3.61/1.72	   f(b) -> f(a)
3.61/1.72	
3.61/1.72	The set Q consists of the following terms:
3.61/1.72	
3.61/1.72	   a
3.61/1.72	   f(b)
3.61/1.72	
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(9) DependencyPairsProof (EQUIVALENT)
3.61/1.72	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(10)
3.61/1.72	Obligation:
3.61/1.72	Q DP problem:
3.61/1.72	The TRS P consists of the following rules:
3.61/1.72	
3.61/1.72	   F(b) -> F(a)
3.61/1.72	   F(b) -> A
3.61/1.72	
3.61/1.72	The TRS R consists of the following rules:
3.61/1.72	
3.61/1.72	   a -> U1(b)
3.61/1.72	   f(b) -> f(a)
3.61/1.72	
3.61/1.72	The set Q consists of the following terms:
3.61/1.72	
3.61/1.72	   a
3.61/1.72	   f(b)
3.61/1.72	
3.61/1.72	We have to consider all minimal (P,Q,R)-chains.
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(11) DependencyGraphProof (EQUIVALENT)
3.61/1.72	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(12)
3.61/1.72	TRUE
3.64/1.75	EOF