3.64/1.76	YES
3.64/1.77	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.64/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.64/1.77	
3.64/1.77	
3.64/1.77	Termination w.r.t. Q of the given QTRS could be proven:
3.64/1.77	
3.64/1.77	(0) QTRS
3.64/1.77	(1) QTRSRRRProof [EQUIVALENT, 75 ms]
3.64/1.77	(2) QTRS
3.64/1.77	(3) QTRSRRRProof [EQUIVALENT, 0 ms]
3.64/1.77	(4) QTRS
3.64/1.77	(5) QTRSRRRProof [EQUIVALENT, 2 ms]
3.64/1.77	(6) QTRS
3.64/1.77	(7) QTRSRRRProof [EQUIVALENT, 0 ms]
3.64/1.77	(8) QTRS
3.64/1.77	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.64/1.77	(10) YES
3.64/1.77	
3.64/1.77	
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(0)
3.64/1.77	Obligation:
3.64/1.77	Q restricted rewrite system:
3.64/1.77	The TRS R consists of the following rules:
3.64/1.77	
3.64/1.77	   a__f(X) -> a__if(mark(X), c, f(true))
3.64/1.77	   a__if(true, X, Y) -> mark(X)
3.64/1.77	   a__if(false, X, Y) -> mark(Y)
3.64/1.77	   mark(f(X)) -> a__f(mark(X))
3.64/1.77	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
3.64/1.77	   mark(c) -> c
3.64/1.77	   mark(true) -> true
3.64/1.77	   mark(false) -> false
3.64/1.77	   a__f(X) -> f(X)
3.64/1.77	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.64/1.77	
3.64/1.77	The set Q consists of the following terms:
3.64/1.77	
3.64/1.77	   a__f(x0)
3.64/1.77	   mark(f(x0))
3.64/1.77	   mark(if(x0, x1, x2))
3.64/1.77	   mark(c)
3.64/1.77	   mark(true)
3.64/1.77	   mark(false)
3.64/1.77	   a__if(x0, x1, x2)
3.64/1.77	
3.64/1.77	
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(1) QTRSRRRProof (EQUIVALENT)
3.64/1.77	Used ordering:
3.64/1.77	Polynomial interpretation [POLO]:
3.64/1.77	
3.64/1.77	   POL(a__f(x_1)) = 2*x_1
3.64/1.77	   POL(a__if(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3
3.64/1.77	   POL(c) = 0
3.64/1.77	   POL(f(x_1)) = 2*x_1
3.64/1.77	   POL(false) = 2
3.64/1.77	   POL(if(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3
3.64/1.77	   POL(mark(x_1)) = x_1
3.64/1.77	   POL(true) = 0
3.64/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.64/1.77	
3.64/1.77	   a__if(false, X, Y) -> mark(Y)
3.64/1.77	
3.64/1.77	
3.64/1.77	
3.64/1.77	
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(2)
3.64/1.77	Obligation:
3.64/1.77	Q restricted rewrite system:
3.64/1.77	The TRS R consists of the following rules:
3.64/1.77	
3.64/1.77	   a__f(X) -> a__if(mark(X), c, f(true))
3.64/1.77	   a__if(true, X, Y) -> mark(X)
3.64/1.77	   mark(f(X)) -> a__f(mark(X))
3.64/1.77	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
3.64/1.77	   mark(c) -> c
3.64/1.77	   mark(true) -> true
3.64/1.77	   mark(false) -> false
3.64/1.77	   a__f(X) -> f(X)
3.64/1.77	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.64/1.77	
3.64/1.77	The set Q consists of the following terms:
3.64/1.77	
3.64/1.77	   a__f(x0)
3.64/1.77	   mark(f(x0))
3.64/1.77	   mark(if(x0, x1, x2))
3.64/1.77	   mark(c)
3.64/1.77	   mark(true)
3.64/1.77	   mark(false)
3.64/1.77	   a__if(x0, x1, x2)
3.64/1.77	
3.64/1.77	
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(3) QTRSRRRProof (EQUIVALENT)
3.64/1.77	Used ordering:
3.64/1.77	Polynomial interpretation [POLO]:
3.64/1.77	
3.64/1.77	   POL(a__f(x_1)) = 2*x_1
3.64/1.77	   POL(a__if(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
3.64/1.77	   POL(c) = 0
3.64/1.77	   POL(f(x_1)) = 2*x_1
3.64/1.77	   POL(false) = 2
3.64/1.77	   POL(if(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
3.64/1.77	   POL(mark(x_1)) = 2*x_1
3.64/1.77	   POL(true) = 0
3.64/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.64/1.77	
3.64/1.77	   mark(false) -> false
3.64/1.77	
3.64/1.77	
3.64/1.77	
3.64/1.77	
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(4)
3.64/1.77	Obligation:
3.64/1.77	Q restricted rewrite system:
3.64/1.77	The TRS R consists of the following rules:
3.64/1.77	
3.64/1.77	   a__f(X) -> a__if(mark(X), c, f(true))
3.64/1.77	   a__if(true, X, Y) -> mark(X)
3.64/1.77	   mark(f(X)) -> a__f(mark(X))
3.64/1.77	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
3.64/1.77	   mark(c) -> c
3.64/1.77	   mark(true) -> true
3.64/1.77	   a__f(X) -> f(X)
3.64/1.77	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.64/1.77	
3.64/1.77	The set Q consists of the following terms:
3.64/1.77	
3.64/1.77	   a__f(x0)
3.64/1.77	   mark(f(x0))
3.64/1.77	   mark(if(x0, x1, x2))
3.64/1.77	   mark(c)
3.64/1.77	   mark(true)
3.64/1.77	   mark(false)
3.64/1.77	   a__if(x0, x1, x2)
3.64/1.77	
3.64/1.77	
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(5) QTRSRRRProof (EQUIVALENT)
3.64/1.77	Used ordering:
3.64/1.77	Polynomial interpretation [POLO]:
3.64/1.77	
3.64/1.77	   POL(a__f(x_1)) = 1 + 2*x_1
3.64/1.77	   POL(a__if(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3
3.64/1.77	   POL(c) = 0
3.64/1.77	   POL(f(x_1)) = 1 + 2*x_1
3.64/1.77	   POL(if(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3
3.64/1.77	   POL(mark(x_1)) = 2*x_1
3.64/1.77	   POL(true) = 0
3.64/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.64/1.77	
3.64/1.77	   mark(f(X)) -> a__f(mark(X))
3.64/1.77	
3.64/1.77	
3.64/1.77	
3.64/1.77	
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(6)
3.64/1.77	Obligation:
3.64/1.77	Q restricted rewrite system:
3.64/1.77	The TRS R consists of the following rules:
3.64/1.77	
3.64/1.77	   a__f(X) -> a__if(mark(X), c, f(true))
3.64/1.77	   a__if(true, X, Y) -> mark(X)
3.64/1.77	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
3.64/1.77	   mark(c) -> c
3.64/1.77	   mark(true) -> true
3.64/1.77	   a__f(X) -> f(X)
3.64/1.77	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.64/1.77	
3.64/1.77	The set Q consists of the following terms:
3.64/1.77	
3.64/1.77	   a__f(x0)
3.64/1.77	   mark(f(x0))
3.64/1.77	   mark(if(x0, x1, x2))
3.64/1.77	   mark(c)
3.64/1.77	   mark(true)
3.64/1.77	   mark(false)
3.64/1.77	   a__if(x0, x1, x2)
3.64/1.77	
3.64/1.77	
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(7) QTRSRRRProof (EQUIVALENT)
3.64/1.77	Used ordering:
3.64/1.77	Knuth-Bendix order [KBO] with precedence:mark_1 > a__if_3 > if_3 > f_1 > true > c > a__f_1
3.64/1.77	
3.64/1.77	and weight map:
3.64/1.77	
3.64/1.77	   c=1
3.64/1.77	   true=1
3.64/1.77	   a__f_1=4
3.64/1.77	   mark_1=0
3.64/1.77	   f_1=1
3.64/1.77	   a__if_3=0
3.64/1.77	   if_3=0
3.64/1.77	
3.64/1.77	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.64/1.77	
3.64/1.77	   a__f(X) -> a__if(mark(X), c, f(true))
3.64/1.77	   a__if(true, X, Y) -> mark(X)
3.64/1.77	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
3.64/1.77	   mark(c) -> c
3.64/1.77	   mark(true) -> true
3.64/1.77	   a__f(X) -> f(X)
3.64/1.77	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.64/1.77	
3.64/1.77	
3.64/1.77	
3.64/1.77	
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(8)
3.64/1.77	Obligation:
3.64/1.77	Q restricted rewrite system:
3.64/1.77	R is empty.
3.64/1.77	The set Q consists of the following terms:
3.64/1.77	
3.64/1.77	   a__f(x0)
3.64/1.77	   mark(f(x0))
3.64/1.77	   mark(if(x0, x1, x2))
3.64/1.77	   mark(c)
3.64/1.77	   mark(true)
3.64/1.77	   mark(false)
3.64/1.77	   a__if(x0, x1, x2)
3.64/1.77	
3.64/1.77	
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(9) RisEmptyProof (EQUIVALENT)
3.64/1.77	The TRS R is empty. Hence, termination is trivially proven.
3.64/1.77	----------------------------------------
3.64/1.77	
3.64/1.77	(10)
3.64/1.77	YES
3.95/1.80	EOF