3.21/1.81	YES
3.55/1.82	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.55/1.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.55/1.82	
3.55/1.82	
3.55/1.82	Quasi decreasingness of the given CTRS could be proven:
3.55/1.82	
3.55/1.82	(0) CTRS
3.55/1.82	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.55/1.82	(2) QTRS
3.55/1.82	(3) QTRSRRRProof [EQUIVALENT, 45 ms]
3.55/1.82	(4) QTRS
3.55/1.82	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.55/1.82	(6) QTRS
3.55/1.82	(7) QTRSRRRProof [EQUIVALENT, 0 ms]
3.55/1.82	(8) QTRS
3.55/1.82	(9) QTRSRRRProof [EQUIVALENT, 0 ms]
3.55/1.82	(10) QTRS
3.55/1.82	(11) RisEmptyProof [EQUIVALENT, 0 ms]
3.55/1.82	(12) YES
3.55/1.82	
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(0)
3.55/1.82	Obligation:
3.55/1.82	Conditional term rewrite system:
3.55/1.82	The TRS R consists of the following rules:
3.55/1.82	
3.55/1.82	   A -> B
3.55/1.82	
3.55/1.82	The conditional TRS C consists of the following conditional rules:
3.55/1.82	
3.55/1.82	   f(x) -> x <= x -> a
3.55/1.82	   g(x) -> C <= A -> B
3.55/1.82	
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(1) CTRSToQTRSProof (SOUND)
3.55/1.82	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(2)
3.55/1.82	Obligation:
3.55/1.82	Q restricted rewrite system:
3.55/1.82	The TRS R consists of the following rules:
3.55/1.82	
3.55/1.82	   f(x) -> U1(x, x)
3.55/1.82	   U1(a, x) -> x
3.55/1.82	   g(x) -> U2(A)
3.55/1.82	   U2(B) -> C
3.55/1.82	   A -> B
3.55/1.82	
3.55/1.82	Q is empty.
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(3) QTRSRRRProof (EQUIVALENT)
3.55/1.82	Used ordering:
3.55/1.82	Polynomial interpretation [POLO]:
3.55/1.82	
3.55/1.82	   POL(A) = 1
3.55/1.82	   POL(B) = 1
3.55/1.82	   POL(C) = 2
3.55/1.82	   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
3.55/1.82	   POL(U2(x_1)) = 2*x_1
3.55/1.82	   POL(a) = 1
3.55/1.82	   POL(f(x_1)) = 2 + 2*x_1
3.55/1.82	   POL(g(x_1)) = 2 + x_1
3.55/1.82	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.55/1.82	
3.55/1.82	   U1(a, x) -> x
3.55/1.82	
3.55/1.82	
3.55/1.82	
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(4)
3.55/1.82	Obligation:
3.55/1.82	Q restricted rewrite system:
3.55/1.82	The TRS R consists of the following rules:
3.55/1.82	
3.55/1.82	   f(x) -> U1(x, x)
3.55/1.82	   g(x) -> U2(A)
3.55/1.82	   U2(B) -> C
3.55/1.82	   A -> B
3.55/1.82	
3.55/1.82	Q is empty.
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(5) QTRSRRRProof (EQUIVALENT)
3.55/1.82	Used ordering:
3.55/1.82	Polynomial interpretation [POLO]:
3.55/1.82	
3.55/1.82	   POL(A) = 1
3.55/1.82	   POL(B) = 0
3.55/1.82	   POL(C) = 0
3.55/1.82	   POL(U1(x_1, x_2)) = 1 + x_1 + x_2
3.55/1.82	   POL(U2(x_1)) = 2*x_1
3.55/1.82	   POL(f(x_1)) = 1 + 2*x_1
3.55/1.82	   POL(g(x_1)) = 2 + x_1
3.55/1.82	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.55/1.82	
3.55/1.82	   A -> B
3.55/1.82	
3.55/1.82	
3.55/1.82	
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(6)
3.55/1.82	Obligation:
3.55/1.82	Q restricted rewrite system:
3.55/1.82	The TRS R consists of the following rules:
3.55/1.82	
3.55/1.82	   f(x) -> U1(x, x)
3.55/1.82	   g(x) -> U2(A)
3.55/1.82	   U2(B) -> C
3.55/1.82	
3.55/1.82	Q is empty.
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(7) QTRSRRRProof (EQUIVALENT)
3.55/1.82	Used ordering:
3.55/1.82	f/1(YES)
3.55/1.82	U1/2(YES,YES)
3.55/1.82	g/1(YES)
3.55/1.82	U2/1)YES(
3.55/1.82	A/0)
3.55/1.82	B/0)
3.55/1.82	C/0)
3.55/1.82	
3.55/1.82	Quasi precedence:
3.55/1.82	f_1 > [U1_2, A]
3.55/1.82	g_1 > [U1_2, A]
3.55/1.82	[B, C] > [U1_2, A]
3.55/1.82	
3.55/1.82	
3.55/1.82	Status:
3.55/1.82	f_1: multiset status
3.55/1.82	U1_2: multiset status
3.55/1.82	g_1: multiset status
3.55/1.82	A: multiset status
3.55/1.82	B: multiset status
3.55/1.82	C: multiset status
3.55/1.82	
3.55/1.82	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.55/1.82	
3.55/1.82	   f(x) -> U1(x, x)
3.55/1.82	   g(x) -> U2(A)
3.55/1.82	
3.55/1.82	
3.55/1.82	
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(8)
3.55/1.82	Obligation:
3.55/1.82	Q restricted rewrite system:
3.55/1.82	The TRS R consists of the following rules:
3.55/1.82	
3.55/1.82	   U2(B) -> C
3.55/1.82	
3.55/1.82	Q is empty.
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(9) QTRSRRRProof (EQUIVALENT)
3.55/1.82	Used ordering:
3.55/1.82	Polynomial interpretation [POLO]:
3.55/1.82	
3.55/1.82	   POL(B) = 1
3.55/1.82	   POL(C) = 0
3.55/1.82	   POL(U2(x_1)) = 2 + 2*x_1
3.55/1.82	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.55/1.82	
3.55/1.82	   U2(B) -> C
3.55/1.82	
3.55/1.82	
3.55/1.82	
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(10)
3.55/1.82	Obligation:
3.55/1.82	Q restricted rewrite system:
3.55/1.82	R is empty.
3.55/1.82	Q is empty.
3.55/1.82	
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(11) RisEmptyProof (EQUIVALENT)
3.55/1.82	The TRS R is empty. Hence, termination is trivially proven.
3.55/1.82	----------------------------------------
3.55/1.82	
3.55/1.82	(12)
3.55/1.82	YES
3.55/1.85	EOF