3.58/1.74	YES
3.58/1.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.58/1.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.58/1.75	
3.58/1.75	
3.58/1.75	Quasi decreasingness of the given CTRS could be proven:
3.58/1.75	
3.58/1.75	(0) CTRS
3.58/1.75	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.58/1.75	(2) QTRS
3.58/1.75	(3) QTRSRRRProof [EQUIVALENT, 63 ms]
3.58/1.75	(4) QTRS
3.58/1.75	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.58/1.75	(6) QTRS
3.58/1.75	(7) QTRSRRRProof [EQUIVALENT, 8 ms]
3.58/1.75	(8) QTRS
3.58/1.75	(9) QTRSRRRProof [EQUIVALENT, 0 ms]
3.58/1.75	(10) QTRS
3.58/1.75	(11) RisEmptyProof [EQUIVALENT, 0 ms]
3.58/1.75	(12) YES
3.58/1.75	
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(0)
3.58/1.75	Obligation:
3.58/1.75	Conditional term rewrite system:
3.58/1.75	The TRS R consists of the following rules:
3.58/1.75	
3.58/1.75	   a -> c
3.58/1.75	   a -> d
3.58/1.75	   b -> c
3.58/1.75	   b -> d
3.58/1.75	   s(c) -> t(k)
3.58/1.75	   s(c) -> t(l)
3.58/1.75	   g(x, x) -> h(x, x)
3.58/1.75	
3.58/1.75	The conditional TRS C consists of the following conditional rules:
3.58/1.75	
3.58/1.75	   f(x) -> pair(x, y) <= s(x) -> t(y)
3.58/1.75	
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(1) CTRSToQTRSProof (SOUND)
3.58/1.75	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(2)
3.58/1.75	Obligation:
3.58/1.75	Q restricted rewrite system:
3.58/1.75	The TRS R consists of the following rules:
3.58/1.75	
3.58/1.75	   f(x) -> U1(s(x), x)
3.58/1.75	   U1(t(y), x) -> pair(x, y)
3.58/1.75	   a -> c
3.58/1.75	   a -> d
3.58/1.75	   b -> c
3.58/1.75	   b -> d
3.58/1.75	   s(c) -> t(k)
3.58/1.75	   s(c) -> t(l)
3.58/1.75	   g(x, x) -> h(x, x)
3.58/1.75	
3.58/1.75	Q is empty.
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(3) QTRSRRRProof (EQUIVALENT)
3.58/1.75	Used ordering:
3.58/1.75	Polynomial interpretation [POLO]:
3.58/1.75	
3.58/1.75	   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
3.58/1.75	   POL(a) = 1
3.58/1.75	   POL(b) = 1
3.58/1.75	   POL(c) = 0
3.58/1.75	   POL(d) = 0
3.58/1.75	   POL(f(x_1)) = 2 + 2*x_1
3.58/1.75	   POL(g(x_1, x_2)) = 1 + x_1 + x_2
3.58/1.75	   POL(h(x_1, x_2)) = 1 + x_1 + x_2
3.58/1.75	   POL(k) = 0
3.58/1.75	   POL(l) = 0
3.58/1.75	   POL(pair(x_1, x_2)) = 2 + x_1 + 2*x_2
3.58/1.75	   POL(s(x_1)) = x_1
3.58/1.75	   POL(t(x_1)) = 2*x_1
3.58/1.75	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.58/1.75	
3.58/1.75	   a -> c
3.58/1.75	   a -> d
3.58/1.75	   b -> c
3.58/1.75	   b -> d
3.58/1.75	
3.58/1.75	
3.58/1.75	
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(4)
3.58/1.75	Obligation:
3.58/1.75	Q restricted rewrite system:
3.58/1.75	The TRS R consists of the following rules:
3.58/1.75	
3.58/1.75	   f(x) -> U1(s(x), x)
3.58/1.75	   U1(t(y), x) -> pair(x, y)
3.58/1.75	   s(c) -> t(k)
3.58/1.75	   s(c) -> t(l)
3.58/1.75	   g(x, x) -> h(x, x)
3.58/1.75	
3.58/1.75	Q is empty.
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(5) QTRSRRRProof (EQUIVALENT)
3.58/1.75	Used ordering:
3.58/1.75	Polynomial interpretation [POLO]:
3.58/1.75	
3.58/1.75	   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
3.58/1.75	   POL(c) = 0
3.58/1.75	   POL(f(x_1)) = 2 + 2*x_1
3.58/1.75	   POL(g(x_1, x_2)) = 1 + x_1 + x_2
3.58/1.75	   POL(h(x_1, x_2)) = x_1 + x_2
3.58/1.75	   POL(k) = 0
3.58/1.75	   POL(l) = 0
3.58/1.75	   POL(pair(x_1, x_2)) = 2 + x_1 + x_2
3.58/1.75	   POL(s(x_1)) = x_1
3.58/1.75	   POL(t(x_1)) = x_1
3.58/1.75	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.58/1.75	
3.58/1.75	   g(x, x) -> h(x, x)
3.58/1.75	
3.58/1.75	
3.58/1.75	
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(6)
3.58/1.75	Obligation:
3.58/1.75	Q restricted rewrite system:
3.58/1.75	The TRS R consists of the following rules:
3.58/1.75	
3.58/1.75	   f(x) -> U1(s(x), x)
3.58/1.75	   U1(t(y), x) -> pair(x, y)
3.58/1.75	   s(c) -> t(k)
3.58/1.75	   s(c) -> t(l)
3.58/1.75	
3.58/1.75	Q is empty.
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(7) QTRSRRRProof (EQUIVALENT)
3.58/1.75	Used ordering:
3.58/1.75	Polynomial interpretation [POLO]:
3.58/1.75	
3.58/1.75	   POL(U1(x_1, x_2)) = x_1 + x_2
3.58/1.75	   POL(c) = 0
3.58/1.75	   POL(f(x_1)) = 2 + 2*x_1
3.58/1.75	   POL(k) = 0
3.58/1.75	   POL(l) = 0
3.58/1.75	   POL(pair(x_1, x_2)) = x_1 + 2*x_2
3.58/1.75	   POL(s(x_1)) = 1 + x_1
3.58/1.75	   POL(t(x_1)) = 2*x_1
3.58/1.75	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.58/1.75	
3.58/1.75	   f(x) -> U1(s(x), x)
3.58/1.75	   s(c) -> t(k)
3.58/1.75	   s(c) -> t(l)
3.58/1.75	
3.58/1.75	
3.58/1.75	
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(8)
3.58/1.75	Obligation:
3.58/1.75	Q restricted rewrite system:
3.58/1.75	The TRS R consists of the following rules:
3.58/1.75	
3.58/1.75	   U1(t(y), x) -> pair(x, y)
3.58/1.75	
3.58/1.75	Q is empty.
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(9) QTRSRRRProof (EQUIVALENT)
3.58/1.75	Used ordering:
3.58/1.75	Knuth-Bendix order [KBO] with precedence:U1_2 > pair_2 > t_1
3.58/1.75	
3.58/1.75	and weight map:
3.58/1.75	
3.58/1.75	   t_1=1
3.58/1.75	   U1_2=0
3.58/1.75	   pair_2=1
3.58/1.75	
3.58/1.75	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.58/1.75	
3.58/1.75	   U1(t(y), x) -> pair(x, y)
3.58/1.75	
3.58/1.75	
3.58/1.75	
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(10)
3.58/1.75	Obligation:
3.58/1.75	Q restricted rewrite system:
3.58/1.75	R is empty.
3.58/1.75	Q is empty.
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(11) RisEmptyProof (EQUIVALENT)
3.58/1.75	The TRS R is empty. Hence, termination is trivially proven.
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(12)
3.58/1.75	YES
3.82/1.77	EOF