4.18/1.86	YES
4.18/1.87	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.18/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.18/1.87	
4.18/1.87	
4.18/1.87	Quasi decreasingness of the given CTRS could be proven:
4.18/1.87	
4.18/1.87	(0) CTRS
4.18/1.87	(1) CTRSToQTRSProof [SOUND, 0 ms]
4.18/1.87	(2) QTRS
4.18/1.87	(3) QTRSRRRProof [EQUIVALENT, 51 ms]
4.18/1.87	(4) QTRS
4.18/1.87	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
4.18/1.87	(6) QTRS
4.18/1.87	(7) QTRSRRRProof [EQUIVALENT, 8 ms]
4.18/1.87	(8) QTRS
4.18/1.87	(9) QTRSRRRProof [EQUIVALENT, 0 ms]
4.18/1.87	(10) QTRS
4.18/1.87	(11) QTRSRRRProof [EQUIVALENT, 0 ms]
4.18/1.87	(12) QTRS
4.18/1.87	(13) QTRSRRRProof [EQUIVALENT, 1 ms]
4.18/1.87	(14) QTRS
4.18/1.87	(15) QTRSRRRProof [EQUIVALENT, 0 ms]
4.18/1.87	(16) QTRS
4.18/1.87	(17) RisEmptyProof [EQUIVALENT, 0 ms]
4.18/1.87	(18) YES
4.18/1.87	
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(0)
4.18/1.87	Obligation:
4.18/1.87	Conditional term rewrite system:
4.18/1.87	The TRS R consists of the following rules:
4.18/1.87	
4.18/1.87	   a -> c
4.18/1.87	   a -> d
4.18/1.87	   b -> c
4.18/1.87	   b -> d
4.18/1.87	   c -> e
4.18/1.87	   c -> l
4.18/1.87	   k -> l
4.18/1.87	   k -> m
4.18/1.87	   d -> m
4.18/1.87	
4.18/1.87	The conditional TRS C consists of the following conditional rules:
4.18/1.87	
4.18/1.87	   f(x) -> x <= x -> e
4.18/1.87	   g(d, x, y) -> A <= y -> x
4.18/1.87	   h(x, y) -> g(x, y, f(k)) <= y -> x
4.18/1.87	
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(1) CTRSToQTRSProof (SOUND)
4.18/1.87	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(2)
4.18/1.87	Obligation:
4.18/1.87	Q restricted rewrite system:
4.18/1.87	The TRS R consists of the following rules:
4.18/1.87	
4.18/1.87	   f(x) -> U1(x, x)
4.18/1.87	   U1(e, x) -> x
4.18/1.87	   g(d, x, y) -> U2(y)
4.18/1.87	   U2(x) -> A
4.18/1.87	   h(x, y) -> U3(y, x, y)
4.18/1.87	   U3(x, x, y) -> g(x, y, f(k))
4.18/1.87	   a -> c
4.18/1.87	   a -> d
4.18/1.87	   b -> c
4.18/1.87	   b -> d
4.18/1.87	   c -> e
4.18/1.87	   c -> l
4.18/1.87	   k -> l
4.18/1.87	   k -> m
4.18/1.87	   d -> m
4.18/1.87	
4.18/1.87	Q is empty.
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(3) QTRSRRRProof (EQUIVALENT)
4.18/1.87	Used ordering:
4.18/1.87	Polynomial interpretation [POLO]:
4.18/1.87	
4.18/1.87	   POL(A) = 0
4.18/1.87	   POL(U1(x_1, x_2)) = x_1 + x_2
4.18/1.87	   POL(U2(x_1)) = 1 + x_1
4.18/1.87	   POL(U3(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3
4.18/1.87	   POL(a) = 2
4.18/1.87	   POL(b) = 2
4.18/1.87	   POL(c) = 0
4.18/1.87	   POL(d) = 1
4.18/1.87	   POL(e) = 0
4.18/1.87	   POL(f(x_1)) = 2*x_1
4.18/1.87	   POL(g(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
4.18/1.87	   POL(h(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
4.18/1.87	   POL(k) = 1
4.18/1.87	   POL(l) = 0
4.18/1.87	   POL(m) = 0
4.18/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.18/1.87	
4.18/1.87	   g(d, x, y) -> U2(y)
4.18/1.87	   U2(x) -> A
4.18/1.87	   a -> c
4.18/1.87	   a -> d
4.18/1.87	   b -> c
4.18/1.87	   b -> d
4.18/1.87	   k -> l
4.18/1.87	   k -> m
4.18/1.87	   d -> m
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(4)
4.18/1.87	Obligation:
4.18/1.87	Q restricted rewrite system:
4.18/1.87	The TRS R consists of the following rules:
4.18/1.87	
4.18/1.87	   f(x) -> U1(x, x)
4.18/1.87	   U1(e, x) -> x
4.18/1.87	   h(x, y) -> U3(y, x, y)
4.18/1.87	   U3(x, x, y) -> g(x, y, f(k))
4.18/1.87	   c -> e
4.18/1.87	   c -> l
4.18/1.87	
4.18/1.87	Q is empty.
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(5) QTRSRRRProof (EQUIVALENT)
4.18/1.87	Used ordering:
4.18/1.87	Polynomial interpretation [POLO]:
4.18/1.87	
4.18/1.87	   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
4.18/1.87	   POL(U3(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3
4.18/1.87	   POL(c) = 1
4.18/1.87	   POL(e) = 1
4.18/1.87	   POL(f(x_1)) = 2 + 2*x_1
4.18/1.87	   POL(g(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
4.18/1.87	   POL(h(x_1, x_2)) = 2 + x_1 + 2*x_2
4.18/1.87	   POL(k) = 0
4.18/1.87	   POL(l) = 1
4.18/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.18/1.87	
4.18/1.87	   U1(e, x) -> x
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(6)
4.18/1.87	Obligation:
4.18/1.87	Q restricted rewrite system:
4.18/1.87	The TRS R consists of the following rules:
4.18/1.87	
4.18/1.87	   f(x) -> U1(x, x)
4.18/1.87	   h(x, y) -> U3(y, x, y)
4.18/1.87	   U3(x, x, y) -> g(x, y, f(k))
4.18/1.87	   c -> e
4.18/1.87	   c -> l
4.18/1.87	
4.18/1.87	Q is empty.
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(7) QTRSRRRProof (EQUIVALENT)
4.18/1.87	Used ordering:
4.18/1.87	Polynomial interpretation [POLO]:
4.18/1.87	
4.18/1.87	   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
4.18/1.87	   POL(U3(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3
4.18/1.87	   POL(c) = 2
4.18/1.87	   POL(e) = 2
4.18/1.87	   POL(f(x_1)) = 2 + 2*x_1
4.18/1.87	   POL(g(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
4.18/1.87	   POL(h(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
4.18/1.87	   POL(k) = 0
4.18/1.87	   POL(l) = 0
4.18/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.18/1.87	
4.18/1.87	   c -> l
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(8)
4.18/1.87	Obligation:
4.18/1.87	Q restricted rewrite system:
4.18/1.87	The TRS R consists of the following rules:
4.18/1.87	
4.18/1.87	   f(x) -> U1(x, x)
4.18/1.87	   h(x, y) -> U3(y, x, y)
4.18/1.87	   U3(x, x, y) -> g(x, y, f(k))
4.18/1.87	   c -> e
4.18/1.87	
4.18/1.87	Q is empty.
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(9) QTRSRRRProof (EQUIVALENT)
4.18/1.87	Used ordering:
4.18/1.87	Polynomial interpretation [POLO]:
4.18/1.87	
4.18/1.87	   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
4.18/1.87	   POL(U3(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3
4.18/1.87	   POL(c) = 2
4.18/1.87	   POL(e) = 0
4.18/1.87	   POL(f(x_1)) = 2 + 2*x_1
4.18/1.87	   POL(g(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
4.18/1.87	   POL(h(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
4.18/1.87	   POL(k) = 0
4.18/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.18/1.87	
4.18/1.87	   c -> e
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(10)
4.18/1.87	Obligation:
4.18/1.87	Q restricted rewrite system:
4.18/1.87	The TRS R consists of the following rules:
4.18/1.87	
4.18/1.87	   f(x) -> U1(x, x)
4.18/1.87	   h(x, y) -> U3(y, x, y)
4.18/1.87	   U3(x, x, y) -> g(x, y, f(k))
4.18/1.87	
4.18/1.87	Q is empty.
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(11) QTRSRRRProof (EQUIVALENT)
4.18/1.87	Used ordering:
4.18/1.87	Polynomial interpretation [POLO]:
4.18/1.87	
4.18/1.87	   POL(U1(x_1, x_2)) = 1 + x_1 + x_2
4.18/1.87	   POL(U3(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3
4.18/1.87	   POL(f(x_1)) = 2 + 2*x_1
4.18/1.87	   POL(g(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
4.18/1.87	   POL(h(x_1, x_2)) = 2 + x_1 + 2*x_2
4.18/1.87	   POL(k) = 0
4.18/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.18/1.87	
4.18/1.87	   f(x) -> U1(x, x)
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(12)
4.18/1.87	Obligation:
4.18/1.87	Q restricted rewrite system:
4.18/1.87	The TRS R consists of the following rules:
4.18/1.87	
4.18/1.87	   h(x, y) -> U3(y, x, y)
4.18/1.87	   U3(x, x, y) -> g(x, y, f(k))
4.18/1.87	
4.18/1.87	Q is empty.
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(13) QTRSRRRProof (EQUIVALENT)
4.18/1.87	Used ordering:
4.18/1.87	Polynomial interpretation [POLO]:
4.18/1.87	
4.18/1.87	   POL(U3(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
4.18/1.87	   POL(f(x_1)) = 2*x_1
4.18/1.87	   POL(g(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + 2*x_3
4.18/1.87	   POL(h(x_1, x_2)) = 2 + x_1 + 2*x_2
4.18/1.87	   POL(k) = 0
4.18/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.18/1.87	
4.18/1.87	   h(x, y) -> U3(y, x, y)
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(14)
4.18/1.87	Obligation:
4.18/1.87	Q restricted rewrite system:
4.18/1.87	The TRS R consists of the following rules:
4.18/1.87	
4.18/1.87	   U3(x, x, y) -> g(x, y, f(k))
4.18/1.87	
4.18/1.87	Q is empty.
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(15) QTRSRRRProof (EQUIVALENT)
4.18/1.87	Used ordering:
4.18/1.87	Knuth-Bendix order [KBO] with precedence:k > f_1 > U3_3 > g_3
4.18/1.87	
4.18/1.87	and weight map:
4.18/1.87	
4.18/1.87	   k=1
4.18/1.87	   f_1=1
4.18/1.87	   U3_3=1
4.18/1.87	   g_3=0
4.18/1.87	
4.18/1.87	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.18/1.87	
4.18/1.87	   U3(x, x, y) -> g(x, y, f(k))
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(16)
4.18/1.87	Obligation:
4.18/1.87	Q restricted rewrite system:
4.18/1.87	R is empty.
4.18/1.87	Q is empty.
4.18/1.87	
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(17) RisEmptyProof (EQUIVALENT)
4.18/1.87	The TRS R is empty. Hence, termination is trivially proven.
4.18/1.87	----------------------------------------
4.18/1.87	
4.18/1.87	(18)
4.18/1.87	YES
4.18/1.90	EOF