3.42/1.76	YES
3.42/1.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.42/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.42/1.77	
3.42/1.77	
3.42/1.77	Quasi decreasingness of the given CTRS could be proven:
3.42/1.77	
3.42/1.77	(0) CTRS
3.42/1.77	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.42/1.77	(2) QTRS
3.42/1.77	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.42/1.77	(4) QDP
3.42/1.77	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.42/1.77	(6) TRUE
3.42/1.77	
3.42/1.77	
3.42/1.77	----------------------------------------
3.42/1.77	
3.42/1.77	(0)
3.42/1.77	Obligation:
3.42/1.77	Conditional term rewrite system:
3.42/1.77	The TRS R consists of the following rules:
3.42/1.77	
3.42/1.77	   p(q(x)) -> p(r(x))
3.42/1.77	   q(h(x)) -> r(x)
3.42/1.77	   s(x) -> 1
3.42/1.77	
3.42/1.77	The conditional TRS C consists of the following conditional rules:
3.42/1.77	
3.42/1.77	   r(x) -> r(h(x)) <= s(x) -> 0
3.42/1.77	
3.42/1.77	
3.42/1.77	----------------------------------------
3.42/1.77	
3.42/1.77	(1) CTRSToQTRSProof (SOUND)
3.42/1.77	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.42/1.77	----------------------------------------
3.42/1.77	
3.42/1.77	(2)
3.42/1.77	Obligation:
3.42/1.77	Q restricted rewrite system:
3.42/1.77	The TRS R consists of the following rules:
3.42/1.77	
3.42/1.77	   r(x) -> U1(s(x), x)
3.42/1.77	   U1(0, x) -> r(h(x))
3.42/1.77	   p(q(x)) -> p(r(x))
3.42/1.77	   q(h(x)) -> r(x)
3.42/1.77	   s(x) -> 1
3.42/1.77	
3.42/1.77	Q is empty.
3.42/1.77	
3.42/1.77	----------------------------------------
3.42/1.77	
3.42/1.77	(3) DependencyPairsProof (EQUIVALENT)
3.42/1.77	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
3.42/1.77	----------------------------------------
3.42/1.77	
3.42/1.77	(4)
3.42/1.77	Obligation:
3.42/1.77	Q DP problem:
3.42/1.77	The TRS P consists of the following rules:
3.42/1.77	
3.42/1.77	   R(x) -> U1^1(s(x), x)
3.42/1.77	   R(x) -> S(x)
3.42/1.77	   U1^1(0, x) -> R(h(x))
3.42/1.77	   P(q(x)) -> P(r(x))
3.42/1.77	   P(q(x)) -> R(x)
3.42/1.77	   Q(h(x)) -> R(x)
3.42/1.77	
3.42/1.77	The TRS R consists of the following rules:
3.42/1.77	
3.42/1.77	   r(x) -> U1(s(x), x)
3.42/1.77	   U1(0, x) -> r(h(x))
3.42/1.77	   p(q(x)) -> p(r(x))
3.42/1.77	   q(h(x)) -> r(x)
3.42/1.77	   s(x) -> 1
3.42/1.77	
3.42/1.77	Q is empty.
3.42/1.77	We have to consider all minimal (P,Q,R)-chains.
3.42/1.77	----------------------------------------
3.42/1.77	
3.42/1.77	(5) DependencyGraphProof (EQUIVALENT)
3.42/1.77	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes.
3.42/1.77	----------------------------------------
3.42/1.77	
3.42/1.77	(6)
3.42/1.77	TRUE
3.49/1.80	EOF