4.12/2.23	YES
4.12/2.23	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.12/2.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.12/2.23	
4.12/2.23	
4.12/2.23	Termination w.r.t. Q of the given QTRS could be proven:
4.12/2.23	
4.12/2.23	(0) QTRS
4.12/2.23	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
4.12/2.23	(2) QDP
4.12/2.23	(3) UsableRulesProof [EQUIVALENT, 0 ms]
4.12/2.23	(4) QDP
4.12/2.23	(5) QReductionProof [EQUIVALENT, 0 ms]
4.12/2.23	(6) QDP
4.12/2.23	(7) TransformationProof [EQUIVALENT, 0 ms]
4.12/2.23	(8) QDP
4.12/2.23	(9) TransformationProof [EQUIVALENT, 0 ms]
4.12/2.23	(10) QDP
4.12/2.23	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
4.12/2.23	(12) TRUE
4.12/2.23	
4.12/2.23	
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(0)
4.12/2.23	Obligation:
4.12/2.23	Q restricted rewrite system:
4.12/2.23	The TRS R consists of the following rules:
4.12/2.23	
4.12/2.23	   f(a(x), y) -> g(x, y)
4.12/2.23	   g(x, y) -> h(x, y)
4.12/2.23	   h(b, y) -> f(y, y)
4.12/2.23	   a(b) -> c
4.12/2.23	
4.12/2.23	The set Q consists of the following terms:
4.12/2.23	
4.12/2.23	   f(a(x0), x1)
4.12/2.23	   g(x0, x1)
4.12/2.23	   h(b, x0)
4.12/2.23	   a(b)
4.12/2.23	
4.12/2.23	
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(1) DependencyPairsProof (EQUIVALENT)
4.12/2.23	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(2)
4.12/2.23	Obligation:
4.12/2.23	Q DP problem:
4.12/2.23	The TRS P consists of the following rules:
4.12/2.23	
4.12/2.23	   F(a(x), y) -> G(x, y)
4.12/2.23	   G(x, y) -> H(x, y)
4.12/2.23	   H(b, y) -> F(y, y)
4.12/2.23	
4.12/2.23	The TRS R consists of the following rules:
4.12/2.23	
4.12/2.23	   f(a(x), y) -> g(x, y)
4.12/2.23	   g(x, y) -> h(x, y)
4.12/2.23	   h(b, y) -> f(y, y)
4.12/2.23	   a(b) -> c
4.12/2.23	
4.12/2.23	The set Q consists of the following terms:
4.12/2.23	
4.12/2.23	   f(a(x0), x1)
4.12/2.23	   g(x0, x1)
4.12/2.23	   h(b, x0)
4.12/2.23	   a(b)
4.12/2.23	
4.12/2.23	We have to consider all minimal (P,Q,R)-chains.
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(3) UsableRulesProof (EQUIVALENT)
4.12/2.23	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(4)
4.12/2.23	Obligation:
4.12/2.23	Q DP problem:
4.12/2.23	The TRS P consists of the following rules:
4.12/2.23	
4.12/2.23	   F(a(x), y) -> G(x, y)
4.12/2.23	   G(x, y) -> H(x, y)
4.12/2.23	   H(b, y) -> F(y, y)
4.12/2.23	
4.12/2.23	R is empty.
4.12/2.23	The set Q consists of the following terms:
4.12/2.23	
4.12/2.23	   f(a(x0), x1)
4.12/2.23	   g(x0, x1)
4.12/2.23	   h(b, x0)
4.12/2.23	   a(b)
4.12/2.23	
4.12/2.23	We have to consider all minimal (P,Q,R)-chains.
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(5) QReductionProof (EQUIVALENT)
4.12/2.23	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.12/2.23	
4.12/2.23	   f(a(x0), x1)
4.12/2.23	   g(x0, x1)
4.12/2.23	   h(b, x0)
4.12/2.23	
4.12/2.23	
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(6)
4.12/2.23	Obligation:
4.12/2.23	Q DP problem:
4.12/2.23	The TRS P consists of the following rules:
4.12/2.23	
4.12/2.23	   F(a(x), y) -> G(x, y)
4.12/2.23	   G(x, y) -> H(x, y)
4.12/2.23	   H(b, y) -> F(y, y)
4.12/2.23	
4.12/2.23	R is empty.
4.12/2.23	The set Q consists of the following terms:
4.12/2.23	
4.12/2.23	   a(b)
4.12/2.23	
4.12/2.23	We have to consider all minimal (P,Q,R)-chains.
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(7) TransformationProof (EQUIVALENT)
4.12/2.23	By instantiating [LPAR04] the rule F(a(x), y) -> G(x, y) we obtained the following new rules [LPAR04]:
4.12/2.23	
4.12/2.23	   (F(a(x0), a(x0)) -> G(x0, a(x0)),F(a(x0), a(x0)) -> G(x0, a(x0)))
4.12/2.23	
4.12/2.23	
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(8)
4.12/2.23	Obligation:
4.12/2.23	Q DP problem:
4.12/2.23	The TRS P consists of the following rules:
4.12/2.23	
4.12/2.23	   G(x, y) -> H(x, y)
4.12/2.23	   H(b, y) -> F(y, y)
4.12/2.23	   F(a(x0), a(x0)) -> G(x0, a(x0))
4.12/2.23	
4.12/2.23	R is empty.
4.12/2.23	The set Q consists of the following terms:
4.12/2.23	
4.12/2.23	   a(b)
4.12/2.23	
4.12/2.23	We have to consider all minimal (P,Q,R)-chains.
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(9) TransformationProof (EQUIVALENT)
4.12/2.23	By instantiating [LPAR04] the rule G(x, y) -> H(x, y) we obtained the following new rules [LPAR04]:
4.12/2.23	
4.12/2.23	   (G(z0, a(z0)) -> H(z0, a(z0)),G(z0, a(z0)) -> H(z0, a(z0)))
4.12/2.23	
4.12/2.23	
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(10)
4.12/2.23	Obligation:
4.12/2.23	Q DP problem:
4.12/2.23	The TRS P consists of the following rules:
4.12/2.23	
4.12/2.23	   H(b, y) -> F(y, y)
4.12/2.23	   F(a(x0), a(x0)) -> G(x0, a(x0))
4.12/2.23	   G(z0, a(z0)) -> H(z0, a(z0))
4.12/2.23	
4.12/2.23	R is empty.
4.12/2.23	The set Q consists of the following terms:
4.12/2.23	
4.12/2.23	   a(b)
4.12/2.23	
4.12/2.23	We have to consider all minimal (P,Q,R)-chains.
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(11) DependencyGraphProof (EQUIVALENT)
4.12/2.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
4.12/2.23	----------------------------------------
4.12/2.23	
4.12/2.23	(12)
4.12/2.23	TRUE
4.35/2.25	EOF