3.44/1.68	YES
3.44/1.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.44/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.44/1.69	
3.44/1.69	
3.44/1.69	Termination w.r.t. Q of the given QTRS could be proven:
3.44/1.69	
3.44/1.69	(0) QTRS
3.44/1.69	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
3.44/1.69	(2) QDP
3.44/1.69	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
3.44/1.69	(4) QDP
3.44/1.69	(5) UsableRulesProof [EQUIVALENT, 0 ms]
3.44/1.69	(6) QDP
3.44/1.69	(7) QReductionProof [EQUIVALENT, 0 ms]
3.44/1.69	(8) QDP
3.44/1.69	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.44/1.69	(10) YES
3.44/1.69	
3.44/1.69	
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(0)
3.44/1.69	Obligation:
3.44/1.69	Q restricted rewrite system:
3.44/1.69	The TRS R consists of the following rules:
3.44/1.69	
3.44/1.69	   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
3.44/1.69	   a__g(b) -> c
3.44/1.69	   a__b -> c
3.44/1.69	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
3.44/1.69	   mark(g(X)) -> a__g(mark(X))
3.44/1.69	   mark(b) -> a__b
3.44/1.69	   mark(c) -> c
3.44/1.69	   a__f(X1, X2, X3) -> f(X1, X2, X3)
3.44/1.69	   a__g(X) -> g(X)
3.44/1.69	   a__b -> b
3.44/1.69	
3.44/1.69	The set Q consists of the following terms:
3.44/1.69	
3.44/1.69	   a__b
3.44/1.69	   mark(f(x0, x1, x2))
3.44/1.69	   mark(g(x0))
3.44/1.69	   mark(b)
3.44/1.69	   mark(c)
3.44/1.69	   a__f(x0, x1, x2)
3.44/1.69	   a__g(x0)
3.44/1.69	
3.44/1.69	
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(1) DependencyPairsProof (EQUIVALENT)
3.44/1.69	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(2)
3.44/1.69	Obligation:
3.44/1.69	Q DP problem:
3.44/1.69	The TRS P consists of the following rules:
3.44/1.69	
3.44/1.69	   A__F(X, g(X), Y) -> A__F(Y, Y, Y)
3.44/1.69	   MARK(f(X1, X2, X3)) -> A__F(X1, X2, X3)
3.44/1.69	   MARK(g(X)) -> A__G(mark(X))
3.44/1.69	   MARK(g(X)) -> MARK(X)
3.44/1.69	   MARK(b) -> A__B
3.44/1.69	
3.44/1.69	The TRS R consists of the following rules:
3.44/1.69	
3.44/1.69	   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
3.44/1.69	   a__g(b) -> c
3.44/1.69	   a__b -> c
3.44/1.69	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
3.44/1.69	   mark(g(X)) -> a__g(mark(X))
3.44/1.69	   mark(b) -> a__b
3.44/1.69	   mark(c) -> c
3.44/1.69	   a__f(X1, X2, X3) -> f(X1, X2, X3)
3.44/1.69	   a__g(X) -> g(X)
3.44/1.69	   a__b -> b
3.44/1.69	
3.44/1.69	The set Q consists of the following terms:
3.44/1.69	
3.44/1.69	   a__b
3.44/1.69	   mark(f(x0, x1, x2))
3.44/1.69	   mark(g(x0))
3.44/1.69	   mark(b)
3.44/1.69	   mark(c)
3.44/1.69	   a__f(x0, x1, x2)
3.44/1.69	   a__g(x0)
3.44/1.69	
3.44/1.69	We have to consider all minimal (P,Q,R)-chains.
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(3) DependencyGraphProof (EQUIVALENT)
3.44/1.69	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(4)
3.44/1.69	Obligation:
3.44/1.69	Q DP problem:
3.44/1.69	The TRS P consists of the following rules:
3.44/1.69	
3.44/1.69	   MARK(g(X)) -> MARK(X)
3.44/1.69	
3.44/1.69	The TRS R consists of the following rules:
3.44/1.69	
3.44/1.69	   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
3.44/1.69	   a__g(b) -> c
3.44/1.69	   a__b -> c
3.44/1.69	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
3.44/1.69	   mark(g(X)) -> a__g(mark(X))
3.44/1.69	   mark(b) -> a__b
3.44/1.69	   mark(c) -> c
3.44/1.69	   a__f(X1, X2, X3) -> f(X1, X2, X3)
3.44/1.69	   a__g(X) -> g(X)
3.44/1.69	   a__b -> b
3.44/1.69	
3.44/1.69	The set Q consists of the following terms:
3.44/1.69	
3.44/1.69	   a__b
3.44/1.69	   mark(f(x0, x1, x2))
3.44/1.69	   mark(g(x0))
3.44/1.69	   mark(b)
3.44/1.69	   mark(c)
3.44/1.69	   a__f(x0, x1, x2)
3.44/1.69	   a__g(x0)
3.44/1.69	
3.44/1.69	We have to consider all minimal (P,Q,R)-chains.
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(5) UsableRulesProof (EQUIVALENT)
3.44/1.69	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.
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(6)
3.44/1.69	Obligation:
3.44/1.69	Q DP problem:
3.44/1.69	The TRS P consists of the following rules:
3.44/1.69	
3.44/1.69	   MARK(g(X)) -> MARK(X)
3.44/1.69	
3.44/1.69	R is empty.
3.44/1.69	The set Q consists of the following terms:
3.44/1.69	
3.44/1.69	   a__b
3.44/1.69	   mark(f(x0, x1, x2))
3.44/1.69	   mark(g(x0))
3.44/1.69	   mark(b)
3.44/1.69	   mark(c)
3.44/1.69	   a__f(x0, x1, x2)
3.44/1.69	   a__g(x0)
3.44/1.69	
3.44/1.69	We have to consider all minimal (P,Q,R)-chains.
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(7) QReductionProof (EQUIVALENT)
3.44/1.69	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
3.44/1.69	
3.44/1.69	   a__b
3.44/1.69	   mark(f(x0, x1, x2))
3.44/1.69	   mark(g(x0))
3.44/1.69	   mark(b)
3.44/1.69	   mark(c)
3.44/1.69	   a__f(x0, x1, x2)
3.44/1.69	   a__g(x0)
3.44/1.69	
3.44/1.69	
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(8)
3.44/1.69	Obligation:
3.44/1.69	Q DP problem:
3.44/1.69	The TRS P consists of the following rules:
3.44/1.69	
3.44/1.69	   MARK(g(X)) -> MARK(X)
3.44/1.69	
3.44/1.69	R is empty.
3.44/1.69	Q is empty.
3.44/1.69	We have to consider all minimal (P,Q,R)-chains.
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(9) QDPSizeChangeProof (EQUIVALENT)
3.44/1.69	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
3.44/1.69	
3.44/1.69	From the DPs we obtained the following set of size-change graphs:
3.44/1.69	*MARK(g(X)) -> MARK(X)
3.44/1.69	The graph contains the following edges 1 > 1
3.44/1.69	
3.44/1.69	
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(10)
3.44/1.69	YES
3.57/1.72	EOF