11.26/3.74	YES
11.31/3.80	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
11.31/3.80	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.31/3.80	
11.31/3.80	
11.31/3.80	Termination w.r.t. Q of the given QTRS could be proven:
11.31/3.80	
11.31/3.80	(0) QTRS
11.31/3.80	(1) QTRS Reverse [SOUND, 0 ms]
11.31/3.80	(2) QTRS
11.31/3.80	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
11.31/3.80	(4) QDP
11.31/3.80	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
11.31/3.80	(6) AND
11.31/3.80	    (7) QDP
11.31/3.80	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
11.31/3.80	        (9) QDP
11.31/3.80	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.31/3.80	        (11) YES
11.31/3.80	    (12) QDP
11.31/3.80	        (13) UsableRulesProof [EQUIVALENT, 0 ms]
11.31/3.80	        (14) QDP
11.31/3.80	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.31/3.80	        (16) YES
11.31/3.80	    (17) QDP
11.31/3.80	        (18) UsableRulesProof [EQUIVALENT, 0 ms]
11.31/3.80	        (19) QDP
11.31/3.80	        (20) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.31/3.80	        (21) YES
11.31/3.80	    (22) QDP
11.31/3.80	        (23) UsableRulesProof [EQUIVALENT, 0 ms]
11.31/3.80	        (24) QDP
11.31/3.80	        (25) QDPOrderProof [EQUIVALENT, 10 ms]
11.31/3.80	        (26) QDP
11.31/3.80	        (27) DependencyGraphProof [EQUIVALENT, 0 ms]
11.31/3.80	        (28) AND
11.31/3.80	            (29) QDP
11.31/3.80	                (30) MRRProof [EQUIVALENT, 12 ms]
11.31/3.80	                (31) QDP
11.31/3.80	                (32) PisEmptyProof [EQUIVALENT, 0 ms]
11.31/3.80	                (33) YES
11.31/3.80	            (34) QDP
11.31/3.80	                (35) MRRProof [EQUIVALENT, 12 ms]
11.31/3.80	                (36) QDP
11.31/3.80	                (37) PisEmptyProof [EQUIVALENT, 0 ms]
11.31/3.80	                (38) YES
11.31/3.80	
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(0)
11.31/3.80	Obligation:
11.31/3.80	Q restricted rewrite system:
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   active(g(X)) -> mark(h(X))
11.31/3.80	   active(c) -> mark(d)
11.31/3.80	   active(h(d)) -> mark(g(c))
11.31/3.80	   proper(g(X)) -> g(proper(X))
11.31/3.80	   proper(h(X)) -> h(proper(X))
11.31/3.80	   proper(c) -> ok(c)
11.31/3.80	   proper(d) -> ok(d)
11.31/3.80	   g(ok(X)) -> ok(g(X))
11.31/3.80	   h(ok(X)) -> ok(h(X))
11.31/3.80	   top(mark(X)) -> top(proper(X))
11.31/3.80	   top(ok(X)) -> top(active(X))
11.31/3.80	
11.31/3.80	The set Q consists of the following terms:
11.31/3.80	
11.31/3.80	   active(g(x0))
11.31/3.80	   active(c)
11.31/3.80	   active(h(d))
11.31/3.80	   proper(g(x0))
11.31/3.80	   proper(h(x0))
11.31/3.80	   proper(c)
11.31/3.80	   proper(d)
11.31/3.80	   g(ok(x0))
11.31/3.80	   h(ok(x0))
11.31/3.80	   top(mark(x0))
11.31/3.80	   top(ok(x0))
11.31/3.80	
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(1) QTRS Reverse (SOUND)
11.31/3.80	We applied the QTRS Reverse Processor [REVERSE].
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(2)
11.31/3.80	Obligation:
11.31/3.80	Q restricted rewrite system:
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   c'(active(x)) -> d'(mark(x))
11.31/3.80	   d'(h(active(x))) -> c'(g(mark(x)))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   c'(proper(x)) -> c'(ok(x))
11.31/3.80	   d'(proper(x)) -> d'(ok(x))
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(3) DependencyPairsProof (EQUIVALENT)
11.31/3.80	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(4)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   G(active(X)) -> H(mark(X))
11.31/3.80	   G(active(X)) -> MARK(X)
11.31/3.80	   C'(active(x)) -> D'(mark(x))
11.31/3.80	   C'(active(x)) -> MARK(x)
11.31/3.80	   D'(h(active(x))) -> C'(g(mark(x)))
11.31/3.80	   D'(h(active(x))) -> G(mark(x))
11.31/3.80	   D'(h(active(x))) -> MARK(x)
11.31/3.80	   G(proper(X)) -> G(X)
11.31/3.80	   H(proper(X)) -> H(X)
11.31/3.80	   C'(proper(x)) -> C'(ok(x))
11.31/3.80	   C'(proper(x)) -> OK(x)
11.31/3.80	   D'(proper(x)) -> D'(ok(x))
11.31/3.80	   D'(proper(x)) -> OK(x)
11.31/3.80	   OK(g(X)) -> G(ok(X))
11.31/3.80	   OK(g(X)) -> OK(X)
11.31/3.80	   OK(h(X)) -> H(ok(X))
11.31/3.80	   OK(h(X)) -> OK(X)
11.31/3.80	
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   c'(active(x)) -> d'(mark(x))
11.31/3.80	   d'(h(active(x))) -> c'(g(mark(x)))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   c'(proper(x)) -> c'(ok(x))
11.31/3.80	   d'(proper(x)) -> d'(ok(x))
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(5) DependencyGraphProof (EQUIVALENT)
11.31/3.80	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 9 less nodes.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(6)
11.31/3.80	Complex Obligation (AND)
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(7)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   H(proper(X)) -> H(X)
11.31/3.80	
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   c'(active(x)) -> d'(mark(x))
11.31/3.80	   d'(h(active(x))) -> c'(g(mark(x)))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   c'(proper(x)) -> c'(ok(x))
11.31/3.80	   d'(proper(x)) -> d'(ok(x))
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(8) UsableRulesProof (EQUIVALENT)
11.31/3.80	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(9)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   H(proper(X)) -> H(X)
11.31/3.80	
11.31/3.80	R is empty.
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(10) QDPSizeChangeProof (EQUIVALENT)
11.31/3.80	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. 
11.31/3.80	
11.31/3.80	From the DPs we obtained the following set of size-change graphs:
11.31/3.80	*H(proper(X)) -> H(X)
11.31/3.80	The graph contains the following edges 1 > 1
11.31/3.80	
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(11)
11.31/3.80	YES
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(12)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   G(proper(X)) -> G(X)
11.31/3.80	
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   c'(active(x)) -> d'(mark(x))
11.31/3.80	   d'(h(active(x))) -> c'(g(mark(x)))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   c'(proper(x)) -> c'(ok(x))
11.31/3.80	   d'(proper(x)) -> d'(ok(x))
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(13) UsableRulesProof (EQUIVALENT)
11.31/3.80	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(14)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   G(proper(X)) -> G(X)
11.31/3.80	
11.31/3.80	R is empty.
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(15) QDPSizeChangeProof (EQUIVALENT)
11.31/3.80	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. 
11.31/3.80	
11.31/3.80	From the DPs we obtained the following set of size-change graphs:
11.31/3.80	*G(proper(X)) -> G(X)
11.31/3.80	The graph contains the following edges 1 > 1
11.31/3.80	
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(16)
11.31/3.80	YES
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(17)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   OK(h(X)) -> OK(X)
11.31/3.80	   OK(g(X)) -> OK(X)
11.31/3.80	
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   c'(active(x)) -> d'(mark(x))
11.31/3.80	   d'(h(active(x))) -> c'(g(mark(x)))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   c'(proper(x)) -> c'(ok(x))
11.31/3.80	   d'(proper(x)) -> d'(ok(x))
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(18) UsableRulesProof (EQUIVALENT)
11.31/3.80	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(19)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   OK(h(X)) -> OK(X)
11.31/3.80	   OK(g(X)) -> OK(X)
11.31/3.80	
11.31/3.80	R is empty.
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(20) QDPSizeChangeProof (EQUIVALENT)
11.31/3.80	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. 
11.31/3.80	
11.31/3.80	From the DPs we obtained the following set of size-change graphs:
11.31/3.80	*OK(h(X)) -> OK(X)
11.31/3.80	The graph contains the following edges 1 > 1
11.31/3.80	
11.31/3.80	
11.31/3.80	*OK(g(X)) -> OK(X)
11.31/3.80	The graph contains the following edges 1 > 1
11.31/3.80	
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(21)
11.31/3.80	YES
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(22)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   D'(h(active(x))) -> C'(g(mark(x)))
11.31/3.80	   C'(active(x)) -> D'(mark(x))
11.31/3.80	   D'(proper(x)) -> D'(ok(x))
11.31/3.80	   C'(proper(x)) -> C'(ok(x))
11.31/3.80	
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   c'(active(x)) -> d'(mark(x))
11.31/3.80	   d'(h(active(x))) -> c'(g(mark(x)))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   c'(proper(x)) -> c'(ok(x))
11.31/3.80	   d'(proper(x)) -> d'(ok(x))
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(23) UsableRulesProof (EQUIVALENT)
11.31/3.80	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(24)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   D'(h(active(x))) -> C'(g(mark(x)))
11.31/3.80	   C'(active(x)) -> D'(mark(x))
11.31/3.80	   D'(proper(x)) -> D'(ok(x))
11.31/3.80	   C'(proper(x)) -> C'(ok(x))
11.31/3.80	
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(25) QDPOrderProof (EQUIVALENT)
11.31/3.80	We use the reduction pair processor [LPAR04,JAR06].
11.31/3.80	
11.31/3.80	
11.31/3.80	The following pairs can be oriented strictly and are deleted.
11.31/3.80	
11.31/3.80	   C'(active(x)) -> D'(mark(x))
11.31/3.80	The remaining pairs can at least be oriented weakly.
11.31/3.80	Used ordering:  Polynomial interpretation [POLO]:
11.31/3.80	
11.31/3.80	   POL(C'(x_1)) = x_1
11.31/3.80	   POL(D'(x_1)) = 0
11.31/3.80	   POL(active(x_1)) = 1
11.31/3.80	   POL(g(x_1)) = 0
11.31/3.80	   POL(h(x_1)) = 0
11.31/3.80	   POL(mark(x_1)) = 0
11.31/3.80	   POL(ok(x_1)) = x_1
11.31/3.80	   POL(proper(x_1)) = x_1
11.31/3.80	   POL(top(x_1)) = 1 + x_1
11.31/3.80	
11.31/3.80	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
11.31/3.80	
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(26)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   D'(h(active(x))) -> C'(g(mark(x)))
11.31/3.80	   D'(proper(x)) -> D'(ok(x))
11.31/3.80	   C'(proper(x)) -> C'(ok(x))
11.31/3.80	
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(27) DependencyGraphProof (EQUIVALENT)
11.31/3.80	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(28)
11.31/3.80	Complex Obligation (AND)
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(29)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   C'(proper(x)) -> C'(ok(x))
11.31/3.80	
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(30) MRRProof (EQUIVALENT)
11.31/3.80	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
11.31/3.80	
11.31/3.80	Strictly oriented dependency pairs:
11.31/3.80	
11.31/3.80	   C'(proper(x)) -> C'(ok(x))
11.31/3.80	
11.31/3.80	Strictly oriented rules of the TRS R:
11.31/3.80	
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	
11.31/3.80	Used ordering: Polynomial interpretation [POLO]:
11.31/3.80	
11.31/3.80	   POL(C'(x_1)) = 3*x_1
11.31/3.80	   POL(active(x_1)) = 2*x_1
11.31/3.80	   POL(g(x_1)) = 1 + 3*x_1
11.31/3.80	   POL(h(x_1)) = x_1
11.31/3.80	   POL(mark(x_1)) = 1 + 2*x_1
11.31/3.80	   POL(ok(x_1)) = 2*x_1
11.31/3.80	   POL(proper(x_1)) = 1 + 2*x_1
11.31/3.80	   POL(top(x_1)) = 2*x_1
11.31/3.80	
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(31)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	P is empty.
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(32) PisEmptyProof (EQUIVALENT)
11.31/3.80	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(33)
11.31/3.80	YES
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(34)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	The TRS P consists of the following rules:
11.31/3.80	
11.31/3.80	   D'(proper(x)) -> D'(ok(x))
11.31/3.80	
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(35) MRRProof (EQUIVALENT)
11.31/3.80	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
11.31/3.80	
11.31/3.80	Strictly oriented dependency pairs:
11.31/3.80	
11.31/3.80	   D'(proper(x)) -> D'(ok(x))
11.31/3.80	
11.31/3.80	Strictly oriented rules of the TRS R:
11.31/3.80	
11.31/3.80	   ok(g(X)) -> g(ok(X))
11.31/3.80	   g(proper(X)) -> proper(g(X))
11.31/3.80	
11.31/3.80	Used ordering: Polynomial interpretation [POLO]:
11.31/3.80	
11.31/3.80	   POL(D'(x_1)) = 3*x_1
11.31/3.80	   POL(active(x_1)) = 2*x_1
11.31/3.80	   POL(g(x_1)) = 1 + 3*x_1
11.31/3.80	   POL(h(x_1)) = x_1
11.31/3.80	   POL(mark(x_1)) = 1 + 2*x_1
11.31/3.80	   POL(ok(x_1)) = 2*x_1
11.31/3.80	   POL(proper(x_1)) = 1 + 2*x_1
11.31/3.80	   POL(top(x_1)) = 2*x_1
11.31/3.80	
11.31/3.80	
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(36)
11.31/3.80	Obligation:
11.31/3.80	Q DP problem:
11.31/3.80	P is empty.
11.31/3.80	The TRS R consists of the following rules:
11.31/3.80	
11.31/3.80	   ok(h(X)) -> h(ok(X))
11.31/3.80	   ok(top(X)) -> active(top(X))
11.31/3.80	   h(proper(X)) -> proper(h(X))
11.31/3.80	   g(active(X)) -> h(mark(X))
11.31/3.80	   mark(top(X)) -> proper(top(X))
11.31/3.80	
11.31/3.80	Q is empty.
11.31/3.80	We have to consider all minimal (P,Q,R)-chains.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(37) PisEmptyProof (EQUIVALENT)
11.31/3.80	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.31/3.80	----------------------------------------
11.31/3.80	
11.31/3.80	(38)
11.31/3.80	YES
11.72/3.97	EOF