122.65/31.92	YES
122.76/31.92	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
122.76/31.92	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
122.76/31.92	
122.76/31.92	
122.76/31.92	Outermost Termination of the given OTRS could be proven:
122.76/31.92	
122.76/31.92	(0) OTRS
122.76/31.92	(1) Raffelsieper-Zantema-Transformation [SOUND, 0 ms]
122.76/31.92	(2) QTRS
122.76/31.92	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
122.76/31.92	(4) QDP
122.76/31.92	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
122.76/31.92	(6) AND
122.76/31.92	    (7) QDP
122.76/31.92	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
122.76/31.92	        (9) QDP
122.76/31.92	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
122.76/31.92	        (11) YES
122.76/31.92	    (12) QDP
122.76/31.92	        (13) TransformationProof [EQUIVALENT, 2 ms]
122.76/31.92	        (14) QDP
122.76/31.92	        (15) DependencyGraphProof [EQUIVALENT, 0 ms]
122.76/31.92	        (16) QDP
122.76/31.92	        (17) QDPOrderProof [EQUIVALENT, 29 ms]
122.76/31.92	        (18) QDP
122.76/31.92	        (19) DependencyGraphProof [EQUIVALENT, 0 ms]
122.76/31.92	        (20) QDP
122.76/31.92	        (21) QDPOrderProof [EQUIVALENT, 6 ms]
122.76/31.92	        (22) QDP
122.76/31.92	        (23) QDPOrderProof [EQUIVALENT, 6043 ms]
122.76/31.92	        (24) QDP
122.76/31.92	        (25) PisEmptyProof [EQUIVALENT, 0 ms]
122.76/31.92	        (26) YES
122.76/31.92	
122.76/31.92	
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(0)
122.76/31.92	Obligation:
122.76/31.92	Term rewrite system R:
122.76/31.92	The TRS R consists of the following rules:
122.76/31.92	
122.76/31.92	   eq(n__0, n__0) -> true
122.76/31.92	   eq(n__s(X), n__s(Y)) -> eq(activate(X), activate(Y))
122.76/31.92	   eq(X, Y) -> false
122.76/31.92	   inf(X) -> cons(X, n__inf(s(X)))
122.76/31.92	   take(0, X) -> nil
122.76/31.92	   take(s(X), cons(Y, L)) -> cons(activate(Y), n__take(activate(X), activate(L)))
122.76/31.92	   length(nil) -> 0
122.76/31.92	   length(cons(X, L)) -> s(n__length(activate(L)))
122.76/31.92	   0 -> n__0
122.76/31.92	   s(X) -> n__s(X)
122.76/31.92	   inf(X) -> n__inf(X)
122.76/31.92	   take(X1, X2) -> n__take(X1, X2)
122.76/31.92	   length(X) -> n__length(X)
122.76/31.92	   activate(n__0) -> 0
122.76/31.92	   activate(n__s(X)) -> s(X)
122.76/31.92	   activate(n__inf(X)) -> inf(X)
122.76/31.92	   activate(n__take(X1, X2)) -> take(X1, X2)
122.76/31.92	   activate(n__length(X)) -> length(X)
122.76/31.92	   activate(X) -> X
122.76/31.92	
122.76/31.92	
122.76/31.92	
122.76/31.92	Outermost Strategy.
122.76/31.92	
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(1) Raffelsieper-Zantema-Transformation (SOUND)
122.76/31.92	We applied the Raffelsieper-Zantema transformation  to transform the outermost TRS to a standard TRS.
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(2)
122.76/31.92	Obligation:
122.76/31.92	Q restricted rewrite system:
122.76/31.92	The TRS R consists of the following rules:
122.76/31.92	
122.76/31.92	   down(eq(n__0, n__0)) -> up(true)
122.76/31.92	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.76/31.92	   down(eq(X, Y)) -> up(false)
122.76/31.92	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.76/31.92	   down(take(0, X)) -> up(nil)
122.76/31.92	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.76/31.92	   down(length(nil)) -> up(0)
122.76/31.92	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.76/31.92	   down(0) -> up(n__0)
122.76/31.92	   down(s(X)) -> up(n__s(X))
122.76/31.92	   down(inf(X)) -> up(n__inf(X))
122.76/31.92	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.76/31.92	   down(length(X)) -> up(n__length(X))
122.76/31.92	   down(activate(n__0)) -> up(0)
122.76/31.92	   down(activate(n__s(X))) -> up(s(X))
122.76/31.92	   down(activate(n__inf(X))) -> up(inf(X))
122.76/31.92	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.76/31.92	   down(activate(n__length(X))) -> up(length(X))
122.76/31.92	   down(activate(X)) -> up(X)
122.76/31.92	   top(up(x)) -> top(down(x))
122.76/31.92	   down(n__s(y2)) -> n__s_flat(down(y2))
122.76/31.92	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.76/31.92	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.76/31.92	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.76/31.92	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.76/31.92	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.76/31.92	   down(n__length(y14)) -> n__length_flat(down(y14))
122.76/31.92	   eq_flat(up(x_1), block(x_2)) -> up(eq(x_1, x_2))
122.76/31.92	   eq_flat(block(x_1), up(x_2)) -> up(eq(x_1, x_2))
122.76/31.92	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.76/31.92	   activate_flat(up(x_1)) -> up(activate(x_1))
122.76/31.92	   inf_flat(up(x_1)) -> up(inf(x_1))
122.76/31.92	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.76/31.92	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.76/31.92	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.76/31.92	   s_flat(up(x_1)) -> up(s(x_1))
122.76/31.92	   take_flat(up(x_1), block(x_2)) -> up(take(x_1, x_2))
122.76/31.92	   take_flat(block(x_1), up(x_2)) -> up(take(x_1, x_2))
122.76/31.92	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.76/31.92	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.76/31.92	   length_flat(up(x_1)) -> up(length(x_1))
122.76/31.92	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.76/31.92	
122.76/31.92	Q is empty.
122.76/31.92	
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(3) DependencyPairsProof (EQUIVALENT)
122.76/31.92	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(4)
122.76/31.92	Obligation:
122.76/31.92	Q DP problem:
122.76/31.92	The TRS P consists of the following rules:
122.76/31.92	
122.76/31.92	   TOP(up(x)) -> TOP(down(x))
122.76/31.92	   TOP(up(x)) -> DOWN(x)
122.76/31.92	   DOWN(n__s(y2)) -> N__S_FLAT(down(y2))
122.76/31.92	   DOWN(n__s(y2)) -> DOWN(y2)
122.76/31.92	   DOWN(cons(y5, y6)) -> CONS_FLAT(down(y5), block(y6))
122.76/31.92	   DOWN(cons(y5, y6)) -> DOWN(y5)
122.76/31.92	   DOWN(cons(y5, y6)) -> CONS_FLAT(block(y5), down(y6))
122.76/31.92	   DOWN(cons(y5, y6)) -> DOWN(y6)
122.76/31.92	   DOWN(n__inf(y7)) -> N__INF_FLAT(down(y7))
122.76/31.92	   DOWN(n__inf(y7)) -> DOWN(y7)
122.76/31.92	   DOWN(n__take(y11, y12)) -> N__TAKE_FLAT(down(y11), block(y12))
122.76/31.92	   DOWN(n__take(y11, y12)) -> DOWN(y11)
122.76/31.92	   DOWN(n__take(y11, y12)) -> N__TAKE_FLAT(block(y11), down(y12))
122.76/31.92	   DOWN(n__take(y11, y12)) -> DOWN(y12)
122.76/31.92	   DOWN(n__length(y14)) -> N__LENGTH_FLAT(down(y14))
122.76/31.92	   DOWN(n__length(y14)) -> DOWN(y14)
122.76/31.92	
122.76/31.92	The TRS R consists of the following rules:
122.76/31.92	
122.76/31.92	   down(eq(n__0, n__0)) -> up(true)
122.76/31.92	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.76/31.92	   down(eq(X, Y)) -> up(false)
122.76/31.92	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.76/31.92	   down(take(0, X)) -> up(nil)
122.76/31.92	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.76/31.92	   down(length(nil)) -> up(0)
122.76/31.92	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.76/31.92	   down(0) -> up(n__0)
122.76/31.92	   down(s(X)) -> up(n__s(X))
122.76/31.92	   down(inf(X)) -> up(n__inf(X))
122.76/31.92	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.76/31.92	   down(length(X)) -> up(n__length(X))
122.76/31.92	   down(activate(n__0)) -> up(0)
122.76/31.92	   down(activate(n__s(X))) -> up(s(X))
122.76/31.92	   down(activate(n__inf(X))) -> up(inf(X))
122.76/31.92	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.76/31.92	   down(activate(n__length(X))) -> up(length(X))
122.76/31.92	   down(activate(X)) -> up(X)
122.76/31.92	   top(up(x)) -> top(down(x))
122.76/31.92	   down(n__s(y2)) -> n__s_flat(down(y2))
122.76/31.92	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.76/31.92	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.76/31.92	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.76/31.92	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.76/31.92	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.76/31.92	   down(n__length(y14)) -> n__length_flat(down(y14))
122.76/31.92	   eq_flat(up(x_1), block(x_2)) -> up(eq(x_1, x_2))
122.76/31.92	   eq_flat(block(x_1), up(x_2)) -> up(eq(x_1, x_2))
122.76/31.92	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.76/31.92	   activate_flat(up(x_1)) -> up(activate(x_1))
122.76/31.92	   inf_flat(up(x_1)) -> up(inf(x_1))
122.76/31.92	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.76/31.92	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.76/31.92	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.76/31.92	   s_flat(up(x_1)) -> up(s(x_1))
122.76/31.92	   take_flat(up(x_1), block(x_2)) -> up(take(x_1, x_2))
122.76/31.92	   take_flat(block(x_1), up(x_2)) -> up(take(x_1, x_2))
122.76/31.92	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.76/31.92	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.76/31.92	   length_flat(up(x_1)) -> up(length(x_1))
122.76/31.92	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.76/31.92	
122.76/31.92	Q is empty.
122.76/31.92	We have to consider all minimal (P,Q,R)-chains.
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(5) DependencyGraphProof (EQUIVALENT)
122.76/31.92	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes.
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(6)
122.76/31.92	Complex Obligation (AND)
122.76/31.92	
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(7)
122.76/31.92	Obligation:
122.76/31.92	Q DP problem:
122.76/31.92	The TRS P consists of the following rules:
122.76/31.92	
122.76/31.92	   DOWN(cons(y5, y6)) -> DOWN(y5)
122.76/31.92	   DOWN(n__s(y2)) -> DOWN(y2)
122.76/31.92	   DOWN(cons(y5, y6)) -> DOWN(y6)
122.76/31.92	   DOWN(n__inf(y7)) -> DOWN(y7)
122.76/31.92	   DOWN(n__take(y11, y12)) -> DOWN(y11)
122.76/31.92	   DOWN(n__take(y11, y12)) -> DOWN(y12)
122.76/31.92	   DOWN(n__length(y14)) -> DOWN(y14)
122.76/31.92	
122.76/31.92	The TRS R consists of the following rules:
122.76/31.92	
122.76/31.92	   down(eq(n__0, n__0)) -> up(true)
122.76/31.92	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.76/31.92	   down(eq(X, Y)) -> up(false)
122.76/31.92	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.76/31.92	   down(take(0, X)) -> up(nil)
122.76/31.92	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.76/31.92	   down(length(nil)) -> up(0)
122.76/31.92	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.76/31.92	   down(0) -> up(n__0)
122.76/31.92	   down(s(X)) -> up(n__s(X))
122.76/31.92	   down(inf(X)) -> up(n__inf(X))
122.76/31.92	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.76/31.92	   down(length(X)) -> up(n__length(X))
122.76/31.92	   down(activate(n__0)) -> up(0)
122.76/31.92	   down(activate(n__s(X))) -> up(s(X))
122.76/31.92	   down(activate(n__inf(X))) -> up(inf(X))
122.76/31.92	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.76/31.92	   down(activate(n__length(X))) -> up(length(X))
122.76/31.92	   down(activate(X)) -> up(X)
122.76/31.92	   top(up(x)) -> top(down(x))
122.76/31.92	   down(n__s(y2)) -> n__s_flat(down(y2))
122.76/31.92	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.76/31.92	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.76/31.92	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.76/31.92	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.76/31.92	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.76/31.92	   down(n__length(y14)) -> n__length_flat(down(y14))
122.76/31.92	   eq_flat(up(x_1), block(x_2)) -> up(eq(x_1, x_2))
122.76/31.92	   eq_flat(block(x_1), up(x_2)) -> up(eq(x_1, x_2))
122.76/31.92	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.76/31.92	   activate_flat(up(x_1)) -> up(activate(x_1))
122.76/31.92	   inf_flat(up(x_1)) -> up(inf(x_1))
122.76/31.92	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.76/31.92	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.76/31.92	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.76/31.92	   s_flat(up(x_1)) -> up(s(x_1))
122.76/31.92	   take_flat(up(x_1), block(x_2)) -> up(take(x_1, x_2))
122.76/31.92	   take_flat(block(x_1), up(x_2)) -> up(take(x_1, x_2))
122.76/31.92	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.76/31.92	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.76/31.92	   length_flat(up(x_1)) -> up(length(x_1))
122.76/31.92	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.76/31.92	
122.76/31.92	Q is empty.
122.76/31.92	We have to consider all minimal (P,Q,R)-chains.
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(8) UsableRulesProof (EQUIVALENT)
122.76/31.92	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.
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(9)
122.76/31.92	Obligation:
122.76/31.92	Q DP problem:
122.76/31.92	The TRS P consists of the following rules:
122.76/31.92	
122.76/31.92	   DOWN(cons(y5, y6)) -> DOWN(y5)
122.76/31.92	   DOWN(n__s(y2)) -> DOWN(y2)
122.76/31.92	   DOWN(cons(y5, y6)) -> DOWN(y6)
122.76/31.92	   DOWN(n__inf(y7)) -> DOWN(y7)
122.76/31.92	   DOWN(n__take(y11, y12)) -> DOWN(y11)
122.76/31.92	   DOWN(n__take(y11, y12)) -> DOWN(y12)
122.76/31.92	   DOWN(n__length(y14)) -> DOWN(y14)
122.76/31.92	
122.76/31.92	R is empty.
122.76/31.92	Q is empty.
122.76/31.92	We have to consider all minimal (P,Q,R)-chains.
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(10) QDPSizeChangeProof (EQUIVALENT)
122.76/31.92	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. 
122.76/31.92	
122.76/31.92	From the DPs we obtained the following set of size-change graphs:
122.76/31.92	*DOWN(cons(y5, y6)) -> DOWN(y5)
122.76/31.92	The graph contains the following edges 1 > 1
122.76/31.92	
122.76/31.92	
122.76/31.92	*DOWN(n__s(y2)) -> DOWN(y2)
122.76/31.92	The graph contains the following edges 1 > 1
122.76/31.92	
122.76/31.92	
122.76/31.92	*DOWN(cons(y5, y6)) -> DOWN(y6)
122.76/31.92	The graph contains the following edges 1 > 1
122.76/31.92	
122.76/31.92	
122.76/31.92	*DOWN(n__inf(y7)) -> DOWN(y7)
122.76/31.92	The graph contains the following edges 1 > 1
122.76/31.92	
122.76/31.92	
122.76/31.92	*DOWN(n__take(y11, y12)) -> DOWN(y11)
122.76/31.92	The graph contains the following edges 1 > 1
122.76/31.92	
122.76/31.92	
122.76/31.92	*DOWN(n__take(y11, y12)) -> DOWN(y12)
122.76/31.92	The graph contains the following edges 1 > 1
122.76/31.92	
122.76/31.92	
122.76/31.92	*DOWN(n__length(y14)) -> DOWN(y14)
122.76/31.92	The graph contains the following edges 1 > 1
122.76/31.92	
122.76/31.92	
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(11)
122.76/31.92	YES
122.76/31.92	
122.76/31.92	----------------------------------------
122.76/31.92	
122.76/31.92	(12)
122.76/31.92	Obligation:
122.76/31.92	Q DP problem:
122.76/31.92	The TRS P consists of the following rules:
122.76/31.92	
122.76/31.92	   TOP(up(x)) -> TOP(down(x))
122.76/31.92	
122.76/31.92	The TRS R consists of the following rules:
122.76/31.92	
122.76/31.92	   down(eq(n__0, n__0)) -> up(true)
122.76/31.92	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.76/31.92	   down(eq(X, Y)) -> up(false)
122.76/31.92	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.76/31.92	   down(take(0, X)) -> up(nil)
122.76/31.92	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.76/31.92	   down(length(nil)) -> up(0)
122.76/31.92	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.76/31.92	   down(0) -> up(n__0)
122.76/31.92	   down(s(X)) -> up(n__s(X))
122.76/31.92	   down(inf(X)) -> up(n__inf(X))
122.76/31.92	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.76/31.92	   down(length(X)) -> up(n__length(X))
122.76/31.92	   down(activate(n__0)) -> up(0)
122.76/31.93	   down(activate(n__s(X))) -> up(s(X))
122.76/31.93	   down(activate(n__inf(X))) -> up(inf(X))
122.76/31.93	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.76/31.93	   down(activate(n__length(X))) -> up(length(X))
122.76/31.93	   down(activate(X)) -> up(X)
122.76/31.93	   top(up(x)) -> top(down(x))
122.76/31.93	   down(n__s(y2)) -> n__s_flat(down(y2))
122.76/31.93	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.76/31.93	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.76/31.93	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.76/31.93	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.76/31.93	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.76/31.93	   down(n__length(y14)) -> n__length_flat(down(y14))
122.76/31.93	   eq_flat(up(x_1), block(x_2)) -> up(eq(x_1, x_2))
122.76/31.93	   eq_flat(block(x_1), up(x_2)) -> up(eq(x_1, x_2))
122.76/31.93	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.76/31.93	   activate_flat(up(x_1)) -> up(activate(x_1))
122.76/31.93	   inf_flat(up(x_1)) -> up(inf(x_1))
122.76/31.93	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.76/31.93	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.76/31.93	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.76/31.93	   s_flat(up(x_1)) -> up(s(x_1))
122.76/31.93	   take_flat(up(x_1), block(x_2)) -> up(take(x_1, x_2))
122.76/31.93	   take_flat(block(x_1), up(x_2)) -> up(take(x_1, x_2))
122.76/31.93	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.76/31.93	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.76/31.93	   length_flat(up(x_1)) -> up(length(x_1))
122.76/31.93	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.76/31.93	
122.76/31.93	Q is empty.
122.76/31.93	We have to consider all minimal (P,Q,R)-chains.
122.76/31.93	----------------------------------------
122.76/31.93	
122.76/31.93	(13) TransformationProof (EQUIVALENT)
122.76/31.93	By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]:
122.76/31.93	
122.76/31.93	   (TOP(up(eq(n__0, n__0))) -> TOP(up(true)),TOP(up(eq(n__0, n__0))) -> TOP(up(true)))
122.76/31.93	   (TOP(up(eq(n__s(x0), n__s(x1)))) -> TOP(up(eq(activate(x0), activate(x1)))),TOP(up(eq(n__s(x0), n__s(x1)))) -> TOP(up(eq(activate(x0), activate(x1)))))
122.76/31.93	   (TOP(up(eq(x0, x1))) -> TOP(up(false)),TOP(up(eq(x0, x1))) -> TOP(up(false)))
122.76/31.93	   (TOP(up(inf(x0))) -> TOP(up(cons(x0, n__inf(s(x0))))),TOP(up(inf(x0))) -> TOP(up(cons(x0, n__inf(s(x0))))))
122.76/31.93	   (TOP(up(take(0, x0))) -> TOP(up(nil)),TOP(up(take(0, x0))) -> TOP(up(nil)))
122.76/31.93	   (TOP(up(take(s(x0), cons(x1, x2)))) -> TOP(up(cons(activate(x1), n__take(activate(x0), activate(x2))))),TOP(up(take(s(x0), cons(x1, x2)))) -> TOP(up(cons(activate(x1), n__take(activate(x0), activate(x2))))))
122.76/31.93	   (TOP(up(length(nil))) -> TOP(up(0)),TOP(up(length(nil))) -> TOP(up(0)))
122.76/31.93	   (TOP(up(length(cons(x0, x1)))) -> TOP(up(s(n__length(activate(x1))))),TOP(up(length(cons(x0, x1)))) -> TOP(up(s(n__length(activate(x1))))))
122.76/31.93	   (TOP(up(0)) -> TOP(up(n__0)),TOP(up(0)) -> TOP(up(n__0)))
122.76/31.93	   (TOP(up(s(x0))) -> TOP(up(n__s(x0))),TOP(up(s(x0))) -> TOP(up(n__s(x0))))
122.76/31.93	   (TOP(up(inf(x0))) -> TOP(up(n__inf(x0))),TOP(up(inf(x0))) -> TOP(up(n__inf(x0))))
122.76/31.93	   (TOP(up(take(x0, x1))) -> TOP(up(n__take(x0, x1))),TOP(up(take(x0, x1))) -> TOP(up(n__take(x0, x1))))
122.76/31.93	   (TOP(up(length(x0))) -> TOP(up(n__length(x0))),TOP(up(length(x0))) -> TOP(up(n__length(x0))))
122.76/31.93	   (TOP(up(activate(n__0))) -> TOP(up(0)),TOP(up(activate(n__0))) -> TOP(up(0)))
122.76/31.93	   (TOP(up(activate(n__s(x0)))) -> TOP(up(s(x0))),TOP(up(activate(n__s(x0)))) -> TOP(up(s(x0))))
122.76/31.93	   (TOP(up(activate(n__inf(x0)))) -> TOP(up(inf(x0))),TOP(up(activate(n__inf(x0)))) -> TOP(up(inf(x0))))
122.76/31.93	   (TOP(up(activate(n__take(x0, x1)))) -> TOP(up(take(x0, x1))),TOP(up(activate(n__take(x0, x1)))) -> TOP(up(take(x0, x1))))
122.76/31.93	   (TOP(up(activate(n__length(x0)))) -> TOP(up(length(x0))),TOP(up(activate(n__length(x0)))) -> TOP(up(length(x0))))
122.76/31.93	   (TOP(up(activate(x0))) -> TOP(up(x0)),TOP(up(activate(x0))) -> TOP(up(x0)))
122.76/31.93	   (TOP(up(n__s(x0))) -> TOP(n__s_flat(down(x0))),TOP(up(n__s(x0))) -> TOP(n__s_flat(down(x0))))
122.76/31.93	   (TOP(up(cons(x0, x1))) -> TOP(cons_flat(down(x0), block(x1))),TOP(up(cons(x0, x1))) -> TOP(cons_flat(down(x0), block(x1))))
122.76/31.93	   (TOP(up(cons(x0, x1))) -> TOP(cons_flat(block(x0), down(x1))),TOP(up(cons(x0, x1))) -> TOP(cons_flat(block(x0), down(x1))))
122.76/31.93	   (TOP(up(n__inf(x0))) -> TOP(n__inf_flat(down(x0))),TOP(up(n__inf(x0))) -> TOP(n__inf_flat(down(x0))))
122.76/31.93	   (TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(down(x0), block(x1))),TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(down(x0), block(x1))))
122.81/31.93	   (TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(block(x0), down(x1))),TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(block(x0), down(x1))))
122.81/31.93	   (TOP(up(n__length(x0))) -> TOP(n__length_flat(down(x0))),TOP(up(n__length(x0))) -> TOP(n__length_flat(down(x0))))
122.81/31.93	
122.81/31.93	
122.81/31.93	----------------------------------------
122.81/31.93	
122.81/31.93	(14)
122.81/31.93	Obligation:
122.81/31.93	Q DP problem:
122.81/31.93	The TRS P consists of the following rules:
122.81/31.93	
122.81/31.93	   TOP(up(eq(n__0, n__0))) -> TOP(up(true))
122.81/31.93	   TOP(up(eq(n__s(x0), n__s(x1)))) -> TOP(up(eq(activate(x0), activate(x1))))
122.81/31.93	   TOP(up(eq(x0, x1))) -> TOP(up(false))
122.81/31.93	   TOP(up(inf(x0))) -> TOP(up(cons(x0, n__inf(s(x0)))))
122.81/31.93	   TOP(up(take(0, x0))) -> TOP(up(nil))
122.81/31.93	   TOP(up(take(s(x0), cons(x1, x2)))) -> TOP(up(cons(activate(x1), n__take(activate(x0), activate(x2)))))
122.81/31.93	   TOP(up(length(nil))) -> TOP(up(0))
122.81/31.93	   TOP(up(length(cons(x0, x1)))) -> TOP(up(s(n__length(activate(x1)))))
122.81/31.93	   TOP(up(0)) -> TOP(up(n__0))
122.81/31.93	   TOP(up(s(x0))) -> TOP(up(n__s(x0)))
122.81/31.93	   TOP(up(inf(x0))) -> TOP(up(n__inf(x0)))
122.81/31.93	   TOP(up(take(x0, x1))) -> TOP(up(n__take(x0, x1)))
122.81/31.93	   TOP(up(length(x0))) -> TOP(up(n__length(x0)))
122.81/31.93	   TOP(up(activate(n__0))) -> TOP(up(0))
122.81/31.93	   TOP(up(activate(n__s(x0)))) -> TOP(up(s(x0)))
122.81/31.93	   TOP(up(activate(n__inf(x0)))) -> TOP(up(inf(x0)))
122.81/31.93	   TOP(up(activate(n__take(x0, x1)))) -> TOP(up(take(x0, x1)))
122.81/31.93	   TOP(up(activate(n__length(x0)))) -> TOP(up(length(x0)))
122.81/31.93	   TOP(up(activate(x0))) -> TOP(up(x0))
122.81/31.93	   TOP(up(n__s(x0))) -> TOP(n__s_flat(down(x0)))
122.81/31.93	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(down(x0), block(x1)))
122.81/31.93	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(block(x0), down(x1)))
122.81/31.93	   TOP(up(n__inf(x0))) -> TOP(n__inf_flat(down(x0)))
122.81/31.93	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(down(x0), block(x1)))
122.81/31.93	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(block(x0), down(x1)))
122.81/31.93	   TOP(up(n__length(x0))) -> TOP(n__length_flat(down(x0)))
122.81/31.93	
122.81/31.93	The TRS R consists of the following rules:
122.81/31.93	
122.81/31.93	   down(eq(n__0, n__0)) -> up(true)
122.81/31.93	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.81/31.93	   down(eq(X, Y)) -> up(false)
122.81/31.93	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.81/31.93	   down(take(0, X)) -> up(nil)
122.81/31.93	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.81/31.93	   down(length(nil)) -> up(0)
122.81/31.93	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.81/31.93	   down(0) -> up(n__0)
122.81/31.93	   down(s(X)) -> up(n__s(X))
122.81/31.93	   down(inf(X)) -> up(n__inf(X))
122.81/31.93	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.81/31.93	   down(length(X)) -> up(n__length(X))
122.81/31.93	   down(activate(n__0)) -> up(0)
122.81/31.93	   down(activate(n__s(X))) -> up(s(X))
122.81/31.93	   down(activate(n__inf(X))) -> up(inf(X))
122.81/31.93	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.81/31.93	   down(activate(n__length(X))) -> up(length(X))
122.81/31.93	   down(activate(X)) -> up(X)
122.81/31.93	   top(up(x)) -> top(down(x))
122.81/31.93	   down(n__s(y2)) -> n__s_flat(down(y2))
122.81/31.93	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.81/31.93	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.81/31.93	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.81/31.93	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.81/31.93	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.81/31.93	   down(n__length(y14)) -> n__length_flat(down(y14))
122.81/31.93	   eq_flat(up(x_1), block(x_2)) -> up(eq(x_1, x_2))
122.81/31.93	   eq_flat(block(x_1), up(x_2)) -> up(eq(x_1, x_2))
122.81/31.93	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.81/31.93	   activate_flat(up(x_1)) -> up(activate(x_1))
122.81/31.93	   inf_flat(up(x_1)) -> up(inf(x_1))
122.81/31.93	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.81/31.93	   s_flat(up(x_1)) -> up(s(x_1))
122.81/31.93	   take_flat(up(x_1), block(x_2)) -> up(take(x_1, x_2))
122.81/31.93	   take_flat(block(x_1), up(x_2)) -> up(take(x_1, x_2))
122.81/31.93	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   length_flat(up(x_1)) -> up(length(x_1))
122.81/31.93	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.81/31.93	
122.81/31.93	Q is empty.
122.81/31.93	We have to consider all minimal (P,Q,R)-chains.
122.81/31.93	----------------------------------------
122.81/31.93	
122.81/31.93	(15) DependencyGraphProof (EQUIVALENT)
122.81/31.93	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.
122.81/31.93	----------------------------------------
122.81/31.93	
122.81/31.93	(16)
122.81/31.93	Obligation:
122.81/31.93	Q DP problem:
122.81/31.93	The TRS P consists of the following rules:
122.81/31.93	
122.81/31.93	   TOP(up(inf(x0))) -> TOP(up(cons(x0, n__inf(s(x0)))))
122.81/31.93	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(down(x0), block(x1)))
122.81/31.93	   TOP(up(take(s(x0), cons(x1, x2)))) -> TOP(up(cons(activate(x1), n__take(activate(x0), activate(x2)))))
122.81/31.93	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(block(x0), down(x1)))
122.81/31.93	   TOP(up(length(cons(x0, x1)))) -> TOP(up(s(n__length(activate(x1)))))
122.81/31.93	   TOP(up(s(x0))) -> TOP(up(n__s(x0)))
122.81/31.93	   TOP(up(n__s(x0))) -> TOP(n__s_flat(down(x0)))
122.81/31.93	   TOP(up(inf(x0))) -> TOP(up(n__inf(x0)))
122.81/31.93	   TOP(up(n__inf(x0))) -> TOP(n__inf_flat(down(x0)))
122.81/31.93	   TOP(up(take(x0, x1))) -> TOP(up(n__take(x0, x1)))
122.81/31.93	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(down(x0), block(x1)))
122.81/31.93	   TOP(up(length(x0))) -> TOP(up(n__length(x0)))
122.81/31.93	   TOP(up(n__length(x0))) -> TOP(n__length_flat(down(x0)))
122.81/31.93	   TOP(up(activate(n__s(x0)))) -> TOP(up(s(x0)))
122.81/31.93	   TOP(up(activate(n__inf(x0)))) -> TOP(up(inf(x0)))
122.81/31.93	   TOP(up(activate(n__take(x0, x1)))) -> TOP(up(take(x0, x1)))
122.81/31.93	   TOP(up(activate(n__length(x0)))) -> TOP(up(length(x0)))
122.81/31.93	   TOP(up(activate(x0))) -> TOP(up(x0))
122.81/31.93	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(block(x0), down(x1)))
122.81/31.93	
122.81/31.93	The TRS R consists of the following rules:
122.81/31.93	
122.81/31.93	   down(eq(n__0, n__0)) -> up(true)
122.81/31.93	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.81/31.93	   down(eq(X, Y)) -> up(false)
122.81/31.93	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.81/31.93	   down(take(0, X)) -> up(nil)
122.81/31.93	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.81/31.93	   down(length(nil)) -> up(0)
122.81/31.93	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.81/31.93	   down(0) -> up(n__0)
122.81/31.93	   down(s(X)) -> up(n__s(X))
122.81/31.93	   down(inf(X)) -> up(n__inf(X))
122.81/31.93	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.81/31.93	   down(length(X)) -> up(n__length(X))
122.81/31.93	   down(activate(n__0)) -> up(0)
122.81/31.93	   down(activate(n__s(X))) -> up(s(X))
122.81/31.93	   down(activate(n__inf(X))) -> up(inf(X))
122.81/31.93	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.81/31.93	   down(activate(n__length(X))) -> up(length(X))
122.81/31.93	   down(activate(X)) -> up(X)
122.81/31.93	   top(up(x)) -> top(down(x))
122.81/31.93	   down(n__s(y2)) -> n__s_flat(down(y2))
122.81/31.93	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.81/31.93	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.81/31.93	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.81/31.93	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.81/31.93	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.81/31.93	   down(n__length(y14)) -> n__length_flat(down(y14))
122.81/31.93	   eq_flat(up(x_1), block(x_2)) -> up(eq(x_1, x_2))
122.81/31.93	   eq_flat(block(x_1), up(x_2)) -> up(eq(x_1, x_2))
122.81/31.93	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.81/31.93	   activate_flat(up(x_1)) -> up(activate(x_1))
122.81/31.93	   inf_flat(up(x_1)) -> up(inf(x_1))
122.81/31.93	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.81/31.93	   s_flat(up(x_1)) -> up(s(x_1))
122.81/31.93	   take_flat(up(x_1), block(x_2)) -> up(take(x_1, x_2))
122.81/31.93	   take_flat(block(x_1), up(x_2)) -> up(take(x_1, x_2))
122.81/31.93	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   length_flat(up(x_1)) -> up(length(x_1))
122.81/31.93	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.81/31.93	
122.81/31.93	Q is empty.
122.81/31.93	We have to consider all minimal (P,Q,R)-chains.
122.81/31.93	----------------------------------------
122.81/31.93	
122.81/31.93	(17) QDPOrderProof (EQUIVALENT)
122.81/31.93	We use the reduction pair processor [LPAR04,JAR06].
122.81/31.93	
122.81/31.93	
122.81/31.93	The following pairs can be oriented strictly and are deleted.
122.81/31.93	
122.81/31.93	   TOP(up(inf(x0))) -> TOP(up(cons(x0, n__inf(s(x0)))))
122.81/31.93	   TOP(up(take(s(x0), cons(x1, x2)))) -> TOP(up(cons(activate(x1), n__take(activate(x0), activate(x2)))))
122.81/31.93	   TOP(up(length(cons(x0, x1)))) -> TOP(up(s(n__length(activate(x1)))))
122.81/31.93	   TOP(up(inf(x0))) -> TOP(up(n__inf(x0)))
122.81/31.93	   TOP(up(take(x0, x1))) -> TOP(up(n__take(x0, x1)))
122.81/31.93	   TOP(up(length(x0))) -> TOP(up(n__length(x0)))
122.81/31.93	   TOP(up(activate(n__s(x0)))) -> TOP(up(s(x0)))
122.81/31.93	   TOP(up(activate(x0))) -> TOP(up(x0))
122.81/31.93	The remaining pairs can at least be oriented weakly.
122.81/31.93	Used ordering:  Polynomial interpretation [POLO]:
122.81/31.93	
122.81/31.93	   POL(0) = 0
122.81/31.93	   POL(TOP(x_1)) = x_1
122.81/31.93	   POL(activate(x_1)) = 1 + x_1
122.81/31.93	   POL(block(x_1)) = 0
122.81/31.93	   POL(cons(x_1, x_2)) = 0
122.81/31.93	   POL(cons_flat(x_1, x_2)) = 0
122.81/31.93	   POL(down(x_1)) = 0
122.81/31.93	   POL(eq(x_1, x_2)) = 0
122.81/31.93	   POL(false) = 1
122.81/31.93	   POL(inf(x_1)) = 1
122.81/31.93	   POL(length(x_1)) = 1
122.81/31.93	   POL(n__0) = 0
122.81/31.93	   POL(n__inf(x_1)) = 0
122.81/31.93	   POL(n__inf_flat(x_1)) = 0
122.81/31.93	   POL(n__length(x_1)) = 0
122.81/31.93	   POL(n__length_flat(x_1)) = 0
122.81/31.93	   POL(n__s(x_1)) = 0
122.81/31.93	   POL(n__s_flat(x_1)) = 0
122.81/31.93	   POL(n__take(x_1, x_2)) = 0
122.81/31.93	   POL(n__take_flat(x_1, x_2)) = 0
122.81/31.93	   POL(nil) = 0
122.81/31.93	   POL(s(x_1)) = 0
122.81/31.93	   POL(take(x_1, x_2)) = 1
122.81/31.93	   POL(true) = 1
122.81/31.93	   POL(up(x_1)) = x_1
122.81/31.93	
122.81/31.93	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
122.81/31.93	
122.81/31.93	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.81/31.93	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.81/31.93	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.81/31.93	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	
122.81/31.93	
122.81/31.93	----------------------------------------
122.81/31.93	
122.81/31.93	(18)
122.81/31.93	Obligation:
122.81/31.93	Q DP problem:
122.81/31.93	The TRS P consists of the following rules:
122.81/31.93	
122.81/31.93	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(down(x0), block(x1)))
122.81/31.93	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(block(x0), down(x1)))
122.81/31.93	   TOP(up(s(x0))) -> TOP(up(n__s(x0)))
122.81/31.93	   TOP(up(n__s(x0))) -> TOP(n__s_flat(down(x0)))
122.81/31.93	   TOP(up(n__inf(x0))) -> TOP(n__inf_flat(down(x0)))
122.81/31.93	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(down(x0), block(x1)))
122.81/31.93	   TOP(up(n__length(x0))) -> TOP(n__length_flat(down(x0)))
122.81/31.93	   TOP(up(activate(n__inf(x0)))) -> TOP(up(inf(x0)))
122.81/31.93	   TOP(up(activate(n__take(x0, x1)))) -> TOP(up(take(x0, x1)))
122.81/31.93	   TOP(up(activate(n__length(x0)))) -> TOP(up(length(x0)))
122.81/31.93	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(block(x0), down(x1)))
122.81/31.93	
122.81/31.93	The TRS R consists of the following rules:
122.81/31.93	
122.81/31.93	   down(eq(n__0, n__0)) -> up(true)
122.81/31.93	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.81/31.93	   down(eq(X, Y)) -> up(false)
122.81/31.93	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.81/31.93	   down(take(0, X)) -> up(nil)
122.81/31.93	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.81/31.93	   down(length(nil)) -> up(0)
122.81/31.93	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.81/31.93	   down(0) -> up(n__0)
122.81/31.93	   down(s(X)) -> up(n__s(X))
122.81/31.93	   down(inf(X)) -> up(n__inf(X))
122.81/31.93	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.81/31.93	   down(length(X)) -> up(n__length(X))
122.81/31.93	   down(activate(n__0)) -> up(0)
122.81/31.93	   down(activate(n__s(X))) -> up(s(X))
122.81/31.93	   down(activate(n__inf(X))) -> up(inf(X))
122.81/31.93	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.81/31.93	   down(activate(n__length(X))) -> up(length(X))
122.81/31.93	   down(activate(X)) -> up(X)
122.81/31.93	   top(up(x)) -> top(down(x))
122.81/31.93	   down(n__s(y2)) -> n__s_flat(down(y2))
122.81/31.93	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.81/31.93	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.81/31.93	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.81/31.93	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.81/31.93	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.81/31.93	   down(n__length(y14)) -> n__length_flat(down(y14))
122.81/31.93	   eq_flat(up(x_1), block(x_2)) -> up(eq(x_1, x_2))
122.81/31.93	   eq_flat(block(x_1), up(x_2)) -> up(eq(x_1, x_2))
122.81/31.93	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.81/31.93	   activate_flat(up(x_1)) -> up(activate(x_1))
122.81/31.93	   inf_flat(up(x_1)) -> up(inf(x_1))
122.81/31.93	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.81/31.93	   s_flat(up(x_1)) -> up(s(x_1))
122.81/31.93	   take_flat(up(x_1), block(x_2)) -> up(take(x_1, x_2))
122.81/31.93	   take_flat(block(x_1), up(x_2)) -> up(take(x_1, x_2))
122.81/31.93	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   length_flat(up(x_1)) -> up(length(x_1))
122.81/31.93	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.81/31.93	
122.81/31.93	Q is empty.
122.81/31.93	We have to consider all minimal (P,Q,R)-chains.
122.81/31.93	----------------------------------------
122.81/31.93	
122.81/31.93	(19) DependencyGraphProof (EQUIVALENT)
122.81/31.93	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
122.81/31.93	----------------------------------------
122.81/31.93	
122.81/31.93	(20)
122.81/31.93	Obligation:
122.81/31.93	Q DP problem:
122.81/31.93	The TRS P consists of the following rules:
122.81/31.93	
122.81/31.93	   TOP(up(s(x0))) -> TOP(up(n__s(x0)))
122.81/31.93	   TOP(up(n__s(x0))) -> TOP(n__s_flat(down(x0)))
122.81/31.93	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(down(x0), block(x1)))
122.81/31.93	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(block(x0), down(x1)))
122.81/31.93	   TOP(up(n__inf(x0))) -> TOP(n__inf_flat(down(x0)))
122.81/31.93	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(down(x0), block(x1)))
122.81/31.93	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(block(x0), down(x1)))
122.81/31.93	   TOP(up(n__length(x0))) -> TOP(n__length_flat(down(x0)))
122.81/31.93	
122.81/31.93	The TRS R consists of the following rules:
122.81/31.93	
122.81/31.93	   down(eq(n__0, n__0)) -> up(true)
122.81/31.93	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.81/31.93	   down(eq(X, Y)) -> up(false)
122.81/31.93	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.81/31.93	   down(take(0, X)) -> up(nil)
122.81/31.93	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.81/31.93	   down(length(nil)) -> up(0)
122.81/31.93	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.81/31.93	   down(0) -> up(n__0)
122.81/31.93	   down(s(X)) -> up(n__s(X))
122.81/31.93	   down(inf(X)) -> up(n__inf(X))
122.81/31.93	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.81/31.93	   down(length(X)) -> up(n__length(X))
122.81/31.93	   down(activate(n__0)) -> up(0)
122.81/31.93	   down(activate(n__s(X))) -> up(s(X))
122.81/31.93	   down(activate(n__inf(X))) -> up(inf(X))
122.81/31.93	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.81/31.93	   down(activate(n__length(X))) -> up(length(X))
122.81/31.93	   down(activate(X)) -> up(X)
122.81/31.93	   top(up(x)) -> top(down(x))
122.81/31.93	   down(n__s(y2)) -> n__s_flat(down(y2))
122.81/31.93	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.81/31.93	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.81/31.93	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.81/31.93	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.81/31.93	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.81/31.93	   down(n__length(y14)) -> n__length_flat(down(y14))
122.81/31.93	   eq_flat(up(x_1), block(x_2)) -> up(eq(x_1, x_2))
122.81/31.93	   eq_flat(block(x_1), up(x_2)) -> up(eq(x_1, x_2))
122.81/31.93	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.81/31.93	   activate_flat(up(x_1)) -> up(activate(x_1))
122.81/31.93	   inf_flat(up(x_1)) -> up(inf(x_1))
122.81/31.93	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.81/31.93	   s_flat(up(x_1)) -> up(s(x_1))
122.81/31.93	   take_flat(up(x_1), block(x_2)) -> up(take(x_1, x_2))
122.81/31.93	   take_flat(block(x_1), up(x_2)) -> up(take(x_1, x_2))
122.81/31.93	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   length_flat(up(x_1)) -> up(length(x_1))
122.81/31.93	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.81/31.93	
122.81/31.93	Q is empty.
122.81/31.93	We have to consider all minimal (P,Q,R)-chains.
122.81/31.93	----------------------------------------
122.81/31.93	
122.81/31.93	(21) QDPOrderProof (EQUIVALENT)
122.81/31.93	We use the reduction pair processor [LPAR04,JAR06].
122.81/31.93	
122.81/31.93	
122.81/31.93	The following pairs can be oriented strictly and are deleted.
122.81/31.93	
122.81/31.93	   TOP(up(s(x0))) -> TOP(up(n__s(x0)))
122.81/31.93	The remaining pairs can at least be oriented weakly.
122.81/31.93	Used ordering:  Polynomial interpretation [POLO]:
122.81/31.93	
122.81/31.93	   POL(0) = 0
122.81/31.93	   POL(TOP(x_1)) = x_1
122.81/31.93	   POL(activate(x_1)) = 0
122.81/31.93	   POL(block(x_1)) = 0
122.81/31.93	   POL(cons(x_1, x_2)) = 0
122.81/31.93	   POL(cons_flat(x_1, x_2)) = 0
122.81/31.93	   POL(down(x_1)) = 0
122.81/31.93	   POL(eq(x_1, x_2)) = 0
122.81/31.93	   POL(false) = 1
122.81/31.93	   POL(inf(x_1)) = x_1
122.81/31.93	   POL(length(x_1)) = 0
122.81/31.93	   POL(n__0) = 0
122.81/31.93	   POL(n__inf(x_1)) = 0
122.81/31.93	   POL(n__inf_flat(x_1)) = 0
122.81/31.93	   POL(n__length(x_1)) = 0
122.81/31.93	   POL(n__length_flat(x_1)) = 0
122.81/31.93	   POL(n__s(x_1)) = 0
122.81/31.93	   POL(n__s_flat(x_1)) = 0
122.81/31.93	   POL(n__take(x_1, x_2)) = 0
122.81/31.93	   POL(n__take_flat(x_1, x_2)) = 0
122.81/31.93	   POL(nil) = 0
122.81/31.93	   POL(s(x_1)) = 1
122.81/31.93	   POL(take(x_1, x_2)) = 0
122.81/31.93	   POL(true) = 1
122.81/31.93	   POL(up(x_1)) = x_1
122.81/31.93	
122.81/31.93	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
122.81/31.93	
122.81/31.93	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.81/31.93	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.81/31.93	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.81/31.93	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.81/31.93	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.81/31.93	
122.81/31.93	
122.81/31.93	----------------------------------------
122.81/31.93	
122.81/31.93	(22)
122.81/31.93	Obligation:
122.81/31.93	Q DP problem:
122.81/31.93	The TRS P consists of the following rules:
122.81/31.93	
122.81/31.93	   TOP(up(n__s(x0))) -> TOP(n__s_flat(down(x0)))
122.81/31.93	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(down(x0), block(x1)))
122.81/31.93	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(block(x0), down(x1)))
122.81/31.93	   TOP(up(n__inf(x0))) -> TOP(n__inf_flat(down(x0)))
122.81/31.93	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(down(x0), block(x1)))
122.81/31.93	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(block(x0), down(x1)))
122.81/31.93	   TOP(up(n__length(x0))) -> TOP(n__length_flat(down(x0)))
122.81/31.93	
122.81/31.93	The TRS R consists of the following rules:
122.81/31.93	
122.81/31.93	   down(eq(n__0, n__0)) -> up(true)
122.81/31.93	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.81/31.93	   down(eq(X, Y)) -> up(false)
122.81/31.93	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.81/31.93	   down(take(0, X)) -> up(nil)
122.81/31.93	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.81/31.93	   down(length(nil)) -> up(0)
122.81/31.93	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.81/31.93	   down(0) -> up(n__0)
122.81/31.94	   down(s(X)) -> up(n__s(X))
122.81/31.94	   down(inf(X)) -> up(n__inf(X))
122.81/31.94	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.81/31.94	   down(length(X)) -> up(n__length(X))
122.81/31.94	   down(activate(n__0)) -> up(0)
122.81/31.94	   down(activate(n__s(X))) -> up(s(X))
122.81/31.94	   down(activate(n__inf(X))) -> up(inf(X))
122.81/31.94	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.81/31.94	   down(activate(n__length(X))) -> up(length(X))
122.81/31.94	   down(activate(X)) -> up(X)
122.81/31.94	   top(up(x)) -> top(down(x))
122.81/31.94	   down(n__s(y2)) -> n__s_flat(down(y2))
122.81/31.94	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.81/31.94	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.81/31.94	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.81/31.94	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.81/31.94	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.81/31.94	   down(n__length(y14)) -> n__length_flat(down(y14))
122.81/31.94	   eq_flat(up(x_1), block(x_2)) -> up(eq(x_1, x_2))
122.81/31.94	   eq_flat(block(x_1), up(x_2)) -> up(eq(x_1, x_2))
122.81/31.94	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.81/31.94	   activate_flat(up(x_1)) -> up(activate(x_1))
122.81/31.94	   inf_flat(up(x_1)) -> up(inf(x_1))
122.81/31.94	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.81/31.94	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.81/31.94	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.81/31.94	   s_flat(up(x_1)) -> up(s(x_1))
122.81/31.94	   take_flat(up(x_1), block(x_2)) -> up(take(x_1, x_2))
122.81/31.94	   take_flat(block(x_1), up(x_2)) -> up(take(x_1, x_2))
122.81/31.94	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.81/31.94	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.81/31.94	   length_flat(up(x_1)) -> up(length(x_1))
122.81/31.94	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.81/31.94	
122.81/31.94	Q is empty.
122.81/31.94	We have to consider all minimal (P,Q,R)-chains.
122.81/31.94	----------------------------------------
122.81/31.94	
122.81/31.94	(23) QDPOrderProof (EQUIVALENT)
122.81/31.94	We use the reduction pair processor [LPAR04,JAR06].
122.81/31.94	
122.81/31.94	
122.81/31.94	The following pairs can be oriented strictly and are deleted.
122.81/31.94	
122.81/31.94	   TOP(up(n__s(x0))) -> TOP(n__s_flat(down(x0)))
122.81/31.94	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(down(x0), block(x1)))
122.81/31.94	   TOP(up(cons(x0, x1))) -> TOP(cons_flat(block(x0), down(x1)))
122.81/31.94	   TOP(up(n__inf(x0))) -> TOP(n__inf_flat(down(x0)))
122.81/31.94	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(down(x0), block(x1)))
122.81/31.94	   TOP(up(n__take(x0, x1))) -> TOP(n__take_flat(block(x0), down(x1)))
122.81/31.94	   TOP(up(n__length(x0))) -> TOP(n__length_flat(down(x0)))
122.81/31.94	The remaining pairs can at least be oriented weakly.
122.81/31.94	Used ordering:  Matrix interpretation [MATRO]:
122.81/31.94	
122.81/31.94	Non-tuple symbols: 
122.81/31.94	<<<
122.81/31.94	 M( n__length_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 1]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( true ) =  	[[0], [0]]
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( n__inf_1(x_1) ) =  	[[2], [0]] 	+ 	[[1, 2], [0, 1]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( n__take_2(x_1, x_2) ) =  	[[2], [1]] 	+ 	[[1, 1], [0, 1]] 	* 	x_1 	+ 	[[2, 0], [0, 1]] 	* 	x_2
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( n__take_flat_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[0, 0], [2, 1]] 	* 	x_1 	+ 	[[0, 0], [2, 1]] 	* 	x_2
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( n__length_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 1]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( n__inf_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 1]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( block_1(x_1) ) =  	[[1], [0]] 	+ 	[[0, 0], [0, 2]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( n__0 ) =  	[[1], [0]]
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( inf_1(x_1) ) =  	[[0], [3]] 	+ 	[[0, 0], [1, 3]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( cons_flat_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[0, 0], [2, 2]] 	* 	x_1 	+ 	[[0, 0], [0, 1]] 	* 	x_2
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( activate_1(x_1) ) =  	[[0], [2]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( n__s_1(x_1) ) =  	[[1], [0]] 	+ 	[[1, 2], [0, 1]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( false ) =  	[[0], [0]]
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( nil ) =  	[[0], [2]]
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( s_1(x_1) ) =  	[[3], [1]] 	+ 	[[1, 0], [1, 1]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( n__s_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 1]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( eq_2(x_1, x_2) ) =  	[[0], [1]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 0], [1, 0]] 	* 	x_2
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( length_1(x_1) ) =  	[[0], [1]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( take_2(x_1, x_2) ) =  	[[0], [3]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1 	+ 	[[0, 0], [1, 1]] 	* 	x_2
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( down_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 2]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( 0 ) =  	[[1], [2]]
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( up_1(x_1) ) =  	[[0], [2]] 	+ 	[[0, 0], [0, 2]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	<<<
122.81/31.94	 M( cons_2(x_1, x_2) ) =  	[[3], [1]] 	+ 	[[2, 0], [0, 2]] 	* 	x_1 	+ 	[[2, 0], [0, 1]] 	* 	x_2
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	Tuple symbols: 
122.81/31.94	<<<
122.81/31.94	 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[0, 1]] 	* 	x_1
122.81/31.94	>>>
122.81/31.94	
122.81/31.94	
122.81/31.94	
122.81/31.94	Matrix type: 
122.81/31.94	
122.81/31.94	We used a basic matrix type which is not further parametrizeable.
122.81/31.94	
122.81/31.94	
122.81/31.94	
122.81/31.94	
122.81/31.94	
122.81/31.94	As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
122.81/31.94	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
122.81/31.94	
122.81/31.94	   down(eq(n__0, n__0)) -> up(true)
122.81/31.94	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.81/31.94	   down(eq(X, Y)) -> up(false)
122.81/31.94	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.81/31.94	   down(take(0, X)) -> up(nil)
122.81/31.94	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.81/31.94	   down(length(nil)) -> up(0)
122.81/31.94	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.81/31.94	   down(0) -> up(n__0)
122.81/31.94	   down(s(X)) -> up(n__s(X))
122.81/31.94	   down(inf(X)) -> up(n__inf(X))
122.81/31.94	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.81/31.94	   down(length(X)) -> up(n__length(X))
122.81/31.94	   down(activate(n__0)) -> up(0)
122.81/31.94	   down(activate(n__s(X))) -> up(s(X))
122.81/31.94	   down(activate(n__inf(X))) -> up(inf(X))
122.81/31.94	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.81/31.94	   down(activate(n__length(X))) -> up(length(X))
122.81/31.94	   down(activate(X)) -> up(X)
122.81/31.94	   down(n__s(y2)) -> n__s_flat(down(y2))
122.81/31.94	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.81/31.94	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.81/31.94	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.81/31.94	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.81/31.94	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.81/31.94	   down(n__length(y14)) -> n__length_flat(down(y14))
122.81/31.94	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.81/31.94	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.81/31.94	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.81/31.94	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.81/31.94	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.81/31.94	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.81/31.94	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.81/31.94	
122.81/31.94	
122.81/31.94	----------------------------------------
122.81/31.94	
122.81/31.94	(24)
122.81/31.94	Obligation:
122.81/31.94	Q DP problem:
122.81/31.94	P is empty.
122.81/31.94	The TRS R consists of the following rules:
122.81/31.94	
122.81/31.94	   down(eq(n__0, n__0)) -> up(true)
122.81/31.94	   down(eq(n__s(X), n__s(Y))) -> up(eq(activate(X), activate(Y)))
122.81/31.94	   down(eq(X, Y)) -> up(false)
122.81/31.94	   down(inf(X)) -> up(cons(X, n__inf(s(X))))
122.81/31.94	   down(take(0, X)) -> up(nil)
122.81/31.94	   down(take(s(X), cons(Y, L))) -> up(cons(activate(Y), n__take(activate(X), activate(L))))
122.81/31.94	   down(length(nil)) -> up(0)
122.81/31.94	   down(length(cons(X, L))) -> up(s(n__length(activate(L))))
122.81/31.94	   down(0) -> up(n__0)
122.81/31.94	   down(s(X)) -> up(n__s(X))
122.81/31.94	   down(inf(X)) -> up(n__inf(X))
122.81/31.94	   down(take(X1, X2)) -> up(n__take(X1, X2))
122.81/31.94	   down(length(X)) -> up(n__length(X))
122.81/31.94	   down(activate(n__0)) -> up(0)
122.81/31.94	   down(activate(n__s(X))) -> up(s(X))
122.81/31.94	   down(activate(n__inf(X))) -> up(inf(X))
122.81/31.94	   down(activate(n__take(X1, X2))) -> up(take(X1, X2))
122.81/31.94	   down(activate(n__length(X))) -> up(length(X))
122.81/31.94	   down(activate(X)) -> up(X)
122.81/31.94	   top(up(x)) -> top(down(x))
122.81/31.94	   down(n__s(y2)) -> n__s_flat(down(y2))
122.81/31.94	   down(cons(y5, y6)) -> cons_flat(down(y5), block(y6))
122.81/31.94	   down(cons(y5, y6)) -> cons_flat(block(y5), down(y6))
122.81/31.94	   down(n__inf(y7)) -> n__inf_flat(down(y7))
122.81/31.94	   down(n__take(y11, y12)) -> n__take_flat(down(y11), block(y12))
122.81/31.94	   down(n__take(y11, y12)) -> n__take_flat(block(y11), down(y12))
122.81/31.94	   down(n__length(y14)) -> n__length_flat(down(y14))
122.81/31.94	   eq_flat(up(x_1), block(x_2)) -> up(eq(x_1, x_2))
122.81/31.94	   eq_flat(block(x_1), up(x_2)) -> up(eq(x_1, x_2))
122.81/31.94	   n__s_flat(up(x_1)) -> up(n__s(x_1))
122.81/31.94	   activate_flat(up(x_1)) -> up(activate(x_1))
122.81/31.94	   inf_flat(up(x_1)) -> up(inf(x_1))
122.81/31.94	   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
122.81/31.94	   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
122.81/31.94	   n__inf_flat(up(x_1)) -> up(n__inf(x_1))
122.81/31.94	   s_flat(up(x_1)) -> up(s(x_1))
122.81/31.94	   take_flat(up(x_1), block(x_2)) -> up(take(x_1, x_2))
122.81/31.94	   take_flat(block(x_1), up(x_2)) -> up(take(x_1, x_2))
122.81/31.94	   n__take_flat(up(x_1), block(x_2)) -> up(n__take(x_1, x_2))
122.81/31.94	   n__take_flat(block(x_1), up(x_2)) -> up(n__take(x_1, x_2))
122.81/31.94	   length_flat(up(x_1)) -> up(length(x_1))
122.81/31.94	   n__length_flat(up(x_1)) -> up(n__length(x_1))
122.81/31.94	
122.81/31.94	Q is empty.
122.81/31.94	We have to consider all minimal (P,Q,R)-chains.
122.81/31.94	----------------------------------------
122.81/31.94	
122.81/31.94	(25) PisEmptyProof (EQUIVALENT)
122.81/31.94	The TRS P is empty. Hence, there is no (P,Q,R) chain.
122.81/31.94	----------------------------------------
122.81/31.94	
122.81/31.94	(26)
122.81/31.94	YES
122.81/32.00	EOF