3.60/2.85	YES
3.60/2.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.60/2.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.60/2.86	
3.60/2.86	
3.60/2.86	Termination w.r.t. Q of the given QTRS could be proven:
3.60/2.86	
3.60/2.86	(0) QTRS
3.60/2.86	(1) QTRSToCSRProof [SOUND, 0 ms]
3.60/2.86	(2) CSR
3.60/2.86	(3) CSRInnermostProof [EQUIVALENT, 0 ms]
3.60/2.86	(4) CSR
3.60/2.86	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.60/2.86	(6) QCSDP
3.60/2.86	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.60/2.86	(8) AND
3.60/2.86	    (9) QCSDP
3.60/2.86	        (10) QCSDPSubtermProof [EQUIVALENT, 4 ms]
3.60/2.86	        (11) QCSDP
3.60/2.86	        (12) PIsEmptyProof [EQUIVALENT, 0 ms]
3.60/2.86	        (13) YES
3.60/2.86	    (14) QCSDP
3.60/2.86	        (15) QCSDPSubtermProof [EQUIVALENT, 9 ms]
3.60/2.86	        (16) QCSDP
3.60/2.86	        (17) PIsEmptyProof [EQUIVALENT, 0 ms]
3.60/2.86	        (18) YES
3.60/2.86	    (19) QCSDP
3.60/2.86	        (20) QCSDPSubtermProof [EQUIVALENT, 3 ms]
3.60/2.86	        (21) QCSDP
3.60/2.86	        (22) PIsEmptyProof [EQUIVALENT, 0 ms]
3.60/2.86	        (23) YES
3.60/2.86	
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(0)
3.60/2.86	Obligation:
3.60/2.86	Q restricted rewrite system:
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   active(f(X)) -> mark(cons(X, f(g(X))))
3.60/2.86	   active(g(0)) -> mark(s(0))
3.60/2.86	   active(g(s(X))) -> mark(s(s(g(X))))
3.60/2.86	   active(sel(0, cons(X, Y))) -> mark(X)
3.60/2.86	   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
3.60/2.86	   active(f(X)) -> f(active(X))
3.60/2.86	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.60/2.86	   active(g(X)) -> g(active(X))
3.60/2.86	   active(s(X)) -> s(active(X))
3.60/2.86	   active(sel(X1, X2)) -> sel(active(X1), X2)
3.60/2.86	   active(sel(X1, X2)) -> sel(X1, active(X2))
3.60/2.86	   f(mark(X)) -> mark(f(X))
3.60/2.86	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.60/2.86	   g(mark(X)) -> mark(g(X))
3.60/2.86	   s(mark(X)) -> mark(s(X))
3.60/2.86	   sel(mark(X1), X2) -> mark(sel(X1, X2))
3.60/2.86	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
3.60/2.86	   proper(f(X)) -> f(proper(X))
3.60/2.86	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.60/2.86	   proper(g(X)) -> g(proper(X))
3.60/2.86	   proper(0) -> ok(0)
3.60/2.86	   proper(s(X)) -> s(proper(X))
3.60/2.86	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
3.60/2.86	   f(ok(X)) -> ok(f(X))
3.60/2.86	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.60/2.86	   g(ok(X)) -> ok(g(X))
3.60/2.86	   s(ok(X)) -> ok(s(X))
3.60/2.86	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
3.60/2.86	   top(mark(X)) -> top(proper(X))
3.60/2.86	   top(ok(X)) -> top(active(X))
3.60/2.86	
3.60/2.86	The set Q consists of the following terms:
3.60/2.86	
3.60/2.86	   active(f(x0))
3.60/2.86	   active(cons(x0, x1))
3.60/2.86	   active(g(x0))
3.60/2.86	   active(s(x0))
3.60/2.86	   active(sel(x0, x1))
3.60/2.86	   f(mark(x0))
3.60/2.86	   cons(mark(x0), x1)
3.60/2.86	   g(mark(x0))
3.60/2.86	   s(mark(x0))
3.60/2.86	   sel(mark(x0), x1)
3.60/2.86	   sel(x0, mark(x1))
3.60/2.86	   proper(f(x0))
3.60/2.86	   proper(cons(x0, x1))
3.60/2.86	   proper(g(x0))
3.60/2.86	   proper(0)
3.60/2.86	   proper(s(x0))
3.60/2.86	   proper(sel(x0, x1))
3.60/2.86	   f(ok(x0))
3.60/2.86	   cons(ok(x0), ok(x1))
3.60/2.86	   g(ok(x0))
3.60/2.86	   s(ok(x0))
3.60/2.86	   sel(ok(x0), ok(x1))
3.60/2.86	   top(mark(x0))
3.60/2.86	   top(ok(x0))
3.60/2.86	
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(1) QTRSToCSRProof (SOUND)
3.60/2.86	The following Q TRS is given: Q restricted rewrite system:
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   active(f(X)) -> mark(cons(X, f(g(X))))
3.60/2.86	   active(g(0)) -> mark(s(0))
3.60/2.86	   active(g(s(X))) -> mark(s(s(g(X))))
3.60/2.86	   active(sel(0, cons(X, Y))) -> mark(X)
3.60/2.86	   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
3.60/2.86	   active(f(X)) -> f(active(X))
3.60/2.86	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.60/2.86	   active(g(X)) -> g(active(X))
3.60/2.86	   active(s(X)) -> s(active(X))
3.60/2.86	   active(sel(X1, X2)) -> sel(active(X1), X2)
3.60/2.86	   active(sel(X1, X2)) -> sel(X1, active(X2))
3.60/2.86	   f(mark(X)) -> mark(f(X))
3.60/2.86	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.60/2.86	   g(mark(X)) -> mark(g(X))
3.60/2.86	   s(mark(X)) -> mark(s(X))
3.60/2.86	   sel(mark(X1), X2) -> mark(sel(X1, X2))
3.60/2.86	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
3.60/2.86	   proper(f(X)) -> f(proper(X))
3.60/2.86	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.60/2.86	   proper(g(X)) -> g(proper(X))
3.60/2.86	   proper(0) -> ok(0)
3.60/2.86	   proper(s(X)) -> s(proper(X))
3.60/2.86	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
3.60/2.86	   f(ok(X)) -> ok(f(X))
3.60/2.86	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.60/2.86	   g(ok(X)) -> ok(g(X))
3.60/2.86	   s(ok(X)) -> ok(s(X))
3.60/2.86	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
3.60/2.86	   top(mark(X)) -> top(proper(X))
3.60/2.86	   top(ok(X)) -> top(active(X))
3.60/2.86	
3.60/2.86	The set Q consists of the following terms:
3.60/2.86	
3.60/2.86	   active(f(x0))
3.60/2.86	   active(cons(x0, x1))
3.60/2.86	   active(g(x0))
3.60/2.86	   active(s(x0))
3.60/2.86	   active(sel(x0, x1))
3.60/2.86	   f(mark(x0))
3.60/2.86	   cons(mark(x0), x1)
3.60/2.86	   g(mark(x0))
3.60/2.86	   s(mark(x0))
3.60/2.86	   sel(mark(x0), x1)
3.60/2.86	   sel(x0, mark(x1))
3.60/2.86	   proper(f(x0))
3.60/2.86	   proper(cons(x0, x1))
3.60/2.86	   proper(g(x0))
3.60/2.86	   proper(0)
3.60/2.86	   proper(s(x0))
3.60/2.86	   proper(sel(x0, x1))
3.60/2.86	   f(ok(x0))
3.60/2.86	   cons(ok(x0), ok(x1))
3.60/2.86	   g(ok(x0))
3.60/2.86	   s(ok(x0))
3.60/2.86	   sel(ok(x0), ok(x1))
3.60/2.86	   top(mark(x0))
3.60/2.86	   top(ok(x0))
3.60/2.86	
3.60/2.86	Special symbols used for the transformation (see [GM04]):
3.60/2.86	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.60/2.86	The replacement map contains the following entries:
3.60/2.86	
3.60/2.86	f: {1}
3.60/2.86	cons: {1}
3.60/2.86	g: {1}
3.60/2.86	0: empty set
3.60/2.86	s: {1}
3.60/2.86	sel: {1, 2}
3.60/2.86	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(2)
3.60/2.86	Obligation:
3.60/2.86	Context-sensitive rewrite system:
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   f(X) -> cons(X, f(g(X)))
3.60/2.86	   g(0) -> s(0)
3.60/2.86	   g(s(X)) -> s(s(g(X)))
3.60/2.86	   sel(0, cons(X, Y)) -> X
3.60/2.86	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.60/2.86	
3.60/2.86	The replacement map contains the following entries:
3.60/2.86	
3.60/2.86	f: {1}
3.60/2.86	cons: {1}
3.60/2.86	g: {1}
3.60/2.86	0: empty set
3.60/2.86	s: {1}
3.60/2.86	sel: {1, 2}
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(3) CSRInnermostProof (EQUIVALENT)
3.60/2.86	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(4)
3.60/2.86	Obligation:
3.60/2.86	Context-sensitive rewrite system:
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   f(X) -> cons(X, f(g(X)))
3.60/2.86	   g(0) -> s(0)
3.60/2.86	   g(s(X)) -> s(s(g(X)))
3.60/2.86	   sel(0, cons(X, Y)) -> X
3.60/2.86	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.60/2.86	
3.60/2.86	The replacement map contains the following entries:
3.60/2.86	
3.60/2.86	f: {1}
3.60/2.86	cons: {1}
3.60/2.86	g: {1}
3.60/2.86	0: empty set
3.60/2.86	s: {1}
3.60/2.86	sel: {1, 2}
3.60/2.86	
3.60/2.86	
3.60/2.86	Innermost Strategy.
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(5) CSDependencyPairsProof (EQUIVALENT)
3.60/2.86	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(6)
3.60/2.86	Obligation:
3.60/2.86	Q-restricted context-sensitive dependency pair problem:
3.60/2.86	The symbols in {f_1, g_1, s_1, sel_2, G_1, SEL_2, F_1} are replacing on all positions.
3.60/2.86	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/2.86	The symbols in {U_1} are not replacing on any position.
3.60/2.86	
3.60/2.86	The ordinary context-sensitive dependency pairs DP_o are:
3.60/2.86	   G(s(X)) -> G(X)
3.60/2.86	   SEL(s(X), cons(Y, Z)) -> SEL(X, Z)
3.60/2.86	
3.60/2.86	The collapsing dependency pairs are DP_c:
3.60/2.86	   SEL(s(X), cons(Y, Z)) -> Z
3.60/2.86	
3.60/2.86	
3.60/2.86	The hidden terms of R are:
3.60/2.86	
3.60/2.86	   f(g(x0))
3.60/2.86	   g(x0)
3.60/2.86	
3.60/2.86	Every hiding context is built from:
3.60/2.86	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@4f65c3e7
3.60/2.86	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@50daa61d
3.60/2.86	
3.60/2.86	Hence, the new unhiding pairs DP_u are :
3.60/2.86	   SEL(s(X), cons(Y, Z)) -> U(Z)
3.60/2.86	   U(g(x_0)) -> U(x_0)
3.60/2.86	   U(f(x_0)) -> U(x_0)
3.60/2.86	   U(f(g(x0))) -> F(g(x0))
3.60/2.86	   U(g(x0)) -> G(x0)
3.60/2.86	
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   f(X) -> cons(X, f(g(X)))
3.60/2.86	   g(0) -> s(0)
3.60/2.86	   g(s(X)) -> s(s(g(X)))
3.60/2.86	   sel(0, cons(X, Y)) -> X
3.60/2.86	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.60/2.86	
3.60/2.86	The set Q consists of the following terms:
3.60/2.86	
3.60/2.86	   f(x0)
3.60/2.86	   g(0)
3.60/2.86	   g(s(x0))
3.60/2.86	   sel(0, cons(x0, x1))
3.60/2.86	   sel(s(x0), cons(x1, x2))
3.60/2.86	
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(7) QCSDependencyGraphProof (EQUIVALENT)
3.60/2.86	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 3 less nodes.
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(8)
3.60/2.86	Complex Obligation (AND)
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(9)
3.60/2.86	Obligation:
3.60/2.86	Q-restricted context-sensitive dependency pair problem:
3.60/2.86	The symbols in {f_1, g_1, s_1, sel_2, G_1} are replacing on all positions.
3.60/2.86	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/2.86	
3.60/2.86	The TRS P consists of the following rules:
3.60/2.86	
3.60/2.86	   G(s(X)) -> G(X)
3.60/2.86	
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   f(X) -> cons(X, f(g(X)))
3.60/2.86	   g(0) -> s(0)
3.60/2.86	   g(s(X)) -> s(s(g(X)))
3.60/2.86	   sel(0, cons(X, Y)) -> X
3.60/2.86	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.60/2.86	
3.60/2.86	The set Q consists of the following terms:
3.60/2.86	
3.60/2.86	   f(x0)
3.60/2.86	   g(0)
3.60/2.86	   g(s(x0))
3.60/2.86	   sel(0, cons(x0, x1))
3.60/2.86	   sel(s(x0), cons(x1, x2))
3.60/2.86	
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(10) QCSDPSubtermProof (EQUIVALENT)
3.60/2.86	We use the subterm processor [DA_EMMES].
3.60/2.86	
3.60/2.86	
3.60/2.86	The following pairs can be oriented strictly and are deleted.
3.60/2.86	
3.60/2.86	   G(s(X)) -> G(X)
3.60/2.86	The remaining pairs can at least be oriented weakly.
3.60/2.86	none
3.60/2.86	Used ordering:  Combined order from the following AFS and order.
3.60/2.86	G(x1)  =  x1
3.60/2.86	
3.60/2.86	
3.60/2.86	Subterm Order
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(11)
3.60/2.86	Obligation:
3.60/2.86	Q-restricted context-sensitive dependency pair problem:
3.60/2.86	The symbols in {f_1, g_1, s_1, sel_2} are replacing on all positions.
3.60/2.86	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/2.86	
3.60/2.86	The TRS P consists of the following rules:
3.60/2.86	none
3.60/2.86	
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   f(X) -> cons(X, f(g(X)))
3.60/2.86	   g(0) -> s(0)
3.60/2.86	   g(s(X)) -> s(s(g(X)))
3.60/2.86	   sel(0, cons(X, Y)) -> X
3.60/2.86	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.60/2.86	
3.60/2.86	The set Q consists of the following terms:
3.60/2.86	
3.60/2.86	   f(x0)
3.60/2.86	   g(0)
3.60/2.86	   g(s(x0))
3.60/2.86	   sel(0, cons(x0, x1))
3.60/2.86	   sel(s(x0), cons(x1, x2))
3.60/2.86	
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(12) PIsEmptyProof (EQUIVALENT)
3.60/2.86	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(13)
3.60/2.86	YES
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(14)
3.60/2.86	Obligation:
3.60/2.86	Q-restricted context-sensitive dependency pair problem:
3.60/2.86	The symbols in {f_1, g_1, s_1, sel_2} are replacing on all positions.
3.60/2.86	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/2.86	The symbols in {U_1} are not replacing on any position.
3.60/2.86	
3.60/2.86	The TRS P consists of the following rules:
3.60/2.86	
3.60/2.86	   U(g(x_0)) -> U(x_0)
3.60/2.86	   U(f(x_0)) -> U(x_0)
3.60/2.86	
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   f(X) -> cons(X, f(g(X)))
3.60/2.86	   g(0) -> s(0)
3.60/2.86	   g(s(X)) -> s(s(g(X)))
3.60/2.86	   sel(0, cons(X, Y)) -> X
3.60/2.86	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.60/2.86	
3.60/2.86	The set Q consists of the following terms:
3.60/2.86	
3.60/2.86	   f(x0)
3.60/2.86	   g(0)
3.60/2.86	   g(s(x0))
3.60/2.86	   sel(0, cons(x0, x1))
3.60/2.86	   sel(s(x0), cons(x1, x2))
3.60/2.86	
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(15) QCSDPSubtermProof (EQUIVALENT)
3.60/2.86	We use the subterm processor [DA_EMMES].
3.60/2.86	
3.60/2.86	
3.60/2.86	The following pairs can be oriented strictly and are deleted.
3.60/2.86	
3.60/2.86	   U(g(x_0)) -> U(x_0)
3.60/2.86	   U(f(x_0)) -> U(x_0)
3.60/2.86	The remaining pairs can at least be oriented weakly.
3.60/2.86	none
3.60/2.86	Used ordering:  Combined order from the following AFS and order.
3.60/2.86	U(x1)  =  x1
3.60/2.86	
3.60/2.86	
3.60/2.86	Subterm Order
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(16)
3.60/2.86	Obligation:
3.60/2.86	Q-restricted context-sensitive dependency pair problem:
3.60/2.86	The symbols in {f_1, g_1, s_1, sel_2} are replacing on all positions.
3.60/2.86	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/2.86	
3.60/2.86	The TRS P consists of the following rules:
3.60/2.86	none
3.60/2.86	
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   f(X) -> cons(X, f(g(X)))
3.60/2.86	   g(0) -> s(0)
3.60/2.86	   g(s(X)) -> s(s(g(X)))
3.60/2.86	   sel(0, cons(X, Y)) -> X
3.60/2.86	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.60/2.86	
3.60/2.86	The set Q consists of the following terms:
3.60/2.86	
3.60/2.86	   f(x0)
3.60/2.86	   g(0)
3.60/2.86	   g(s(x0))
3.60/2.86	   sel(0, cons(x0, x1))
3.60/2.86	   sel(s(x0), cons(x1, x2))
3.60/2.86	
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(17) PIsEmptyProof (EQUIVALENT)
3.60/2.86	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(18)
3.60/2.86	YES
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(19)
3.60/2.86	Obligation:
3.60/2.86	Q-restricted context-sensitive dependency pair problem:
3.60/2.86	The symbols in {f_1, g_1, s_1, sel_2, SEL_2} are replacing on all positions.
3.60/2.86	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/2.86	
3.60/2.86	The TRS P consists of the following rules:
3.60/2.86	
3.60/2.86	   SEL(s(X), cons(Y, Z)) -> SEL(X, Z)
3.60/2.86	
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   f(X) -> cons(X, f(g(X)))
3.60/2.86	   g(0) -> s(0)
3.60/2.86	   g(s(X)) -> s(s(g(X)))
3.60/2.86	   sel(0, cons(X, Y)) -> X
3.60/2.86	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.60/2.86	
3.60/2.86	The set Q consists of the following terms:
3.60/2.86	
3.60/2.86	   f(x0)
3.60/2.86	   g(0)
3.60/2.86	   g(s(x0))
3.60/2.86	   sel(0, cons(x0, x1))
3.60/2.86	   sel(s(x0), cons(x1, x2))
3.60/2.86	
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(20) QCSDPSubtermProof (EQUIVALENT)
3.60/2.86	We use the subterm processor [DA_EMMES].
3.60/2.86	
3.60/2.86	
3.60/2.86	The following pairs can be oriented strictly and are deleted.
3.60/2.86	
3.60/2.86	   SEL(s(X), cons(Y, Z)) -> SEL(X, Z)
3.60/2.86	The remaining pairs can at least be oriented weakly.
3.60/2.86	none
3.60/2.86	Used ordering:  Combined order from the following AFS and order.
3.60/2.86	SEL(x1, x2)  =  x1
3.60/2.86	
3.60/2.86	
3.60/2.86	Subterm Order
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(21)
3.60/2.86	Obligation:
3.60/2.86	Q-restricted context-sensitive dependency pair problem:
3.60/2.86	The symbols in {f_1, g_1, s_1, sel_2} are replacing on all positions.
3.60/2.86	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/2.86	
3.60/2.86	The TRS P consists of the following rules:
3.60/2.86	none
3.60/2.86	
3.60/2.86	The TRS R consists of the following rules:
3.60/2.86	
3.60/2.86	   f(X) -> cons(X, f(g(X)))
3.60/2.86	   g(0) -> s(0)
3.60/2.86	   g(s(X)) -> s(s(g(X)))
3.60/2.86	   sel(0, cons(X, Y)) -> X
3.60/2.86	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.60/2.86	
3.60/2.86	The set Q consists of the following terms:
3.60/2.86	
3.60/2.86	   f(x0)
3.60/2.86	   g(0)
3.60/2.86	   g(s(x0))
3.60/2.86	   sel(0, cons(x0, x1))
3.60/2.86	   sel(s(x0), cons(x1, x2))
3.60/2.86	
3.60/2.86	
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(22) PIsEmptyProof (EQUIVALENT)
3.60/2.86	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.60/2.86	----------------------------------------
3.60/2.86	
3.60/2.86	(23)
3.60/2.86	YES
3.60/2.87	EOF