3.73/1.81	YES
3.73/1.82	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.73/1.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.73/1.82	
3.73/1.82	
3.73/1.82	Termination w.r.t. Q of the given QTRS could be proven:
3.73/1.82	
3.73/1.82	(0) QTRS
3.73/1.82	(1) QTRSToCSRProof [SOUND, 0 ms]
3.73/1.82	(2) CSR
3.73/1.82	(3) CSRInnermostProof [EQUIVALENT, 0 ms]
3.73/1.82	(4) CSR
3.73/1.82	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.73/1.82	(6) QCSDP
3.73/1.82	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.73/1.82	(8) AND
3.73/1.82	    (9) QCSDP
3.73/1.82	        (10) QCSDPSubtermProof [EQUIVALENT, 6 ms]
3.73/1.82	        (11) QCSDP
3.73/1.82	        (12) PIsEmptyProof [EQUIVALENT, 0 ms]
3.73/1.82	        (13) YES
3.73/1.82	    (14) QCSDP
3.73/1.82	        (15) QCSDPSubtermProof [EQUIVALENT, 2 ms]
3.73/1.82	        (16) QCSDP
3.73/1.82	        (17) PIsEmptyProof [EQUIVALENT, 0 ms]
3.73/1.82	        (18) YES
3.73/1.82	    (19) QCSDP
3.73/1.82	        (20) QCSDPSubtermProof [EQUIVALENT, 5 ms]
3.73/1.82	        (21) QCSDP
3.73/1.82	        (22) PIsEmptyProof [EQUIVALENT, 0 ms]
3.73/1.82	        (23) YES
3.73/1.82	
3.73/1.82	
3.73/1.82	----------------------------------------
3.73/1.82	
3.73/1.82	(0)
3.73/1.82	Obligation:
3.73/1.82	Q restricted rewrite system:
3.73/1.82	The TRS R consists of the following rules:
3.73/1.82	
3.73/1.82	   active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
3.73/1.82	   active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
3.73/1.82	   active(add(0, X)) -> mark(X)
3.73/1.82	   active(add(s(X), Y)) -> mark(s(add(X, Y)))
3.73/1.82	   active(sel(0, cons(X, XS))) -> mark(X)
3.73/1.82	   active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
3.73/1.82	   active(fib(X)) -> fib(active(X))
3.73/1.82	   active(sel(X1, X2)) -> sel(active(X1), X2)
3.73/1.82	   active(sel(X1, X2)) -> sel(X1, active(X2))
3.73/1.82	   active(fib1(X1, X2)) -> fib1(active(X1), X2)
3.73/1.82	   active(fib1(X1, X2)) -> fib1(X1, active(X2))
3.73/1.82	   active(s(X)) -> s(active(X))
3.73/1.82	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.73/1.82	   active(add(X1, X2)) -> add(active(X1), X2)
3.73/1.82	   active(add(X1, X2)) -> add(X1, active(X2))
3.73/1.82	   fib(mark(X)) -> mark(fib(X))
3.73/1.82	   sel(mark(X1), X2) -> mark(sel(X1, X2))
3.73/1.82	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
3.73/1.82	   fib1(mark(X1), X2) -> mark(fib1(X1, X2))
3.73/1.82	   fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
3.73/1.82	   s(mark(X)) -> mark(s(X))
3.73/1.82	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.73/1.82	   add(mark(X1), X2) -> mark(add(X1, X2))
3.73/1.82	   add(X1, mark(X2)) -> mark(add(X1, X2))
3.73/1.82	   proper(fib(X)) -> fib(proper(X))
3.73/1.82	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
3.73/1.82	   proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
3.73/1.82	   proper(s(X)) -> s(proper(X))
3.73/1.82	   proper(0) -> ok(0)
3.73/1.82	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.73/1.82	   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
3.73/1.82	   fib(ok(X)) -> ok(fib(X))
3.73/1.82	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
3.73/1.82	   fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
3.73/1.82	   s(ok(X)) -> ok(s(X))
3.73/1.82	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.73/1.82	   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
3.73/1.82	   top(mark(X)) -> top(proper(X))
3.73/1.82	   top(ok(X)) -> top(active(X))
3.73/1.82	
3.73/1.82	The set Q consists of the following terms:
3.73/1.82	
3.73/1.82	   active(fib(x0))
3.73/1.82	   active(fib1(x0, x1))
3.73/1.82	   active(sel(x0, x1))
3.73/1.82	   active(s(x0))
3.73/1.82	   active(cons(x0, x1))
3.73/1.82	   active(add(x0, x1))
3.73/1.82	   fib(mark(x0))
3.73/1.82	   sel(mark(x0), x1)
3.73/1.82	   sel(x0, mark(x1))
3.73/1.82	   fib1(mark(x0), x1)
3.73/1.82	   fib1(x0, mark(x1))
3.73/1.82	   s(mark(x0))
3.73/1.82	   cons(mark(x0), x1)
3.73/1.82	   add(mark(x0), x1)
3.73/1.82	   add(x0, mark(x1))
3.73/1.82	   proper(fib(x0))
3.73/1.82	   proper(sel(x0, x1))
3.73/1.82	   proper(fib1(x0, x1))
3.73/1.82	   proper(s(x0))
3.73/1.82	   proper(0)
3.73/1.82	   proper(cons(x0, x1))
3.73/1.82	   proper(add(x0, x1))
3.73/1.82	   fib(ok(x0))
3.73/1.82	   sel(ok(x0), ok(x1))
3.73/1.82	   fib1(ok(x0), ok(x1))
3.73/1.82	   s(ok(x0))
3.73/1.82	   cons(ok(x0), ok(x1))
3.73/1.82	   add(ok(x0), ok(x1))
3.73/1.82	   top(mark(x0))
3.73/1.82	   top(ok(x0))
3.73/1.82	
3.73/1.82	
3.73/1.82	----------------------------------------
3.73/1.82	
3.73/1.82	(1) QTRSToCSRProof (SOUND)
3.73/1.82	The following Q TRS is given: Q restricted rewrite system:
3.73/1.82	The TRS R consists of the following rules:
3.73/1.82	
3.73/1.82	   active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
3.73/1.82	   active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
3.73/1.82	   active(add(0, X)) -> mark(X)
3.73/1.82	   active(add(s(X), Y)) -> mark(s(add(X, Y)))
3.73/1.82	   active(sel(0, cons(X, XS))) -> mark(X)
3.73/1.82	   active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
3.73/1.82	   active(fib(X)) -> fib(active(X))
3.73/1.82	   active(sel(X1, X2)) -> sel(active(X1), X2)
3.73/1.82	   active(sel(X1, X2)) -> sel(X1, active(X2))
3.73/1.82	   active(fib1(X1, X2)) -> fib1(active(X1), X2)
3.73/1.82	   active(fib1(X1, X2)) -> fib1(X1, active(X2))
3.73/1.82	   active(s(X)) -> s(active(X))
3.73/1.82	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.73/1.82	   active(add(X1, X2)) -> add(active(X1), X2)
3.73/1.83	   active(add(X1, X2)) -> add(X1, active(X2))
3.73/1.83	   fib(mark(X)) -> mark(fib(X))
3.73/1.83	   sel(mark(X1), X2) -> mark(sel(X1, X2))
3.73/1.83	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
3.73/1.83	   fib1(mark(X1), X2) -> mark(fib1(X1, X2))
3.73/1.83	   fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
3.73/1.83	   s(mark(X)) -> mark(s(X))
3.73/1.83	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.73/1.83	   add(mark(X1), X2) -> mark(add(X1, X2))
3.73/1.83	   add(X1, mark(X2)) -> mark(add(X1, X2))
3.73/1.83	   proper(fib(X)) -> fib(proper(X))
3.73/1.83	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
3.73/1.83	   proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
3.73/1.83	   proper(s(X)) -> s(proper(X))
3.73/1.83	   proper(0) -> ok(0)
3.73/1.83	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.73/1.83	   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
3.73/1.83	   fib(ok(X)) -> ok(fib(X))
3.73/1.83	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
3.73/1.83	   fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
3.73/1.83	   s(ok(X)) -> ok(s(X))
3.73/1.83	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.73/1.83	   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
3.73/1.83	   top(mark(X)) -> top(proper(X))
3.73/1.83	   top(ok(X)) -> top(active(X))
3.73/1.83	
3.73/1.83	The set Q consists of the following terms:
3.73/1.83	
3.73/1.83	   active(fib(x0))
3.73/1.83	   active(fib1(x0, x1))
3.73/1.83	   active(sel(x0, x1))
3.73/1.83	   active(s(x0))
3.73/1.83	   active(cons(x0, x1))
3.73/1.83	   active(add(x0, x1))
3.73/1.83	   fib(mark(x0))
3.73/1.83	   sel(mark(x0), x1)
3.73/1.83	   sel(x0, mark(x1))
3.73/1.83	   fib1(mark(x0), x1)
3.73/1.83	   fib1(x0, mark(x1))
3.73/1.83	   s(mark(x0))
3.73/1.83	   cons(mark(x0), x1)
3.73/1.83	   add(mark(x0), x1)
3.73/1.83	   add(x0, mark(x1))
3.73/1.83	   proper(fib(x0))
3.73/1.83	   proper(sel(x0, x1))
3.73/1.83	   proper(fib1(x0, x1))
3.73/1.83	   proper(s(x0))
3.73/1.83	   proper(0)
3.73/1.83	   proper(cons(x0, x1))
3.73/1.83	   proper(add(x0, x1))
3.73/1.83	   fib(ok(x0))
3.73/1.83	   sel(ok(x0), ok(x1))
3.73/1.83	   fib1(ok(x0), ok(x1))
3.73/1.83	   s(ok(x0))
3.73/1.83	   cons(ok(x0), ok(x1))
3.73/1.83	   add(ok(x0), ok(x1))
3.73/1.83	   top(mark(x0))
3.73/1.83	   top(ok(x0))
3.73/1.83	
3.73/1.83	Special symbols used for the transformation (see [GM04]):
3.73/1.83	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.73/1.83	The replacement map contains the following entries:
3.73/1.83	
3.73/1.83	fib: {1}
3.73/1.83	sel: {1, 2}
3.73/1.83	fib1: {1, 2}
3.73/1.83	s: {1}
3.73/1.83	0: empty set
3.73/1.83	cons: {1}
3.73/1.83	add: {1, 2}
3.73/1.83	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.
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(2)
3.73/1.83	Obligation:
3.73/1.83	Context-sensitive rewrite system:
3.73/1.83	The TRS R consists of the following rules:
3.73/1.83	
3.73/1.83	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.73/1.83	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.73/1.83	   add(0, X) -> X
3.73/1.83	   add(s(X), Y) -> s(add(X, Y))
3.73/1.83	   sel(0, cons(X, XS)) -> X
3.73/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.73/1.83	
3.73/1.83	The replacement map contains the following entries:
3.73/1.83	
3.73/1.83	fib: {1}
3.73/1.83	sel: {1, 2}
3.73/1.83	fib1: {1, 2}
3.73/1.83	s: {1}
3.73/1.83	0: empty set
3.73/1.83	cons: {1}
3.73/1.83	add: {1, 2}
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(3) CSRInnermostProof (EQUIVALENT)
3.73/1.83	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(4)
3.73/1.83	Obligation:
3.73/1.83	Context-sensitive rewrite system:
3.73/1.83	The TRS R consists of the following rules:
3.73/1.83	
3.73/1.83	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.73/1.83	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.73/1.83	   add(0, X) -> X
3.73/1.83	   add(s(X), Y) -> s(add(X, Y))
3.73/1.83	   sel(0, cons(X, XS)) -> X
3.73/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.73/1.83	
3.73/1.83	The replacement map contains the following entries:
3.73/1.83	
3.73/1.83	fib: {1}
3.73/1.83	sel: {1, 2}
3.73/1.83	fib1: {1, 2}
3.73/1.83	s: {1}
3.73/1.83	0: empty set
3.73/1.83	cons: {1}
3.73/1.83	add: {1, 2}
3.73/1.83	
3.73/1.83	
3.73/1.83	Innermost Strategy.
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(5) CSDependencyPairsProof (EQUIVALENT)
3.73/1.83	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(6)
3.73/1.83	Obligation:
3.73/1.83	Q-restricted context-sensitive dependency pair problem:
3.73/1.83	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2, SEL_2, FIB_1, FIB1_2, ADD_2} are replacing on all positions.
3.73/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.73/1.83	The symbols in {U_1} are not replacing on any position.
3.73/1.83	
3.73/1.83	The ordinary context-sensitive dependency pairs DP_o are:
3.73/1.83	   FIB(N) -> SEL(N, fib1(s(0), s(0)))
3.73/1.83	   FIB(N) -> FIB1(s(0), s(0))
3.73/1.83	   ADD(s(X), Y) -> ADD(X, Y)
3.73/1.83	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
3.73/1.83	
3.73/1.83	The collapsing dependency pairs are DP_c:
3.73/1.83	   SEL(s(N), cons(X, XS)) -> XS
3.73/1.83	
3.73/1.83	
3.73/1.83	The hidden terms of R are:
3.73/1.83	
3.73/1.83	   fib1(x0, add(x1, x0))
3.73/1.83	   add(x0, x1)
3.73/1.83	
3.73/1.83	Every hiding context is built from:
3.73/1.83	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@57f3cd1f
3.73/1.83	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@382cfe27
3.73/1.83	
3.73/1.83	Hence, the new unhiding pairs DP_u are :
3.73/1.83	   SEL(s(N), cons(X, XS)) -> U(XS)
3.73/1.83	   U(add(x_0, x_1)) -> U(x_0)
3.73/1.83	   U(add(x_0, x_1)) -> U(x_1)
3.73/1.83	   U(fib1(x_0, x_1)) -> U(x_0)
3.73/1.83	   U(fib1(x_0, x_1)) -> U(x_1)
3.73/1.83	   U(fib1(x0, add(x1, x0))) -> FIB1(x0, add(x1, x0))
3.73/1.83	   U(add(x0, x1)) -> ADD(x0, x1)
3.73/1.83	
3.73/1.83	The TRS R consists of the following rules:
3.73/1.83	
3.73/1.83	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.73/1.83	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.73/1.83	   add(0, X) -> X
3.73/1.83	   add(s(X), Y) -> s(add(X, Y))
3.73/1.83	   sel(0, cons(X, XS)) -> X
3.73/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.73/1.83	
3.73/1.83	The set Q consists of the following terms:
3.73/1.83	
3.73/1.83	   fib(x0)
3.73/1.83	   fib1(x0, x1)
3.73/1.83	   add(0, x0)
3.73/1.83	   add(s(x0), x1)
3.73/1.83	   sel(0, cons(x0, x1))
3.73/1.83	   sel(s(x0), cons(x1, x2))
3.73/1.83	
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(7) QCSDependencyGraphProof (EQUIVALENT)
3.73/1.83	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 4 less nodes.
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(8)
3.73/1.83	Complex Obligation (AND)
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(9)
3.73/1.83	Obligation:
3.73/1.83	Q-restricted context-sensitive dependency pair problem:
3.73/1.83	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2, ADD_2} are replacing on all positions.
3.73/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.73/1.83	
3.73/1.83	The TRS P consists of the following rules:
3.73/1.83	
3.73/1.83	   ADD(s(X), Y) -> ADD(X, Y)
3.73/1.83	
3.73/1.83	The TRS R consists of the following rules:
3.73/1.83	
3.73/1.83	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.73/1.83	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.73/1.83	   add(0, X) -> X
3.73/1.83	   add(s(X), Y) -> s(add(X, Y))
3.73/1.83	   sel(0, cons(X, XS)) -> X
3.73/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.73/1.83	
3.73/1.83	The set Q consists of the following terms:
3.73/1.83	
3.73/1.83	   fib(x0)
3.73/1.83	   fib1(x0, x1)
3.73/1.83	   add(0, x0)
3.73/1.83	   add(s(x0), x1)
3.73/1.83	   sel(0, cons(x0, x1))
3.73/1.83	   sel(s(x0), cons(x1, x2))
3.73/1.83	
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(10) QCSDPSubtermProof (EQUIVALENT)
3.73/1.83	We use the subterm processor [DA_EMMES].
3.73/1.83	
3.73/1.83	
3.73/1.83	The following pairs can be oriented strictly and are deleted.
3.73/1.83	
3.73/1.83	   ADD(s(X), Y) -> ADD(X, Y)
3.73/1.83	The remaining pairs can at least be oriented weakly.
3.73/1.83	none
3.73/1.83	Used ordering:  Combined order from the following AFS and order.
3.73/1.83	ADD(x1, x2)  =  x1
3.73/1.83	
3.73/1.83	
3.73/1.83	Subterm Order
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(11)
3.73/1.83	Obligation:
3.73/1.83	Q-restricted context-sensitive dependency pair problem:
3.73/1.83	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2} are replacing on all positions.
3.73/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.73/1.83	
3.73/1.83	The TRS P consists of the following rules:
3.73/1.83	none
3.73/1.83	
3.73/1.83	The TRS R consists of the following rules:
3.73/1.83	
3.73/1.83	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.73/1.83	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.73/1.83	   add(0, X) -> X
3.73/1.83	   add(s(X), Y) -> s(add(X, Y))
3.73/1.83	   sel(0, cons(X, XS)) -> X
3.73/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.73/1.83	
3.73/1.83	The set Q consists of the following terms:
3.73/1.83	
3.73/1.83	   fib(x0)
3.73/1.83	   fib1(x0, x1)
3.73/1.83	   add(0, x0)
3.73/1.83	   add(s(x0), x1)
3.73/1.83	   sel(0, cons(x0, x1))
3.73/1.83	   sel(s(x0), cons(x1, x2))
3.73/1.83	
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(12) PIsEmptyProof (EQUIVALENT)
3.73/1.83	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(13)
3.73/1.83	YES
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(14)
3.73/1.83	Obligation:
3.73/1.83	Q-restricted context-sensitive dependency pair problem:
3.73/1.83	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2} are replacing on all positions.
3.73/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.73/1.83	The symbols in {U_1} are not replacing on any position.
3.73/1.83	
3.73/1.83	The TRS P consists of the following rules:
3.73/1.83	
3.73/1.83	   U(add(x_0, x_1)) -> U(x_0)
3.73/1.83	   U(add(x_0, x_1)) -> U(x_1)
3.73/1.83	   U(fib1(x_0, x_1)) -> U(x_0)
3.73/1.83	   U(fib1(x_0, x_1)) -> U(x_1)
3.73/1.83	
3.73/1.83	The TRS R consists of the following rules:
3.73/1.83	
3.73/1.83	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.73/1.83	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.73/1.83	   add(0, X) -> X
3.73/1.83	   add(s(X), Y) -> s(add(X, Y))
3.73/1.83	   sel(0, cons(X, XS)) -> X
3.73/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.73/1.83	
3.73/1.83	The set Q consists of the following terms:
3.73/1.83	
3.73/1.83	   fib(x0)
3.73/1.83	   fib1(x0, x1)
3.73/1.83	   add(0, x0)
3.73/1.83	   add(s(x0), x1)
3.73/1.83	   sel(0, cons(x0, x1))
3.73/1.83	   sel(s(x0), cons(x1, x2))
3.73/1.83	
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(15) QCSDPSubtermProof (EQUIVALENT)
3.73/1.83	We use the subterm processor [DA_EMMES].
3.73/1.83	
3.73/1.83	
3.73/1.83	The following pairs can be oriented strictly and are deleted.
3.73/1.83	
3.73/1.83	   U(add(x_0, x_1)) -> U(x_0)
3.73/1.83	   U(add(x_0, x_1)) -> U(x_1)
3.73/1.83	   U(fib1(x_0, x_1)) -> U(x_0)
3.73/1.83	   U(fib1(x_0, x_1)) -> U(x_1)
3.73/1.83	The remaining pairs can at least be oriented weakly.
3.73/1.83	none
3.73/1.83	Used ordering:  Combined order from the following AFS and order.
3.73/1.83	U(x1)  =  x1
3.73/1.83	
3.73/1.83	
3.73/1.83	Subterm Order
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(16)
3.73/1.83	Obligation:
3.73/1.83	Q-restricted context-sensitive dependency pair problem:
3.73/1.83	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2} are replacing on all positions.
3.73/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.73/1.83	
3.73/1.83	The TRS P consists of the following rules:
3.73/1.83	none
3.73/1.83	
3.73/1.83	The TRS R consists of the following rules:
3.73/1.83	
3.73/1.83	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.73/1.83	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.73/1.83	   add(0, X) -> X
3.73/1.83	   add(s(X), Y) -> s(add(X, Y))
3.73/1.83	   sel(0, cons(X, XS)) -> X
3.73/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.73/1.83	
3.73/1.83	The set Q consists of the following terms:
3.73/1.83	
3.73/1.83	   fib(x0)
3.73/1.83	   fib1(x0, x1)
3.73/1.83	   add(0, x0)
3.73/1.83	   add(s(x0), x1)
3.73/1.83	   sel(0, cons(x0, x1))
3.73/1.83	   sel(s(x0), cons(x1, x2))
3.73/1.83	
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(17) PIsEmptyProof (EQUIVALENT)
3.73/1.83	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(18)
3.73/1.83	YES
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(19)
3.73/1.83	Obligation:
3.73/1.83	Q-restricted context-sensitive dependency pair problem:
3.73/1.83	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2, SEL_2} are replacing on all positions.
3.73/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.73/1.83	
3.73/1.83	The TRS P consists of the following rules:
3.73/1.83	
3.73/1.83	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
3.73/1.83	
3.73/1.83	The TRS R consists of the following rules:
3.73/1.83	
3.73/1.83	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.73/1.83	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.73/1.83	   add(0, X) -> X
3.73/1.83	   add(s(X), Y) -> s(add(X, Y))
3.73/1.83	   sel(0, cons(X, XS)) -> X
3.73/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.73/1.83	
3.73/1.83	The set Q consists of the following terms:
3.73/1.83	
3.73/1.83	   fib(x0)
3.73/1.83	   fib1(x0, x1)
3.73/1.83	   add(0, x0)
3.73/1.83	   add(s(x0), x1)
3.73/1.83	   sel(0, cons(x0, x1))
3.73/1.83	   sel(s(x0), cons(x1, x2))
3.73/1.83	
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(20) QCSDPSubtermProof (EQUIVALENT)
3.73/1.83	We use the subterm processor [DA_EMMES].
3.73/1.83	
3.73/1.83	
3.73/1.83	The following pairs can be oriented strictly and are deleted.
3.73/1.83	
3.73/1.83	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
3.73/1.83	The remaining pairs can at least be oriented weakly.
3.73/1.83	none
3.73/1.83	Used ordering:  Combined order from the following AFS and order.
3.73/1.83	SEL(x1, x2)  =  x1
3.73/1.83	
3.73/1.83	
3.73/1.83	Subterm Order
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(21)
3.73/1.83	Obligation:
3.73/1.83	Q-restricted context-sensitive dependency pair problem:
3.73/1.83	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2} are replacing on all positions.
3.73/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.73/1.83	
3.73/1.83	The TRS P consists of the following rules:
3.73/1.83	none
3.73/1.83	
3.73/1.83	The TRS R consists of the following rules:
3.73/1.83	
3.73/1.83	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.73/1.83	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.73/1.83	   add(0, X) -> X
3.73/1.83	   add(s(X), Y) -> s(add(X, Y))
3.73/1.83	   sel(0, cons(X, XS)) -> X
3.73/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.73/1.83	
3.73/1.83	The set Q consists of the following terms:
3.73/1.83	
3.73/1.83	   fib(x0)
3.73/1.83	   fib1(x0, x1)
3.73/1.83	   add(0, x0)
3.73/1.83	   add(s(x0), x1)
3.73/1.83	   sel(0, cons(x0, x1))
3.73/1.83	   sel(s(x0), cons(x1, x2))
3.73/1.83	
3.73/1.83	
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(22) PIsEmptyProof (EQUIVALENT)
3.73/1.83	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.73/1.83	----------------------------------------
3.73/1.83	
3.73/1.83	(23)
3.73/1.83	YES
3.73/1.86	EOF