3.11/1.54	YES
3.11/1.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.11/1.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.11/1.55	
3.11/1.55	
3.11/1.55	Termination of the given CSR could be proven:
3.11/1.55	
3.11/1.55	(0) CSR
3.11/1.55	(1) CSRInnermostProof [EQUIVALENT, 0 ms]
3.11/1.55	(2) CSR
3.11/1.55	(3) CSDependencyPairsProof [EQUIVALENT, 18 ms]
3.11/1.55	(4) QCSDP
3.11/1.55	(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.11/1.55	(6) AND
3.11/1.55	    (7) QCSDP
3.11/1.55	        (8) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.11/1.55	        (9) QCSDP
3.11/1.55	        (10) PIsEmptyProof [EQUIVALENT, 0 ms]
3.11/1.55	        (11) YES
3.11/1.55	    (12) QCSDP
3.11/1.55	        (13) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.11/1.55	        (14) QCSDP
3.11/1.55	        (15) PIsEmptyProof [EQUIVALENT, 0 ms]
3.11/1.55	        (16) YES
3.11/1.55	    (17) QCSDP
3.11/1.55	        (18) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.11/1.55	        (19) QCSDP
3.11/1.55	        (20) PIsEmptyProof [EQUIVALENT, 0 ms]
3.11/1.55	        (21) YES
3.11/1.55	
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(0)
3.11/1.55	Obligation:
3.11/1.55	Context-sensitive rewrite system:
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.11/1.55	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.11/1.55	   add(0, X) -> X
3.11/1.55	   add(s(X), Y) -> s(add(X, Y))
3.11/1.55	   sel(0, cons(X, XS)) -> X
3.11/1.55	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.11/1.55	
3.11/1.55	The replacement map contains the following entries:
3.11/1.55	
3.11/1.55	fib: {1}
3.11/1.55	sel: {1, 2}
3.11/1.55	fib1: {1, 2}
3.11/1.55	s: {1}
3.11/1.55	0: empty set
3.11/1.55	cons: {1}
3.11/1.55	add: {1, 2}
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(1) CSRInnermostProof (EQUIVALENT)
3.11/1.55	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(2)
3.11/1.55	Obligation:
3.11/1.55	Context-sensitive rewrite system:
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.11/1.55	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.11/1.55	   add(0, X) -> X
3.11/1.55	   add(s(X), Y) -> s(add(X, Y))
3.11/1.55	   sel(0, cons(X, XS)) -> X
3.11/1.55	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.11/1.55	
3.11/1.55	The replacement map contains the following entries:
3.11/1.55	
3.11/1.55	fib: {1}
3.11/1.55	sel: {1, 2}
3.11/1.55	fib1: {1, 2}
3.11/1.55	s: {1}
3.11/1.55	0: empty set
3.11/1.55	cons: {1}
3.11/1.55	add: {1, 2}
3.11/1.55	
3.11/1.55	
3.11/1.55	Innermost Strategy.
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(3) CSDependencyPairsProof (EQUIVALENT)
3.11/1.55	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(4)
3.11/1.55	Obligation:
3.11/1.55	Q-restricted context-sensitive dependency pair problem:
3.11/1.55	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.
3.11/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
3.11/1.55	The symbols in {U_1} are not replacing on any position.
3.11/1.55	
3.11/1.55	The ordinary context-sensitive dependency pairs DP_o are:
3.11/1.55	   FIB(N) -> SEL(N, fib1(s(0), s(0)))
3.11/1.55	   FIB(N) -> FIB1(s(0), s(0))
3.11/1.55	   ADD(s(X), Y) -> ADD(X, Y)
3.11/1.55	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
3.11/1.55	
3.11/1.55	The collapsing dependency pairs are DP_c:
3.11/1.55	   SEL(s(N), cons(X, XS)) -> XS
3.11/1.55	
3.11/1.55	
3.11/1.55	The hidden terms of R are:
3.11/1.55	
3.11/1.55	   fib1(x0, add(x1, x0))
3.11/1.55	   add(x0, x1)
3.11/1.55	
3.11/1.55	Every hiding context is built from:
3.11/1.55	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@224700b4
3.11/1.55	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@5d4bf07d
3.11/1.55	
3.11/1.55	Hence, the new unhiding pairs DP_u are :
3.11/1.55	   SEL(s(N), cons(X, XS)) -> U(XS)
3.11/1.55	   U(add(x_0, x_1)) -> U(x_0)
3.11/1.55	   U(add(x_0, x_1)) -> U(x_1)
3.11/1.55	   U(fib1(x_0, x_1)) -> U(x_0)
3.11/1.55	   U(fib1(x_0, x_1)) -> U(x_1)
3.11/1.55	   U(fib1(x0, add(x1, x0))) -> FIB1(x0, add(x1, x0))
3.11/1.55	   U(add(x0, x1)) -> ADD(x0, x1)
3.11/1.55	
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.11/1.55	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.11/1.55	   add(0, X) -> X
3.11/1.55	   add(s(X), Y) -> s(add(X, Y))
3.11/1.55	   sel(0, cons(X, XS)) -> X
3.11/1.55	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.11/1.55	
3.11/1.55	The set Q consists of the following terms:
3.11/1.55	
3.11/1.55	   fib(x0)
3.11/1.55	   fib1(x0, x1)
3.11/1.55	   add(0, x0)
3.11/1.55	   add(s(x0), x1)
3.11/1.55	   sel(0, cons(x0, x1))
3.11/1.55	   sel(s(x0), cons(x1, x2))
3.11/1.55	
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(5) QCSDependencyGraphProof (EQUIVALENT)
3.11/1.55	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 4 less nodes.
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(6)
3.11/1.55	Complex Obligation (AND)
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(7)
3.11/1.55	Obligation:
3.11/1.55	Q-restricted context-sensitive dependency pair problem:
3.11/1.55	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2, ADD_2} are replacing on all positions.
3.11/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
3.11/1.55	
3.11/1.55	The TRS P consists of the following rules:
3.11/1.55	
3.11/1.55	   ADD(s(X), Y) -> ADD(X, Y)
3.11/1.55	
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.11/1.55	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.11/1.55	   add(0, X) -> X
3.11/1.55	   add(s(X), Y) -> s(add(X, Y))
3.11/1.55	   sel(0, cons(X, XS)) -> X
3.11/1.55	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.11/1.55	
3.11/1.55	The set Q consists of the following terms:
3.11/1.55	
3.11/1.55	   fib(x0)
3.11/1.55	   fib1(x0, x1)
3.11/1.55	   add(0, x0)
3.11/1.55	   add(s(x0), x1)
3.11/1.55	   sel(0, cons(x0, x1))
3.11/1.55	   sel(s(x0), cons(x1, x2))
3.11/1.55	
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(8) QCSDPSubtermProof (EQUIVALENT)
3.11/1.55	We use the subterm processor [DA_EMMES].
3.11/1.55	
3.11/1.55	
3.11/1.55	The following pairs can be oriented strictly and are deleted.
3.11/1.55	
3.11/1.55	   ADD(s(X), Y) -> ADD(X, Y)
3.11/1.55	The remaining pairs can at least be oriented weakly.
3.11/1.55	none
3.11/1.55	Used ordering:  Combined order from the following AFS and order.
3.11/1.55	ADD(x1, x2)  =  x1
3.11/1.55	
3.11/1.55	
3.11/1.55	Subterm Order
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(9)
3.11/1.55	Obligation:
3.11/1.55	Q-restricted context-sensitive dependency pair problem:
3.11/1.55	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2} are replacing on all positions.
3.11/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
3.11/1.55	
3.11/1.55	The TRS P consists of the following rules:
3.11/1.55	none
3.11/1.55	
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.11/1.55	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.11/1.55	   add(0, X) -> X
3.11/1.55	   add(s(X), Y) -> s(add(X, Y))
3.11/1.55	   sel(0, cons(X, XS)) -> X
3.11/1.55	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.11/1.55	
3.11/1.55	The set Q consists of the following terms:
3.11/1.55	
3.11/1.55	   fib(x0)
3.11/1.55	   fib1(x0, x1)
3.11/1.55	   add(0, x0)
3.11/1.55	   add(s(x0), x1)
3.11/1.55	   sel(0, cons(x0, x1))
3.11/1.55	   sel(s(x0), cons(x1, x2))
3.11/1.55	
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(10) PIsEmptyProof (EQUIVALENT)
3.11/1.55	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(11)
3.11/1.55	YES
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(12)
3.11/1.55	Obligation:
3.11/1.55	Q-restricted context-sensitive dependency pair problem:
3.11/1.55	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2} are replacing on all positions.
3.11/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
3.11/1.55	The symbols in {U_1} are not replacing on any position.
3.11/1.55	
3.11/1.55	The TRS P consists of the following rules:
3.11/1.55	
3.11/1.55	   U(add(x_0, x_1)) -> U(x_0)
3.11/1.55	   U(add(x_0, x_1)) -> U(x_1)
3.11/1.55	   U(fib1(x_0, x_1)) -> U(x_0)
3.11/1.55	   U(fib1(x_0, x_1)) -> U(x_1)
3.11/1.55	
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.11/1.55	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.11/1.55	   add(0, X) -> X
3.11/1.55	   add(s(X), Y) -> s(add(X, Y))
3.11/1.55	   sel(0, cons(X, XS)) -> X
3.11/1.55	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.11/1.55	
3.11/1.55	The set Q consists of the following terms:
3.11/1.55	
3.11/1.55	   fib(x0)
3.11/1.55	   fib1(x0, x1)
3.11/1.55	   add(0, x0)
3.11/1.55	   add(s(x0), x1)
3.11/1.55	   sel(0, cons(x0, x1))
3.11/1.55	   sel(s(x0), cons(x1, x2))
3.11/1.55	
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(13) QCSDPSubtermProof (EQUIVALENT)
3.11/1.55	We use the subterm processor [DA_EMMES].
3.11/1.55	
3.11/1.55	
3.11/1.55	The following pairs can be oriented strictly and are deleted.
3.11/1.55	
3.11/1.55	   U(add(x_0, x_1)) -> U(x_0)
3.11/1.55	   U(add(x_0, x_1)) -> U(x_1)
3.11/1.55	   U(fib1(x_0, x_1)) -> U(x_0)
3.11/1.55	   U(fib1(x_0, x_1)) -> U(x_1)
3.11/1.55	The remaining pairs can at least be oriented weakly.
3.11/1.55	none
3.11/1.55	Used ordering:  Combined order from the following AFS and order.
3.11/1.55	U(x1)  =  x1
3.11/1.55	
3.11/1.55	
3.11/1.55	Subterm Order
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(14)
3.11/1.55	Obligation:
3.11/1.55	Q-restricted context-sensitive dependency pair problem:
3.11/1.55	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2} are replacing on all positions.
3.11/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
3.11/1.55	
3.11/1.55	The TRS P consists of the following rules:
3.11/1.55	none
3.11/1.55	
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.11/1.55	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.11/1.55	   add(0, X) -> X
3.11/1.55	   add(s(X), Y) -> s(add(X, Y))
3.11/1.55	   sel(0, cons(X, XS)) -> X
3.11/1.55	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.11/1.55	
3.11/1.55	The set Q consists of the following terms:
3.11/1.55	
3.11/1.55	   fib(x0)
3.11/1.55	   fib1(x0, x1)
3.11/1.55	   add(0, x0)
3.11/1.55	   add(s(x0), x1)
3.11/1.55	   sel(0, cons(x0, x1))
3.11/1.55	   sel(s(x0), cons(x1, x2))
3.11/1.55	
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(15) PIsEmptyProof (EQUIVALENT)
3.11/1.55	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(16)
3.11/1.55	YES
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(17)
3.11/1.55	Obligation:
3.11/1.55	Q-restricted context-sensitive dependency pair problem:
3.11/1.55	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2, SEL_2} are replacing on all positions.
3.11/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
3.11/1.55	
3.11/1.55	The TRS P consists of the following rules:
3.11/1.55	
3.11/1.55	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
3.11/1.55	
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.11/1.55	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.11/1.55	   add(0, X) -> X
3.11/1.55	   add(s(X), Y) -> s(add(X, Y))
3.11/1.55	   sel(0, cons(X, XS)) -> X
3.11/1.55	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.11/1.55	
3.11/1.55	The set Q consists of the following terms:
3.11/1.55	
3.11/1.55	   fib(x0)
3.11/1.55	   fib1(x0, x1)
3.11/1.55	   add(0, x0)
3.11/1.55	   add(s(x0), x1)
3.11/1.55	   sel(0, cons(x0, x1))
3.11/1.55	   sel(s(x0), cons(x1, x2))
3.11/1.55	
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(18) QCSDPSubtermProof (EQUIVALENT)
3.11/1.55	We use the subterm processor [DA_EMMES].
3.11/1.55	
3.11/1.55	
3.11/1.55	The following pairs can be oriented strictly and are deleted.
3.11/1.55	
3.11/1.55	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
3.11/1.55	The remaining pairs can at least be oriented weakly.
3.11/1.55	none
3.11/1.55	Used ordering:  Combined order from the following AFS and order.
3.11/1.55	SEL(x1, x2)  =  x1
3.11/1.55	
3.11/1.55	
3.11/1.55	Subterm Order
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(19)
3.11/1.55	Obligation:
3.11/1.55	Q-restricted context-sensitive dependency pair problem:
3.11/1.55	The symbols in {fib_1, sel_2, fib1_2, s_1, add_2} are replacing on all positions.
3.11/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
3.11/1.55	
3.11/1.55	The TRS P consists of the following rules:
3.11/1.55	none
3.11/1.55	
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.11/1.55	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.11/1.55	   add(0, X) -> X
3.11/1.55	   add(s(X), Y) -> s(add(X, Y))
3.11/1.55	   sel(0, cons(X, XS)) -> X
3.11/1.55	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.11/1.55	
3.11/1.55	The set Q consists of the following terms:
3.11/1.55	
3.11/1.55	   fib(x0)
3.11/1.55	   fib1(x0, x1)
3.11/1.55	   add(0, x0)
3.11/1.55	   add(s(x0), x1)
3.11/1.55	   sel(0, cons(x0, x1))
3.11/1.55	   sel(s(x0), cons(x1, x2))
3.11/1.55	
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(20) PIsEmptyProof (EQUIVALENT)
3.11/1.55	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(21)
3.11/1.55	YES
3.34/1.58	EOF