3.53/1.59	YES
3.53/1.60	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.53/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.53/1.60	
3.53/1.60	
3.53/1.60	Termination of the given CSR could be proven:
3.53/1.60	
3.53/1.60	(0) CSR
3.53/1.60	(1) CSRInnermostProof [EQUIVALENT, 0 ms]
3.53/1.60	(2) CSR
3.53/1.60	(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.53/1.60	(4) QCSDP
3.53/1.60	(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.53/1.60	(6) AND
3.53/1.60	    (7) QCSDP
3.53/1.60	        (8) QCSDPSubtermProof [EQUIVALENT, 16 ms]
3.53/1.60	        (9) QCSDP
3.53/1.60	        (10) PIsEmptyProof [EQUIVALENT, 0 ms]
3.53/1.60	        (11) YES
3.53/1.60	    (12) QCSDP
3.53/1.60	        (13) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.53/1.60	        (14) QCSDP
3.53/1.60	        (15) PIsEmptyProof [EQUIVALENT, 0 ms]
3.53/1.60	        (16) YES
3.53/1.60	    (17) QCSDP
3.53/1.60	        (18) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.53/1.60	        (19) QCSDP
3.53/1.60	        (20) PIsEmptyProof [EQUIVALENT, 0 ms]
3.53/1.60	        (21) YES
3.53/1.60	    (22) QCSDP
3.53/1.60	        (23) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.53/1.60	        (24) QCSDP
3.53/1.60	        (25) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.53/1.60	        (26) TRUE
3.53/1.60	
3.53/1.60	
3.53/1.60	----------------------------------------
3.53/1.60	
3.53/1.60	(0)
3.53/1.60	Obligation:
3.53/1.60	Context-sensitive rewrite system:
3.53/1.60	The TRS R consists of the following rules:
3.53/1.60	
3.53/1.60	   from(X) -> cons(X, from(s(X)))
3.53/1.60	   2ndspos(0, Z) -> rnil
3.53/1.60	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.60	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.60	   2ndsneg(0, Z) -> rnil
3.53/1.60	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.60	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.60	   pi(X) -> 2ndspos(X, from(0))
3.53/1.60	   plus(0, Y) -> Y
3.53/1.60	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.60	   times(0, Y) -> 0
3.53/1.60	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.60	   square(X) -> times(X, X)
3.53/1.60	
3.53/1.60	The replacement map contains the following entries:
3.53/1.60	
3.53/1.60	from: {1}
3.53/1.60	cons: {1}
3.53/1.60	s: {1}
3.53/1.60	2ndspos: {1, 2}
3.53/1.60	0: empty set
3.53/1.60	rnil: empty set
3.53/1.60	cons2: {2}
3.53/1.60	rcons: {1, 2}
3.53/1.60	posrecip: {1}
3.53/1.60	2ndsneg: {1, 2}
3.53/1.60	negrecip: {1}
3.53/1.60	pi: {1}
3.53/1.60	plus: {1, 2}
3.53/1.60	times: {1, 2}
3.53/1.60	square: {1}
3.53/1.60	
3.53/1.60	----------------------------------------
3.53/1.60	
3.53/1.60	(1) CSRInnermostProof (EQUIVALENT)
3.53/1.60	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.53/1.60	----------------------------------------
3.53/1.60	
3.53/1.60	(2)
3.53/1.60	Obligation:
3.53/1.60	Context-sensitive rewrite system:
3.53/1.60	The TRS R consists of the following rules:
3.53/1.60	
3.53/1.60	   from(X) -> cons(X, from(s(X)))
3.53/1.60	   2ndspos(0, Z) -> rnil
3.53/1.60	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.60	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.60	   2ndsneg(0, Z) -> rnil
3.53/1.60	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.60	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.60	   pi(X) -> 2ndspos(X, from(0))
3.53/1.60	   plus(0, Y) -> Y
3.53/1.60	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.60	   times(0, Y) -> 0
3.53/1.60	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.60	   square(X) -> times(X, X)
3.53/1.60	
3.53/1.60	The replacement map contains the following entries:
3.53/1.60	
3.53/1.60	from: {1}
3.53/1.60	cons: {1}
3.53/1.60	s: {1}
3.53/1.60	2ndspos: {1, 2}
3.53/1.60	0: empty set
3.53/1.60	rnil: empty set
3.53/1.60	cons2: {2}
3.53/1.60	rcons: {1, 2}
3.53/1.60	posrecip: {1}
3.53/1.60	2ndsneg: {1, 2}
3.53/1.60	negrecip: {1}
3.53/1.60	pi: {1}
3.53/1.60	plus: {1, 2}
3.53/1.60	times: {1, 2}
3.53/1.60	square: {1}
3.53/1.60	
3.53/1.60	
3.53/1.60	Innermost Strategy.
3.53/1.60	
3.53/1.60	----------------------------------------
3.53/1.60	
3.53/1.60	(3) CSDependencyPairsProof (EQUIVALENT)
3.53/1.60	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.53/1.60	----------------------------------------
3.53/1.60	
3.53/1.60	(4)
3.53/1.60	Obligation:
3.53/1.60	Q-restricted context-sensitive dependency pair problem:
3.53/1.60	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1, 2NDSPOS_2, 2NDSNEG_2, PI_1, FROM_1, PLUS_2, TIMES_2, SQUARE_1} are replacing on all positions.
3.53/1.60	For all symbols f in {cons_2} we have mu(f) = {1}.
3.53/1.60	For all symbols f in {cons2_2} we have mu(f) = {2}.
3.53/1.60	The symbols in {U_1} are not replacing on any position.
3.53/1.60	
3.53/1.60	The ordinary context-sensitive dependency pairs DP_o are:
3.53/1.60	   2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, Z))
3.53/1.60	   2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, Z)
3.53/1.60	   2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, Z))
3.53/1.60	   2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, Z)
3.53/1.60	   PI(X) -> 2NDSPOS(X, from(0))
3.53/1.60	   PI(X) -> FROM(0)
3.53/1.60	   PLUS(s(X), Y) -> PLUS(X, Y)
3.53/1.60	   TIMES(s(X), Y) -> PLUS(Y, times(X, Y))
3.53/1.60	   TIMES(s(X), Y) -> TIMES(X, Y)
3.53/1.60	   SQUARE(X) -> TIMES(X, X)
3.53/1.60	
3.53/1.60	The collapsing dependency pairs are DP_c:
3.53/1.60	   2NDSPOS(s(N), cons(X, Z)) -> Z
3.53/1.60	   2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> Z
3.53/1.60	   2NDSNEG(s(N), cons(X, Z)) -> Z
3.53/1.60	   2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> Z
3.53/1.60	
3.53/1.60	
3.53/1.60	The hidden terms of R are:
3.53/1.60	
3.53/1.60	   from(s(x0))
3.53/1.60	
3.53/1.60	Every hiding context is built from:
3.53/1.60	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@db8adb4
3.53/1.60	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@438a962b
3.53/1.60	
3.53/1.60	Hence, the new unhiding pairs DP_u are :
3.53/1.60	   2NDSPOS(s(N), cons(X, Z)) -> U(Z)
3.53/1.60	   2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> U(Z)
3.53/1.60	   2NDSNEG(s(N), cons(X, Z)) -> U(Z)
3.53/1.60	   2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> U(Z)
3.53/1.60	   U(s(x_0)) -> U(x_0)
3.53/1.60	   U(from(x_0)) -> U(x_0)
3.53/1.60	   U(from(s(x0))) -> FROM(s(x0))
3.53/1.60	
3.53/1.60	The TRS R consists of the following rules:
3.53/1.60	
3.53/1.60	   from(X) -> cons(X, from(s(X)))
3.53/1.60	   2ndspos(0, Z) -> rnil
3.53/1.60	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.60	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.60	   2ndsneg(0, Z) -> rnil
3.53/1.60	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.60	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.60	   pi(X) -> 2ndspos(X, from(0))
3.53/1.60	   plus(0, Y) -> Y
3.53/1.60	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.60	   times(0, Y) -> 0
3.53/1.60	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.60	   square(X) -> times(X, X)
3.53/1.60	
3.53/1.60	The set Q consists of the following terms:
3.53/1.60	
3.53/1.60	   from(x0)
3.53/1.60	   2ndspos(0, x0)
3.53/1.60	   2ndspos(s(x0), cons(x1, x2))
3.53/1.60	   2ndspos(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.60	   2ndsneg(0, x0)
3.53/1.60	   2ndsneg(s(x0), cons(x1, x2))
3.53/1.60	   2ndsneg(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.60	   pi(x0)
3.53/1.60	   plus(0, x0)
3.53/1.60	   plus(s(x0), x1)
3.53/1.60	   times(0, x0)
3.53/1.60	   times(s(x0), x1)
3.53/1.60	   square(x0)
3.53/1.60	
3.53/1.60	
3.53/1.60	----------------------------------------
3.53/1.60	
3.53/1.60	(5) QCSDependencyGraphProof (EQUIVALENT)
3.53/1.60	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 8 less nodes.
3.53/1.60	The rules 2NDSPOS(s(z0), cons(z1, z2)) -> 2NDSPOS(s(z0), cons2(z1, z2)) and 2NDSPOS(s(x0), cons(x1, x2)) -> 2NDSPOS(s(x0), cons2(x1, x2)) form no chain, because ECap^mu(2NDSPOS(s(z0), cons2(z1, z2))) = 2NDSPOS(s(z0), cons2(z1, x_1)) does not unify with 2NDSPOS(s(x0), cons(x1, x2)). The rules 2NDSPOS(s(z0), cons(z1, z2)) -> 2NDSPOS(s(z0), cons2(z1, z2)) and 2NDSPOS(s(x0), cons(x1, x2)) -> U(x2) form no chain, because ECap^mu(2NDSPOS(s(z0), cons2(z1, z2))) = 2NDSPOS(s(z0), cons2(z1, x_1)) does not unify with 2NDSPOS(s(x0), cons(x1, x2)). The rules 2NDSNEG(s(z0), cons(z1, z2)) -> 2NDSNEG(s(z0), cons2(z1, z2)) and 2NDSNEG(s(x0), cons(x1, x2)) -> 2NDSNEG(s(x0), cons2(x1, x2)) form no chain, because ECap^mu(2NDSNEG(s(z0), cons2(z1, z2))) = 2NDSNEG(s(z0), cons2(z1, x_1)) does not unify with 2NDSNEG(s(x0), cons(x1, x2)). The rules 2NDSNEG(s(z0), cons(z1, z2)) -> 2NDSNEG(s(z0), cons2(z1, z2)) and 2NDSNEG(s(x0), cons(x1, x2)) -> U(x2) form no chain, because ECap^mu(2NDSNEG(s(z0), cons2(z1, z2))) = 2NDSNEG(s(z0), cons2(z1, x_1)) does not unify with 2NDSNEG(s(x0), cons(x1, x2)). The rules PI(x0) -> 2NDSPOS(x0, from(0)) and 2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> 2NDSNEG(z0, z3) form no chain, because ECap^mu_R'(2NDSPOS(s(z0), cons2(z1, cons(z2, z3)))) = 2NDSPOS(s(x_1), cons2(z1, x_3)) does not unify with 2NDSPOS(x0, from(0)). 
3.53/1.60	R' = 
3.53/1.60	   ( cons(X, from(s(X))), from(X))
3.53/1.60	
3.53/1.60	
3.53/1.60	The rules PI(x0) -> 2NDSPOS(x0, from(0)) and 2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> U(z3) form no chain, because ECap^mu_R'(2NDSPOS(s(z0), cons2(z1, cons(z2, z3)))) = 2NDSPOS(s(x_1), cons2(z1, x_3)) does not unify with 2NDSPOS(x0, from(0)). 
3.53/1.60	R' = 
3.53/1.60	   ( cons(X, from(s(X))), from(X))
3.53/1.60	
3.53/1.60	
3.53/1.60	
3.53/1.60	----------------------------------------
3.53/1.60	
3.53/1.60	(6)
3.53/1.60	Complex Obligation (AND)
3.53/1.60	
3.53/1.60	----------------------------------------
3.53/1.60	
3.53/1.60	(7)
3.53/1.60	Obligation:
3.53/1.60	Q-restricted context-sensitive dependency pair problem:
3.53/1.60	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1} are replacing on all positions.
3.53/1.60	For all symbols f in {cons_2} we have mu(f) = {1}.
3.53/1.60	For all symbols f in {cons2_2} we have mu(f) = {2}.
3.53/1.60	The symbols in {U_1} are not replacing on any position.
3.53/1.60	
3.53/1.60	The TRS P consists of the following rules:
3.53/1.60	
3.53/1.60	   U(s(x_0)) -> U(x_0)
3.53/1.60	   U(from(x_0)) -> U(x_0)
3.53/1.60	
3.53/1.60	The TRS R consists of the following rules:
3.53/1.60	
3.53/1.60	   from(X) -> cons(X, from(s(X)))
3.53/1.61	   2ndspos(0, Z) -> rnil
3.53/1.61	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.61	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.61	   2ndsneg(0, Z) -> rnil
3.53/1.61	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.61	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.61	   pi(X) -> 2ndspos(X, from(0))
3.53/1.61	   plus(0, Y) -> Y
3.53/1.61	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.61	   times(0, Y) -> 0
3.53/1.61	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.61	   square(X) -> times(X, X)
3.53/1.61	
3.53/1.61	The set Q consists of the following terms:
3.53/1.61	
3.53/1.61	   from(x0)
3.53/1.61	   2ndspos(0, x0)
3.53/1.61	   2ndspos(s(x0), cons(x1, x2))
3.53/1.61	   2ndspos(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   2ndsneg(0, x0)
3.53/1.61	   2ndsneg(s(x0), cons(x1, x2))
3.53/1.61	   2ndsneg(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   pi(x0)
3.53/1.61	   plus(0, x0)
3.53/1.61	   plus(s(x0), x1)
3.53/1.61	   times(0, x0)
3.53/1.61	   times(s(x0), x1)
3.53/1.61	   square(x0)
3.53/1.61	
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(8) QCSDPSubtermProof (EQUIVALENT)
3.53/1.61	We use the subterm processor [DA_EMMES].
3.53/1.61	
3.53/1.61	
3.53/1.61	The following pairs can be oriented strictly and are deleted.
3.53/1.61	
3.53/1.61	   U(s(x_0)) -> U(x_0)
3.53/1.61	   U(from(x_0)) -> U(x_0)
3.53/1.61	The remaining pairs can at least be oriented weakly.
3.53/1.61	none
3.53/1.61	Used ordering:  Combined order from the following AFS and order.
3.53/1.61	U(x1)  =  x1
3.53/1.61	
3.53/1.61	
3.53/1.61	Subterm Order
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(9)
3.53/1.61	Obligation:
3.53/1.61	Q-restricted context-sensitive dependency pair problem:
3.53/1.61	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1} are replacing on all positions.
3.53/1.61	For all symbols f in {cons_2} we have mu(f) = {1}.
3.53/1.61	For all symbols f in {cons2_2} we have mu(f) = {2}.
3.53/1.61	
3.53/1.61	The TRS P consists of the following rules:
3.53/1.61	none
3.53/1.61	
3.53/1.61	The TRS R consists of the following rules:
3.53/1.61	
3.53/1.61	   from(X) -> cons(X, from(s(X)))
3.53/1.61	   2ndspos(0, Z) -> rnil
3.53/1.61	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.61	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.61	   2ndsneg(0, Z) -> rnil
3.53/1.61	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.61	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.61	   pi(X) -> 2ndspos(X, from(0))
3.53/1.61	   plus(0, Y) -> Y
3.53/1.61	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.61	   times(0, Y) -> 0
3.53/1.61	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.61	   square(X) -> times(X, X)
3.53/1.61	
3.53/1.61	The set Q consists of the following terms:
3.53/1.61	
3.53/1.61	   from(x0)
3.53/1.61	   2ndspos(0, x0)
3.53/1.61	   2ndspos(s(x0), cons(x1, x2))
3.53/1.61	   2ndspos(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   2ndsneg(0, x0)
3.53/1.61	   2ndsneg(s(x0), cons(x1, x2))
3.53/1.61	   2ndsneg(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   pi(x0)
3.53/1.61	   plus(0, x0)
3.53/1.61	   plus(s(x0), x1)
3.53/1.61	   times(0, x0)
3.53/1.61	   times(s(x0), x1)
3.53/1.61	   square(x0)
3.53/1.61	
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(10) PIsEmptyProof (EQUIVALENT)
3.53/1.61	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(11)
3.53/1.61	YES
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(12)
3.53/1.61	Obligation:
3.53/1.61	Q-restricted context-sensitive dependency pair problem:
3.53/1.61	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1, PLUS_2} are replacing on all positions.
3.53/1.61	For all symbols f in {cons_2} we have mu(f) = {1}.
3.53/1.61	For all symbols f in {cons2_2} we have mu(f) = {2}.
3.53/1.61	
3.53/1.61	The TRS P consists of the following rules:
3.53/1.61	
3.53/1.61	   PLUS(s(X), Y) -> PLUS(X, Y)
3.53/1.61	
3.53/1.61	The TRS R consists of the following rules:
3.53/1.61	
3.53/1.61	   from(X) -> cons(X, from(s(X)))
3.53/1.61	   2ndspos(0, Z) -> rnil
3.53/1.61	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.61	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.61	   2ndsneg(0, Z) -> rnil
3.53/1.61	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.61	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.61	   pi(X) -> 2ndspos(X, from(0))
3.53/1.61	   plus(0, Y) -> Y
3.53/1.61	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.61	   times(0, Y) -> 0
3.53/1.61	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.61	   square(X) -> times(X, X)
3.53/1.61	
3.53/1.61	The set Q consists of the following terms:
3.53/1.61	
3.53/1.61	   from(x0)
3.53/1.61	   2ndspos(0, x0)
3.53/1.61	   2ndspos(s(x0), cons(x1, x2))
3.53/1.61	   2ndspos(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   2ndsneg(0, x0)
3.53/1.61	   2ndsneg(s(x0), cons(x1, x2))
3.53/1.61	   2ndsneg(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   pi(x0)
3.53/1.61	   plus(0, x0)
3.53/1.61	   plus(s(x0), x1)
3.53/1.61	   times(0, x0)
3.53/1.61	   times(s(x0), x1)
3.53/1.61	   square(x0)
3.53/1.61	
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(13) QCSDPSubtermProof (EQUIVALENT)
3.53/1.61	We use the subterm processor [DA_EMMES].
3.53/1.61	
3.53/1.61	
3.53/1.61	The following pairs can be oriented strictly and are deleted.
3.53/1.61	
3.53/1.61	   PLUS(s(X), Y) -> PLUS(X, Y)
3.53/1.61	The remaining pairs can at least be oriented weakly.
3.53/1.61	none
3.53/1.61	Used ordering:  Combined order from the following AFS and order.
3.53/1.61	PLUS(x1, x2)  =  x1
3.53/1.61	
3.53/1.61	
3.53/1.61	Subterm Order
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(14)
3.53/1.61	Obligation:
3.53/1.61	Q-restricted context-sensitive dependency pair problem:
3.53/1.61	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1} are replacing on all positions.
3.53/1.61	For all symbols f in {cons_2} we have mu(f) = {1}.
3.53/1.61	For all symbols f in {cons2_2} we have mu(f) = {2}.
3.53/1.61	
3.53/1.61	The TRS P consists of the following rules:
3.53/1.61	none
3.53/1.61	
3.53/1.61	The TRS R consists of the following rules:
3.53/1.61	
3.53/1.61	   from(X) -> cons(X, from(s(X)))
3.53/1.61	   2ndspos(0, Z) -> rnil
3.53/1.61	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.61	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.61	   2ndsneg(0, Z) -> rnil
3.53/1.61	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.61	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.61	   pi(X) -> 2ndspos(X, from(0))
3.53/1.61	   plus(0, Y) -> Y
3.53/1.61	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.61	   times(0, Y) -> 0
3.53/1.61	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.61	   square(X) -> times(X, X)
3.53/1.61	
3.53/1.61	The set Q consists of the following terms:
3.53/1.61	
3.53/1.61	   from(x0)
3.53/1.61	   2ndspos(0, x0)
3.53/1.61	   2ndspos(s(x0), cons(x1, x2))
3.53/1.61	   2ndspos(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   2ndsneg(0, x0)
3.53/1.61	   2ndsneg(s(x0), cons(x1, x2))
3.53/1.61	   2ndsneg(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   pi(x0)
3.53/1.61	   plus(0, x0)
3.53/1.61	   plus(s(x0), x1)
3.53/1.61	   times(0, x0)
3.53/1.61	   times(s(x0), x1)
3.53/1.61	   square(x0)
3.53/1.61	
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(15) PIsEmptyProof (EQUIVALENT)
3.53/1.61	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(16)
3.53/1.61	YES
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(17)
3.53/1.61	Obligation:
3.53/1.61	Q-restricted context-sensitive dependency pair problem:
3.53/1.61	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1, TIMES_2} are replacing on all positions.
3.53/1.61	For all symbols f in {cons_2} we have mu(f) = {1}.
3.53/1.61	For all symbols f in {cons2_2} we have mu(f) = {2}.
3.53/1.61	
3.53/1.61	The TRS P consists of the following rules:
3.53/1.61	
3.53/1.61	   TIMES(s(X), Y) -> TIMES(X, Y)
3.53/1.61	
3.53/1.61	The TRS R consists of the following rules:
3.53/1.61	
3.53/1.61	   from(X) -> cons(X, from(s(X)))
3.53/1.61	   2ndspos(0, Z) -> rnil
3.53/1.61	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.61	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.61	   2ndsneg(0, Z) -> rnil
3.53/1.61	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.61	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.61	   pi(X) -> 2ndspos(X, from(0))
3.53/1.61	   plus(0, Y) -> Y
3.53/1.61	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.61	   times(0, Y) -> 0
3.53/1.61	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.61	   square(X) -> times(X, X)
3.53/1.61	
3.53/1.61	The set Q consists of the following terms:
3.53/1.61	
3.53/1.61	   from(x0)
3.53/1.61	   2ndspos(0, x0)
3.53/1.61	   2ndspos(s(x0), cons(x1, x2))
3.53/1.61	   2ndspos(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   2ndsneg(0, x0)
3.53/1.61	   2ndsneg(s(x0), cons(x1, x2))
3.53/1.61	   2ndsneg(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   pi(x0)
3.53/1.61	   plus(0, x0)
3.53/1.61	   plus(s(x0), x1)
3.53/1.61	   times(0, x0)
3.53/1.61	   times(s(x0), x1)
3.53/1.61	   square(x0)
3.53/1.61	
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(18) QCSDPSubtermProof (EQUIVALENT)
3.53/1.61	We use the subterm processor [DA_EMMES].
3.53/1.61	
3.53/1.61	
3.53/1.61	The following pairs can be oriented strictly and are deleted.
3.53/1.61	
3.53/1.61	   TIMES(s(X), Y) -> TIMES(X, Y)
3.53/1.61	The remaining pairs can at least be oriented weakly.
3.53/1.61	none
3.53/1.61	Used ordering:  Combined order from the following AFS and order.
3.53/1.61	TIMES(x1, x2)  =  x1
3.53/1.61	
3.53/1.61	
3.53/1.61	Subterm Order
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(19)
3.53/1.61	Obligation:
3.53/1.61	Q-restricted context-sensitive dependency pair problem:
3.53/1.61	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1} are replacing on all positions.
3.53/1.61	For all symbols f in {cons_2} we have mu(f) = {1}.
3.53/1.61	For all symbols f in {cons2_2} we have mu(f) = {2}.
3.53/1.61	
3.53/1.61	The TRS P consists of the following rules:
3.53/1.61	none
3.53/1.61	
3.53/1.61	The TRS R consists of the following rules:
3.53/1.61	
3.53/1.61	   from(X) -> cons(X, from(s(X)))
3.53/1.61	   2ndspos(0, Z) -> rnil
3.53/1.61	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.61	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.61	   2ndsneg(0, Z) -> rnil
3.53/1.61	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.61	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.61	   pi(X) -> 2ndspos(X, from(0))
3.53/1.61	   plus(0, Y) -> Y
3.53/1.61	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.61	   times(0, Y) -> 0
3.53/1.61	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.61	   square(X) -> times(X, X)
3.53/1.61	
3.53/1.61	The set Q consists of the following terms:
3.53/1.61	
3.53/1.61	   from(x0)
3.53/1.61	   2ndspos(0, x0)
3.53/1.61	   2ndspos(s(x0), cons(x1, x2))
3.53/1.61	   2ndspos(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   2ndsneg(0, x0)
3.53/1.61	   2ndsneg(s(x0), cons(x1, x2))
3.53/1.61	   2ndsneg(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   pi(x0)
3.53/1.61	   plus(0, x0)
3.53/1.61	   plus(s(x0), x1)
3.53/1.61	   times(0, x0)
3.53/1.61	   times(s(x0), x1)
3.53/1.61	   square(x0)
3.53/1.61	
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(20) PIsEmptyProof (EQUIVALENT)
3.53/1.61	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(21)
3.53/1.61	YES
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(22)
3.53/1.61	Obligation:
3.53/1.61	Q-restricted context-sensitive dependency pair problem:
3.53/1.61	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1, 2NDSNEG_2, 2NDSPOS_2} are replacing on all positions.
3.53/1.61	For all symbols f in {cons_2} we have mu(f) = {1}.
3.53/1.61	For all symbols f in {cons2_2} we have mu(f) = {2}.
3.53/1.61	
3.53/1.61	The TRS P consists of the following rules:
3.53/1.61	
3.53/1.61	   2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, Z)
3.53/1.61	   2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, Z))
3.53/1.61	   2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, Z)
3.53/1.61	   2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, Z))
3.53/1.61	
3.53/1.61	The TRS R consists of the following rules:
3.53/1.61	
3.53/1.61	   from(X) -> cons(X, from(s(X)))
3.53/1.61	   2ndspos(0, Z) -> rnil
3.53/1.61	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.61	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.61	   2ndsneg(0, Z) -> rnil
3.53/1.61	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.61	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.61	   pi(X) -> 2ndspos(X, from(0))
3.53/1.61	   plus(0, Y) -> Y
3.53/1.61	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.61	   times(0, Y) -> 0
3.53/1.61	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.61	   square(X) -> times(X, X)
3.53/1.61	
3.53/1.61	The set Q consists of the following terms:
3.53/1.61	
3.53/1.61	   from(x0)
3.53/1.61	   2ndspos(0, x0)
3.53/1.61	   2ndspos(s(x0), cons(x1, x2))
3.53/1.61	   2ndspos(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   2ndsneg(0, x0)
3.53/1.61	   2ndsneg(s(x0), cons(x1, x2))
3.53/1.61	   2ndsneg(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   pi(x0)
3.53/1.61	   plus(0, x0)
3.53/1.61	   plus(s(x0), x1)
3.53/1.61	   times(0, x0)
3.53/1.61	   times(s(x0), x1)
3.53/1.61	   square(x0)
3.53/1.61	
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(23) QCSDPSubtermProof (EQUIVALENT)
3.53/1.61	We use the subterm processor [DA_EMMES].
3.53/1.61	
3.53/1.61	
3.53/1.61	The following pairs can be oriented strictly and are deleted.
3.53/1.61	
3.53/1.61	   2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, Z)
3.53/1.61	   2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, Z)
3.53/1.61	The remaining pairs can at least be oriented weakly.
3.53/1.61	
3.53/1.61	   2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, Z))
3.53/1.61	   2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, Z))
3.53/1.61	Used ordering:  Combined order from the following AFS and order.
3.53/1.61	2NDSNEG(x1, x2)  =  x1
3.53/1.61	
3.53/1.61	2NDSPOS(x1, x2)  =  x1
3.53/1.61	
3.53/1.61	
3.53/1.61	Subterm Order
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(24)
3.53/1.61	Obligation:
3.53/1.61	Q-restricted context-sensitive dependency pair problem:
3.53/1.61	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1, 2NDSNEG_2, 2NDSPOS_2} are replacing on all positions.
3.53/1.61	For all symbols f in {cons_2} we have mu(f) = {1}.
3.53/1.61	For all symbols f in {cons2_2} we have mu(f) = {2}.
3.53/1.61	
3.53/1.61	The TRS P consists of the following rules:
3.53/1.61	
3.53/1.61	   2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, Z))
3.53/1.61	   2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, Z))
3.53/1.61	
3.53/1.61	The TRS R consists of the following rules:
3.53/1.61	
3.53/1.61	   from(X) -> cons(X, from(s(X)))
3.53/1.61	   2ndspos(0, Z) -> rnil
3.53/1.61	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
3.53/1.61	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.53/1.61	   2ndsneg(0, Z) -> rnil
3.53/1.61	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
3.53/1.61	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.53/1.61	   pi(X) -> 2ndspos(X, from(0))
3.53/1.61	   plus(0, Y) -> Y
3.53/1.61	   plus(s(X), Y) -> s(plus(X, Y))
3.53/1.61	   times(0, Y) -> 0
3.53/1.61	   times(s(X), Y) -> plus(Y, times(X, Y))
3.53/1.61	   square(X) -> times(X, X)
3.53/1.61	
3.53/1.61	The set Q consists of the following terms:
3.53/1.61	
3.53/1.61	   from(x0)
3.53/1.61	   2ndspos(0, x0)
3.53/1.61	   2ndspos(s(x0), cons(x1, x2))
3.53/1.61	   2ndspos(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   2ndsneg(0, x0)
3.53/1.61	   2ndsneg(s(x0), cons(x1, x2))
3.53/1.61	   2ndsneg(s(x0), cons2(x1, cons(x2, x3)))
3.53/1.61	   pi(x0)
3.53/1.61	   plus(0, x0)
3.53/1.61	   plus(s(x0), x1)
3.53/1.61	   times(0, x0)
3.53/1.61	   times(s(x0), x1)
3.53/1.61	   square(x0)
3.53/1.61	
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(25) QCSDependencyGraphProof (EQUIVALENT)
3.53/1.61	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
3.53/1.61	
3.53/1.61	----------------------------------------
3.53/1.61	
3.53/1.61	(26)
3.53/1.61	TRUE
3.80/1.63	EOF