3.23/1.64	YES
3.23/1.66	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.23/1.66	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.23/1.66	
3.23/1.66	
3.23/1.66	Termination of the given CSR could be proven:
3.23/1.66	
3.23/1.66	(0) CSR
3.23/1.66	(1) CSRInnermostProof [EQUIVALENT, 0 ms]
3.23/1.66	(2) CSR
3.23/1.66	(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.23/1.66	(4) QCSDP
3.23/1.66	(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.23/1.66	(6) AND
3.23/1.66	    (7) QCSDP
3.23/1.66	        (8) QCSDPSubtermProof [EQUIVALENT, 16 ms]
3.23/1.66	        (9) QCSDP
3.23/1.66	        (10) PIsEmptyProof [EQUIVALENT, 0 ms]
3.23/1.66	        (11) YES
3.23/1.66	    (12) QCSDP
3.23/1.66	        (13) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.23/1.66	        (14) QCSDP
3.23/1.66	        (15) PIsEmptyProof [EQUIVALENT, 0 ms]
3.23/1.66	        (16) YES
3.23/1.66	    (17) QCSDP
3.23/1.66	        (18) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.23/1.66	        (19) QCSDP
3.23/1.66	        (20) PIsEmptyProof [EQUIVALENT, 0 ms]
3.23/1.66	        (21) YES
3.23/1.66	    (22) QCSDP
3.23/1.66	        (23) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.23/1.66	        (24) QCSDP
3.23/1.66	        (25) PIsEmptyProof [EQUIVALENT, 0 ms]
3.23/1.66	        (26) YES
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(0)
3.23/1.66	Obligation:
3.23/1.66	Context-sensitive rewrite system:
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The replacement map contains the following entries:
3.23/1.66	
3.23/1.66	from: {1}
3.23/1.66	cons: {1}
3.23/1.66	s: {1}
3.23/1.66	2ndspos: {1, 2}
3.23/1.66	0: empty set
3.23/1.66	rnil: empty set
3.23/1.66	rcons: {1, 2}
3.23/1.66	posrecip: {1}
3.23/1.66	2ndsneg: {1, 2}
3.23/1.66	negrecip: {1}
3.23/1.66	pi: {1}
3.23/1.66	plus: {1, 2}
3.23/1.66	times: {1, 2}
3.23/1.66	square: {1}
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(1) CSRInnermostProof (EQUIVALENT)
3.23/1.66	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(2)
3.23/1.66	Obligation:
3.23/1.66	Context-sensitive rewrite system:
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The replacement map contains the following entries:
3.23/1.66	
3.23/1.66	from: {1}
3.23/1.66	cons: {1}
3.23/1.66	s: {1}
3.23/1.66	2ndspos: {1, 2}
3.23/1.66	0: empty set
3.23/1.66	rnil: empty set
3.23/1.66	rcons: {1, 2}
3.23/1.66	posrecip: {1}
3.23/1.66	2ndsneg: {1, 2}
3.23/1.66	negrecip: {1}
3.23/1.66	pi: {1}
3.23/1.66	plus: {1, 2}
3.23/1.66	times: {1, 2}
3.23/1.66	square: {1}
3.23/1.66	
3.23/1.66	
3.23/1.66	Innermost Strategy.
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(3) CSDependencyPairsProof (EQUIVALENT)
3.23/1.66	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(4)
3.23/1.66	Obligation:
3.23/1.66	Q-restricted context-sensitive dependency pair problem:
3.23/1.66	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, PI_1, FROM_1, PLUS_2, TIMES_2, SQUARE_1} are replacing on all positions.
3.23/1.66	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.66	The symbols in {U_1} are not replacing on any position.
3.23/1.66	
3.23/1.66	The ordinary context-sensitive dependency pairs DP_o are:
3.23/1.66	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> 2NDSNEG(N, Z)
3.23/1.66	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> 2NDSPOS(N, Z)
3.23/1.66	   PI(X) -> 2NDSPOS(X, from(0))
3.23/1.66	   PI(X) -> FROM(0)
3.23/1.66	   PLUS(s(X), Y) -> PLUS(X, Y)
3.23/1.66	   TIMES(s(X), Y) -> PLUS(Y, times(X, Y))
3.23/1.66	   TIMES(s(X), Y) -> TIMES(X, Y)
3.23/1.66	   SQUARE(X) -> TIMES(X, X)
3.23/1.66	
3.23/1.66	The collapsing dependency pairs are DP_c:
3.23/1.66	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> Y
3.23/1.66	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> Z
3.23/1.66	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> Y
3.23/1.66	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> Z
3.23/1.66	
3.23/1.66	
3.23/1.66	The hidden terms of R are:
3.23/1.66	
3.23/1.66	   from(s(x0))
3.23/1.66	
3.23/1.66	Every hiding context is built from:
3.23/1.66	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@1b522222
3.23/1.66	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@8b3d607
3.23/1.66	
3.23/1.66	Hence, the new unhiding pairs DP_u are :
3.23/1.66	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> U(Y)
3.23/1.66	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> U(Z)
3.23/1.66	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> U(Y)
3.23/1.66	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> U(Z)
3.23/1.66	   U(s(x_0)) -> U(x_0)
3.23/1.66	   U(from(x_0)) -> U(x_0)
3.23/1.66	   U(from(s(x0))) -> FROM(s(x0))
3.23/1.66	
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The set Q consists of the following terms:
3.23/1.66	
3.23/1.66	   from(x0)
3.23/1.66	   2ndspos(0, x0)
3.23/1.66	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   2ndsneg(0, x0)
3.23/1.66	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   pi(x0)
3.23/1.66	   plus(0, x0)
3.23/1.66	   plus(s(x0), x1)
3.23/1.66	   times(0, x0)
3.23/1.66	   times(s(x0), x1)
3.23/1.66	   square(x0)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(5) QCSDependencyGraphProof (EQUIVALENT)
3.23/1.66	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 7 less nodes.
3.23/1.66	The rules PI(x0) -> 2NDSPOS(x0, from(0)) and 2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> 2NDSNEG(z0, z3) form no chain, because ECap^mu_R'(2NDSPOS(s(z0), cons(z1, cons(z2, z3)))) = 2NDSPOS(s(x_1), cons(x_2, cons(z2, z3))) does not unify with 2NDSPOS(x0, from(0)). 
3.23/1.66	R' = 
3.23/1.66	   ( cons(X, from(s(X))), from(X))
3.23/1.66	
3.23/1.66	
3.23/1.66	The rules PI(x0) -> 2NDSPOS(x0, from(0)) and 2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> U(z2) form no chain, because ECap^mu_R'(2NDSPOS(s(z0), cons(z1, cons(z2, z3)))) = 2NDSPOS(s(x_1), cons(x_2, cons(z2, z3))) does not unify with 2NDSPOS(x0, from(0)). 
3.23/1.66	R' = 
3.23/1.66	   ( cons(X, from(s(X))), from(X))
3.23/1.66	
3.23/1.66	
3.23/1.66	The rules PI(x0) -> 2NDSPOS(x0, from(0)) and 2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> U(z3) form no chain, because ECap^mu_R'(2NDSPOS(s(z0), cons(z1, cons(z2, z3)))) = 2NDSPOS(s(x_1), cons(x_2, cons(z2, z3))) does not unify with 2NDSPOS(x0, from(0)). 
3.23/1.66	R' = 
3.23/1.66	   ( cons(X, from(s(X))), from(X))
3.23/1.66	
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(6)
3.23/1.66	Complex Obligation (AND)
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(7)
3.23/1.66	Obligation:
3.23/1.66	Q-restricted context-sensitive dependency pair problem:
3.23/1.66	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.
3.23/1.66	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.66	The symbols in {U_1} are not replacing on any position.
3.23/1.66	
3.23/1.66	The TRS P consists of the following rules:
3.23/1.66	
3.23/1.66	   U(s(x_0)) -> U(x_0)
3.23/1.66	   U(from(x_0)) -> U(x_0)
3.23/1.66	
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The set Q consists of the following terms:
3.23/1.66	
3.23/1.66	   from(x0)
3.23/1.66	   2ndspos(0, x0)
3.23/1.66	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   2ndsneg(0, x0)
3.23/1.66	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   pi(x0)
3.23/1.66	   plus(0, x0)
3.23/1.66	   plus(s(x0), x1)
3.23/1.66	   times(0, x0)
3.23/1.66	   times(s(x0), x1)
3.23/1.66	   square(x0)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(8) QCSDPSubtermProof (EQUIVALENT)
3.23/1.66	We use the subterm processor [DA_EMMES].
3.23/1.66	
3.23/1.66	
3.23/1.66	The following pairs can be oriented strictly and are deleted.
3.23/1.66	
3.23/1.66	   U(s(x_0)) -> U(x_0)
3.23/1.66	   U(from(x_0)) -> U(x_0)
3.23/1.66	The remaining pairs can at least be oriented weakly.
3.23/1.66	none
3.23/1.66	Used ordering:  Combined order from the following AFS and order.
3.23/1.66	U(x1)  =  x1
3.23/1.66	
3.23/1.66	
3.23/1.66	Subterm Order
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(9)
3.23/1.66	Obligation:
3.23/1.66	Q-restricted context-sensitive dependency pair problem:
3.23/1.66	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.
3.23/1.66	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.66	
3.23/1.66	The TRS P consists of the following rules:
3.23/1.66	none
3.23/1.66	
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The set Q consists of the following terms:
3.23/1.66	
3.23/1.66	   from(x0)
3.23/1.66	   2ndspos(0, x0)
3.23/1.66	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   2ndsneg(0, x0)
3.23/1.66	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   pi(x0)
3.23/1.66	   plus(0, x0)
3.23/1.66	   plus(s(x0), x1)
3.23/1.66	   times(0, x0)
3.23/1.66	   times(s(x0), x1)
3.23/1.66	   square(x0)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(10) PIsEmptyProof (EQUIVALENT)
3.23/1.66	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(11)
3.23/1.66	YES
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(12)
3.23/1.66	Obligation:
3.23/1.66	Q-restricted context-sensitive dependency pair problem:
3.23/1.66	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.
3.23/1.66	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.66	
3.23/1.66	The TRS P consists of the following rules:
3.23/1.66	
3.23/1.66	   PLUS(s(X), Y) -> PLUS(X, Y)
3.23/1.66	
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The set Q consists of the following terms:
3.23/1.66	
3.23/1.66	   from(x0)
3.23/1.66	   2ndspos(0, x0)
3.23/1.66	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   2ndsneg(0, x0)
3.23/1.66	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   pi(x0)
3.23/1.66	   plus(0, x0)
3.23/1.66	   plus(s(x0), x1)
3.23/1.66	   times(0, x0)
3.23/1.66	   times(s(x0), x1)
3.23/1.66	   square(x0)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(13) QCSDPSubtermProof (EQUIVALENT)
3.23/1.66	We use the subterm processor [DA_EMMES].
3.23/1.66	
3.23/1.66	
3.23/1.66	The following pairs can be oriented strictly and are deleted.
3.23/1.66	
3.23/1.66	   PLUS(s(X), Y) -> PLUS(X, Y)
3.23/1.66	The remaining pairs can at least be oriented weakly.
3.23/1.66	none
3.23/1.66	Used ordering:  Combined order from the following AFS and order.
3.23/1.66	PLUS(x1, x2)  =  x1
3.23/1.66	
3.23/1.66	
3.23/1.66	Subterm Order
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(14)
3.23/1.66	Obligation:
3.23/1.66	Q-restricted context-sensitive dependency pair problem:
3.23/1.66	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.
3.23/1.66	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.66	
3.23/1.66	The TRS P consists of the following rules:
3.23/1.66	none
3.23/1.66	
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The set Q consists of the following terms:
3.23/1.66	
3.23/1.66	   from(x0)
3.23/1.66	   2ndspos(0, x0)
3.23/1.66	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   2ndsneg(0, x0)
3.23/1.66	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   pi(x0)
3.23/1.66	   plus(0, x0)
3.23/1.66	   plus(s(x0), x1)
3.23/1.66	   times(0, x0)
3.23/1.66	   times(s(x0), x1)
3.23/1.66	   square(x0)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(15) PIsEmptyProof (EQUIVALENT)
3.23/1.66	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(16)
3.23/1.66	YES
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(17)
3.23/1.66	Obligation:
3.23/1.66	Q-restricted context-sensitive dependency pair problem:
3.23/1.66	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.
3.23/1.66	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.66	
3.23/1.66	The TRS P consists of the following rules:
3.23/1.66	
3.23/1.66	   TIMES(s(X), Y) -> TIMES(X, Y)
3.23/1.66	
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The set Q consists of the following terms:
3.23/1.66	
3.23/1.66	   from(x0)
3.23/1.66	   2ndspos(0, x0)
3.23/1.66	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   2ndsneg(0, x0)
3.23/1.66	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   pi(x0)
3.23/1.66	   plus(0, x0)
3.23/1.66	   plus(s(x0), x1)
3.23/1.66	   times(0, x0)
3.23/1.66	   times(s(x0), x1)
3.23/1.66	   square(x0)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(18) QCSDPSubtermProof (EQUIVALENT)
3.23/1.66	We use the subterm processor [DA_EMMES].
3.23/1.66	
3.23/1.66	
3.23/1.66	The following pairs can be oriented strictly and are deleted.
3.23/1.66	
3.23/1.66	   TIMES(s(X), Y) -> TIMES(X, Y)
3.23/1.66	The remaining pairs can at least be oriented weakly.
3.23/1.66	none
3.23/1.66	Used ordering:  Combined order from the following AFS and order.
3.23/1.66	TIMES(x1, x2)  =  x1
3.23/1.66	
3.23/1.66	
3.23/1.66	Subterm Order
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(19)
3.23/1.66	Obligation:
3.23/1.66	Q-restricted context-sensitive dependency pair problem:
3.23/1.66	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.
3.23/1.66	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.66	
3.23/1.66	The TRS P consists of the following rules:
3.23/1.66	none
3.23/1.66	
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The set Q consists of the following terms:
3.23/1.66	
3.23/1.66	   from(x0)
3.23/1.66	   2ndspos(0, x0)
3.23/1.66	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   2ndsneg(0, x0)
3.23/1.66	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   pi(x0)
3.23/1.66	   plus(0, x0)
3.23/1.66	   plus(s(x0), x1)
3.23/1.66	   times(0, x0)
3.23/1.66	   times(s(x0), x1)
3.23/1.66	   square(x0)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(20) PIsEmptyProof (EQUIVALENT)
3.23/1.66	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(21)
3.23/1.66	YES
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(22)
3.23/1.66	Obligation:
3.23/1.66	Q-restricted context-sensitive dependency pair problem:
3.23/1.66	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} are replacing on all positions.
3.23/1.66	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.66	
3.23/1.66	The TRS P consists of the following rules:
3.23/1.66	
3.23/1.66	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> 2NDSPOS(N, Z)
3.23/1.66	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> 2NDSNEG(N, Z)
3.23/1.66	
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The set Q consists of the following terms:
3.23/1.66	
3.23/1.66	   from(x0)
3.23/1.66	   2ndspos(0, x0)
3.23/1.66	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   2ndsneg(0, x0)
3.23/1.66	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   pi(x0)
3.23/1.66	   plus(0, x0)
3.23/1.66	   plus(s(x0), x1)
3.23/1.66	   times(0, x0)
3.23/1.66	   times(s(x0), x1)
3.23/1.66	   square(x0)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(23) QCSDPSubtermProof (EQUIVALENT)
3.23/1.66	We use the subterm processor [DA_EMMES].
3.23/1.66	
3.23/1.66	
3.23/1.66	The following pairs can be oriented strictly and are deleted.
3.23/1.66	
3.23/1.66	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> 2NDSPOS(N, Z)
3.23/1.66	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> 2NDSNEG(N, Z)
3.23/1.66	The remaining pairs can at least be oriented weakly.
3.23/1.66	none
3.23/1.66	Used ordering:  Combined order from the following AFS and order.
3.23/1.66	2NDSPOS(x1, x2)  =  x1
3.23/1.66	
3.23/1.66	2NDSNEG(x1, x2)  =  x1
3.23/1.66	
3.23/1.66	
3.23/1.66	Subterm Order
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(24)
3.23/1.66	Obligation:
3.23/1.66	Q-restricted context-sensitive dependency pair problem:
3.23/1.66	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.
3.23/1.66	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.66	
3.23/1.66	The TRS P consists of the following rules:
3.23/1.66	none
3.23/1.66	
3.23/1.66	The TRS R consists of the following rules:
3.23/1.66	
3.23/1.66	   from(X) -> cons(X, from(s(X)))
3.23/1.66	   2ndspos(0, Z) -> rnil
3.23/1.66	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
3.23/1.66	   2ndsneg(0, Z) -> rnil
3.23/1.66	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
3.23/1.66	   pi(X) -> 2ndspos(X, from(0))
3.23/1.66	   plus(0, Y) -> Y
3.23/1.66	   plus(s(X), Y) -> s(plus(X, Y))
3.23/1.66	   times(0, Y) -> 0
3.23/1.66	   times(s(X), Y) -> plus(Y, times(X, Y))
3.23/1.66	   square(X) -> times(X, X)
3.23/1.66	
3.23/1.66	The set Q consists of the following terms:
3.23/1.66	
3.23/1.66	   from(x0)
3.23/1.66	   2ndspos(0, x0)
3.23/1.66	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   2ndsneg(0, x0)
3.23/1.66	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
3.23/1.66	   pi(x0)
3.23/1.66	   plus(0, x0)
3.23/1.66	   plus(s(x0), x1)
3.23/1.66	   times(0, x0)
3.23/1.66	   times(s(x0), x1)
3.23/1.66	   square(x0)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(25) PIsEmptyProof (EQUIVALENT)
3.23/1.66	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(26)
3.23/1.66	YES
3.51/1.69	EOF