3.91/1.81	YES
3.91/1.82	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.91/1.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.91/1.82	
3.91/1.82	
3.91/1.82	Termination of the given CSR could be proven:
3.91/1.82	
3.91/1.82	(0) CSR
3.91/1.82	(1) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.91/1.82	(2) QCSDP
3.91/1.82	(3) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.91/1.82	(4) AND
3.91/1.82	    (5) QCSDP
3.91/1.82	        (6) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.91/1.82	        (7) QCSDP
3.91/1.82	        (8) PIsEmptyProof [EQUIVALENT, 0 ms]
3.91/1.82	        (9) YES
3.91/1.82	    (10) QCSDP
3.91/1.82	        (11) QCSDPReductionPairProof [EQUIVALENT, 37 ms]
3.91/1.82	        (12) QCSDP
3.91/1.82	        (13) PIsEmptyProof [EQUIVALENT, 0 ms]
3.91/1.82	        (14) YES
3.91/1.82	    (15) QCSDP
3.91/1.82	        (16) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.91/1.82	        (17) QCSDP
3.91/1.82	        (18) PIsEmptyProof [EQUIVALENT, 0 ms]
3.91/1.82	        (19) YES
3.91/1.82	    (20) QCSDP
3.91/1.82	        (21) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.91/1.82	        (22) QCSDP
3.91/1.82	        (23) PIsEmptyProof [EQUIVALENT, 0 ms]
3.91/1.82	        (24) YES
3.91/1.82	
3.91/1.82	
3.91/1.82	----------------------------------------
3.91/1.82	
3.91/1.82	(0)
3.91/1.82	Obligation:
3.91/1.82	Context-sensitive rewrite system:
3.91/1.82	The TRS R consists of the following rules:
3.91/1.83	
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   sel(0, cons(X, XS)) -> X
3.91/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	The replacement map contains the following entries:
3.91/1.83	
3.91/1.83	from: {1}
3.91/1.83	cons: {1}
3.91/1.83	s: {1}
3.91/1.83	sel: {1, 2}
3.91/1.83	0: empty set
3.91/1.83	minus: {1, 2}
3.91/1.83	quot: {1, 2}
3.91/1.83	zWquot: {1, 2}
3.91/1.83	nil: empty set
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(1) CSDependencyPairsProof (EQUIVALENT)
3.91/1.83	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(2)
3.91/1.83	Obligation:
3.91/1.83	Q-restricted context-sensitive dependency pair problem:
3.91/1.83	The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, SEL_2, MINUS_2, QUOT_2, ZWQUOT_2, FROM_1} are replacing on all positions.
3.91/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.91/1.83	The symbols in {U_1} are not replacing on any position.
3.91/1.83	
3.91/1.83	The ordinary context-sensitive dependency pairs DP_o are:
3.91/1.83	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
3.91/1.83	   MINUS(s(X), s(Y)) -> MINUS(X, Y)
3.91/1.83	   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
3.91/1.83	   QUOT(s(X), s(Y)) -> MINUS(X, Y)
3.91/1.83	   ZWQUOT(cons(X, XS), cons(Y, YS)) -> QUOT(X, Y)
3.91/1.83	
3.91/1.83	The collapsing dependency pairs are DP_c:
3.91/1.83	   SEL(s(N), cons(X, XS)) -> XS
3.91/1.83	
3.91/1.83	
3.91/1.83	The hidden terms of R are:
3.91/1.83	
3.91/1.83	   from(s(x0))
3.91/1.83	   zWquot(x0, x1)
3.91/1.83	
3.91/1.83	Every hiding context is built from:
3.91/1.83	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@1b97b9b8
3.91/1.83	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@55316d92
3.91/1.83	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@1ac8b46
3.91/1.83	
3.91/1.83	Hence, the new unhiding pairs DP_u are :
3.91/1.83	   SEL(s(N), cons(X, XS)) -> U(XS)
3.91/1.83	   U(s(x_0)) -> U(x_0)
3.91/1.83	   U(from(x_0)) -> U(x_0)
3.91/1.83	   U(zWquot(x_0, x_1)) -> U(x_0)
3.91/1.83	   U(zWquot(x_0, x_1)) -> U(x_1)
3.91/1.83	   U(from(s(x0))) -> FROM(s(x0))
3.91/1.83	   U(zWquot(x0, x1)) -> ZWQUOT(x0, x1)
3.91/1.83	
3.91/1.83	The TRS R consists of the following rules:
3.91/1.83	
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   sel(0, cons(X, XS)) -> X
3.91/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	Q is empty.
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(3) QCSDependencyGraphProof (EQUIVALENT)
3.91/1.83	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 5 less nodes.
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(4)
3.91/1.83	Complex Obligation (AND)
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(5)
3.91/1.83	Obligation:
3.91/1.83	Q-restricted context-sensitive dependency pair problem:
3.91/1.83	The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, MINUS_2} are replacing on all positions.
3.91/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.91/1.83	
3.91/1.83	The TRS P consists of the following rules:
3.91/1.83	
3.91/1.83	   MINUS(s(X), s(Y)) -> MINUS(X, Y)
3.91/1.83	
3.91/1.83	The TRS R consists of the following rules:
3.91/1.83	
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   sel(0, cons(X, XS)) -> X
3.91/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	Q is empty.
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(6) QCSDPSubtermProof (EQUIVALENT)
3.91/1.83	We use the subterm processor [DA_EMMES].
3.91/1.83	
3.91/1.83	
3.91/1.83	The following pairs can be oriented strictly and are deleted.
3.91/1.83	
3.91/1.83	   MINUS(s(X), s(Y)) -> MINUS(X, Y)
3.91/1.83	The remaining pairs can at least be oriented weakly.
3.91/1.83	none
3.91/1.83	Used ordering:  Combined order from the following AFS and order.
3.91/1.83	MINUS(x1, x2)  =  x1
3.91/1.83	
3.91/1.83	
3.91/1.83	Subterm Order
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(7)
3.91/1.83	Obligation:
3.91/1.83	Q-restricted context-sensitive dependency pair problem:
3.91/1.83	The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
3.91/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.91/1.83	
3.91/1.83	The TRS P consists of the following rules:
3.91/1.83	none
3.91/1.83	
3.91/1.83	The TRS R consists of the following rules:
3.91/1.83	
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   sel(0, cons(X, XS)) -> X
3.91/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	Q is empty.
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(8) PIsEmptyProof (EQUIVALENT)
3.91/1.83	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(9)
3.91/1.83	YES
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(10)
3.91/1.83	Obligation:
3.91/1.83	Q-restricted context-sensitive dependency pair problem:
3.91/1.83	The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, QUOT_2} are replacing on all positions.
3.91/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.91/1.83	
3.91/1.83	The TRS P consists of the following rules:
3.91/1.83	
3.91/1.83	   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
3.91/1.83	
3.91/1.83	The TRS R consists of the following rules:
3.91/1.83	
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   sel(0, cons(X, XS)) -> X
3.91/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	Q is empty.
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(11) QCSDPReductionPairProof (EQUIVALENT)
3.91/1.83	Using the order
3.91/1.83	
3.91/1.83	Polynomial Order [NEGPOLO,POLO] with Interpretation:
3.91/1.83	
3.91/1.83	POL( minus_2(x_1, x_2) ) = 1
3.91/1.83	POL( 0 ) = 1
3.91/1.83	POL( s_1(x_1) ) = 2
3.91/1.83	POL( quot_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 1}
3.91/1.83	POL( from_1(x_1) ) = 2
3.91/1.83	POL( cons_2(x_1, x_2) ) = 2
3.91/1.83	POL( zWquot_2(x_1, x_2) ) = 2x_1 + 2x_2
3.91/1.83	POL( nil ) = 0
3.91/1.83	POL( QUOT_2(x_1, x_2) ) = 2x_1 + x_2 + 1
3.91/1.83	
3.91/1.83	
3.91/1.83	the following usable rules
3.91/1.83	
3.91/1.83	
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	
3.91/1.83	could all be oriented weakly.
3.91/1.83	
3.91/1.83	Furthermore, the pairs
3.91/1.83	
3.91/1.83	
3.91/1.83	   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
3.91/1.83	
3.91/1.83	
3.91/1.83	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
3.91/1.83	
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(12)
3.91/1.83	Obligation:
3.91/1.83	Q-restricted context-sensitive dependency pair problem:
3.91/1.83	The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
3.91/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.91/1.83	
3.91/1.83	The TRS P consists of the following rules:
3.91/1.83	none
3.91/1.83	
3.91/1.83	The TRS R consists of the following rules:
3.91/1.83	
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   sel(0, cons(X, XS)) -> X
3.91/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	Q is empty.
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(13) PIsEmptyProof (EQUIVALENT)
3.91/1.83	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(14)
3.91/1.83	YES
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(15)
3.91/1.83	Obligation:
3.91/1.83	Q-restricted context-sensitive dependency pair problem:
3.91/1.83	The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
3.91/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.91/1.83	The symbols in {U_1} are not replacing on any position.
3.91/1.83	
3.91/1.83	The TRS P consists of the following rules:
3.91/1.83	
3.91/1.83	   U(s(x_0)) -> U(x_0)
3.91/1.83	   U(from(x_0)) -> U(x_0)
3.91/1.83	   U(zWquot(x_0, x_1)) -> U(x_0)
3.91/1.83	   U(zWquot(x_0, x_1)) -> U(x_1)
3.91/1.83	
3.91/1.83	The TRS R consists of the following rules:
3.91/1.83	
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   sel(0, cons(X, XS)) -> X
3.91/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	Q is empty.
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(16) QCSDPSubtermProof (EQUIVALENT)
3.91/1.83	We use the subterm processor [DA_EMMES].
3.91/1.83	
3.91/1.83	
3.91/1.83	The following pairs can be oriented strictly and are deleted.
3.91/1.83	
3.91/1.83	   U(s(x_0)) -> U(x_0)
3.91/1.83	   U(from(x_0)) -> U(x_0)
3.91/1.83	   U(zWquot(x_0, x_1)) -> U(x_0)
3.91/1.83	   U(zWquot(x_0, x_1)) -> U(x_1)
3.91/1.83	The remaining pairs can at least be oriented weakly.
3.91/1.83	none
3.91/1.83	Used ordering:  Combined order from the following AFS and order.
3.91/1.83	U(x1)  =  x1
3.91/1.83	
3.91/1.83	
3.91/1.83	Subterm Order
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(17)
3.91/1.83	Obligation:
3.91/1.83	Q-restricted context-sensitive dependency pair problem:
3.91/1.83	The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
3.91/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.91/1.83	
3.91/1.83	The TRS P consists of the following rules:
3.91/1.83	none
3.91/1.83	
3.91/1.83	The TRS R consists of the following rules:
3.91/1.83	
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   sel(0, cons(X, XS)) -> X
3.91/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	Q is empty.
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(18) PIsEmptyProof (EQUIVALENT)
3.91/1.83	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(19)
3.91/1.83	YES
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(20)
3.91/1.83	Obligation:
3.91/1.83	Q-restricted context-sensitive dependency pair problem:
3.91/1.83	The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, SEL_2} are replacing on all positions.
3.91/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.91/1.83	
3.91/1.83	The TRS P consists of the following rules:
3.91/1.83	
3.91/1.83	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
3.91/1.83	
3.91/1.83	The TRS R consists of the following rules:
3.91/1.83	
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   sel(0, cons(X, XS)) -> X
3.91/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	Q is empty.
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(21) QCSDPSubtermProof (EQUIVALENT)
3.91/1.83	We use the subterm processor [DA_EMMES].
3.91/1.83	
3.91/1.83	
3.91/1.83	The following pairs can be oriented strictly and are deleted.
3.91/1.83	
3.91/1.83	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
3.91/1.83	The remaining pairs can at least be oriented weakly.
3.91/1.83	none
3.91/1.83	Used ordering:  Combined order from the following AFS and order.
3.91/1.83	SEL(x1, x2)  =  x1
3.91/1.83	
3.91/1.83	
3.91/1.83	Subterm Order
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(22)
3.91/1.83	Obligation:
3.91/1.83	Q-restricted context-sensitive dependency pair problem:
3.91/1.83	The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
3.91/1.83	For all symbols f in {cons_2} we have mu(f) = {1}.
3.91/1.83	
3.91/1.83	The TRS P consists of the following rules:
3.91/1.83	none
3.91/1.83	
3.91/1.83	The TRS R consists of the following rules:
3.91/1.83	
3.91/1.83	   from(X) -> cons(X, from(s(X)))
3.91/1.83	   sel(0, cons(X, XS)) -> X
3.91/1.83	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.91/1.83	   minus(X, 0) -> 0
3.91/1.83	   minus(s(X), s(Y)) -> minus(X, Y)
3.91/1.83	   quot(0, s(Y)) -> 0
3.91/1.83	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.91/1.83	   zWquot(XS, nil) -> nil
3.91/1.83	   zWquot(nil, XS) -> nil
3.91/1.83	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
3.91/1.83	
3.91/1.83	Q is empty.
3.91/1.83	
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(23) PIsEmptyProof (EQUIVALENT)
3.91/1.83	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.91/1.83	----------------------------------------
3.91/1.83	
3.91/1.83	(24)
3.91/1.83	YES
4.21/1.86	EOF