4.41/1.77	YES
4.41/1.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.41/1.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.41/1.78	
4.41/1.78	
4.41/1.78	Termination of the given CSR could be proven:
4.41/1.78	
4.41/1.78	(0) CSR
4.41/1.78	(1) CSDependencyPairsProof [EQUIVALENT, 21 ms]
4.41/1.78	(2) QCSDP
4.41/1.78	(3) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
4.41/1.78	(4) QCSDP
4.41/1.78	(5) QCSDPReductionPairProof [EQUIVALENT, 35 ms]
4.41/1.78	(6) QCSDP
4.41/1.78	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
4.41/1.78	(8) QCSDP
4.41/1.78	(9) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.41/1.78	(10) QCSDP
4.41/1.78	(11) PIsEmptyProof [EQUIVALENT, 0 ms]
4.41/1.78	(12) YES
4.41/1.78	
4.41/1.78	
4.41/1.78	----------------------------------------
4.41/1.78	
4.41/1.78	(0)
4.41/1.78	Obligation:
4.41/1.78	Context-sensitive rewrite system:
4.41/1.78	The TRS R consists of the following rules:
4.41/1.78	
4.41/1.78	   cons_1(x, cons_1(y, z)) -> big_0
4.41/1.78	   cons_1(x, cons_0(y, z)) -> big_0
4.41/1.78	   *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0)
4.41/1.78	   cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x))))
4.41/1.78	
4.41/1.78	The replacement map contains the following entries:
4.41/1.78	
4.41/1.78	cons_1: empty set
4.41/1.78	big_0: empty set
4.41/1.78	cons_0: {1, 2}
4.41/1.78	*top*_0: {1}
4.41/1.78	inf_1: empty set
4.41/1.78	s_0: {1}
4.41/1.78	
4.41/1.78	----------------------------------------
4.41/1.78	
4.41/1.78	(1) CSDependencyPairsProof (EQUIVALENT)
4.41/1.78	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
4.41/1.78	----------------------------------------
4.41/1.78	
4.41/1.78	(2)
4.41/1.78	Obligation:
4.41/1.78	Q-restricted context-sensitive dependency pair problem:
4.41/1.78	The symbols in {cons_0_2, *top*_0_1, s_0_1, *TOP*_0_1, CONS_0_2, S_0_1} are replacing on all positions.
4.41/1.78	The symbols in {cons_1_2, inf_1_1, CONS_1_2, U_1} are not replacing on any position.
4.41/1.78	
4.41/1.78	The ordinary context-sensitive dependency pairs DP_o are:
4.41/1.78	   *TOP*_0(inf_1(x)) -> *TOP*_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   *TOP*_0(inf_1(x)) -> CONS_0(x, inf_1(s_0(x)))
4.41/1.78	   S_0(inf_1(x)) -> S_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   S_0(inf_1(x)) -> CONS_0(x, inf_1(s_0(x)))
4.41/1.78	   CONS_0(inf_1(x), x0) -> CONS_0(cons_0(x, inf_1(s_0(x))), x0)
4.41/1.78	   CONS_0(inf_1(x), x0) -> CONS_0(x, inf_1(s_0(x)))
4.41/1.78	   CONS_0(x0, inf_1(x)) -> CONS_1(x0, cons_0(x, inf_1(s_0(x))))
4.41/1.78	
4.41/1.78	The collapsing dependency pairs are DP_c:
4.41/1.78	   *TOP*_0(inf_1(x)) -> x
4.41/1.78	   S_0(inf_1(x)) -> x
4.41/1.78	   CONS_0(inf_1(x), x0) -> x
4.41/1.78	
4.41/1.78	
4.41/1.78	The hidden terms of R are:
4.41/1.78	
4.41/1.78	   s_0(x0)
4.41/1.78	   cons_0(x0, inf_1(s_0(x0)))
4.41/1.78	
4.41/1.78	Every hiding context is built from:
4.41/1.78	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@3cee6e0c
4.41/1.78	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@68903072
4.41/1.78	
4.41/1.78	Hence, the new unhiding pairs DP_u are :
4.41/1.78	   *TOP*_0(inf_1(x)) -> U(x)
4.41/1.78	   S_0(inf_1(x)) -> U(x)
4.41/1.78	   CONS_0(inf_1(x), x0) -> U(x)
4.41/1.78	   U(s_0(x_0)) -> U(x_0)
4.41/1.78	   U(cons_0(x_0, x_1)) -> U(x_0)
4.41/1.78	   U(s_0(x0)) -> S_0(x0)
4.41/1.78	   U(cons_0(x0, inf_1(s_0(x0)))) -> CONS_0(x0, inf_1(s_0(x0)))
4.41/1.78	
4.41/1.78	The TRS R consists of the following rules:
4.41/1.78	
4.41/1.78	   cons_1(x, cons_1(y, z)) -> big_0
4.41/1.78	   cons_1(x, cons_0(y, z)) -> big_0
4.41/1.78	   *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0)
4.41/1.78	   cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x))))
4.41/1.78	
4.41/1.78	Q is empty.
4.41/1.78	
4.41/1.78	----------------------------------------
4.41/1.78	
4.41/1.78	(3) QCSDependencyGraphProof (EQUIVALENT)
4.41/1.78	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 5 less nodes.
4.41/1.78	The rules *TOP*_0(inf_1(x0)) -> *TOP*_0(cons_0(x0, inf_1(s_0(x0)))) and *TOP*_0(inf_1(z0)) -> *TOP*_0(cons_0(z0, inf_1(s_0(z0)))) form no chain, because ECap^mu_R'(*TOP*_0(inf_1(z0))) = *TOP*_0(inf_1(z0)) does not unify with *TOP*_0(cons_0(x0, inf_1(s_0(x0)))). 
4.41/1.78	R' = 
4.41/1.78	   ( big_0, cons_1(x, cons_1(y, z)))
4.41/1.78	   ( big_0, cons_1(x, cons_0(y, z)))
4.41/1.78	   ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x)))
4.41/1.78	   ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x)))
4.41/1.78	   ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0))
4.41/1.78	   ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x)))
4.41/1.78	
4.41/1.78	
4.41/1.78	The rules *TOP*_0(inf_1(x0)) -> *TOP*_0(cons_0(x0, inf_1(s_0(x0)))) and *TOP*_0(inf_1(z0)) -> CONS_0(z0, inf_1(s_0(z0))) form no chain, because ECap^mu_R'(*TOP*_0(inf_1(z0))) = *TOP*_0(inf_1(z0)) does not unify with *TOP*_0(cons_0(x0, inf_1(s_0(x0)))). 
4.41/1.78	R' = 
4.41/1.78	   ( big_0, cons_1(x, cons_1(y, z)))
4.41/1.78	   ( big_0, cons_1(x, cons_0(y, z)))
4.41/1.78	   ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x)))
4.41/1.78	   ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x)))
4.41/1.78	   ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0))
4.41/1.78	   ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x)))
4.41/1.78	
4.41/1.78	
4.41/1.78	The rules *TOP*_0(inf_1(x0)) -> *TOP*_0(cons_0(x0, inf_1(s_0(x0)))) and *TOP*_0(inf_1(z0)) -> U(z0) form no chain, because ECap^mu_R'(*TOP*_0(inf_1(z0))) = *TOP*_0(inf_1(z0)) does not unify with *TOP*_0(cons_0(x0, inf_1(s_0(x0)))). 
4.41/1.78	R' = 
4.41/1.78	   ( big_0, cons_1(x, cons_1(y, z)))
4.41/1.78	   ( big_0, cons_1(x, cons_0(y, z)))
4.41/1.78	   ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x)))
4.41/1.78	   ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x)))
4.41/1.78	   ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0))
4.41/1.78	   ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x)))
4.41/1.78	
4.41/1.78	
4.41/1.78	The rules S_0(inf_1(x0)) -> S_0(cons_0(x0, inf_1(s_0(x0)))) and S_0(inf_1(z0)) -> S_0(cons_0(z0, inf_1(s_0(z0)))) form no chain, because ECap^mu_R'(S_0(inf_1(z0))) = S_0(inf_1(z0)) does not unify with S_0(cons_0(x0, inf_1(s_0(x0)))). 
4.41/1.78	R' = 
4.41/1.78	   ( big_0, cons_1(x, cons_1(y, z)))
4.41/1.78	   ( big_0, cons_1(x, cons_0(y, z)))
4.41/1.78	   ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x)))
4.41/1.78	   ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x)))
4.41/1.78	   ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0))
4.41/1.78	   ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x)))
4.41/1.78	
4.41/1.78	
4.41/1.78	The rules S_0(inf_1(x0)) -> S_0(cons_0(x0, inf_1(s_0(x0)))) and S_0(inf_1(z0)) -> CONS_0(z0, inf_1(s_0(z0))) form no chain, because ECap^mu_R'(S_0(inf_1(z0))) = S_0(inf_1(z0)) does not unify with S_0(cons_0(x0, inf_1(s_0(x0)))). 
4.41/1.78	R' = 
4.41/1.78	   ( big_0, cons_1(x, cons_1(y, z)))
4.41/1.78	   ( big_0, cons_1(x, cons_0(y, z)))
4.41/1.78	   ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x)))
4.41/1.78	   ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x)))
4.41/1.78	   ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0))
4.41/1.78	   ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x)))
4.41/1.78	
4.41/1.78	
4.41/1.78	The rules S_0(inf_1(x0)) -> S_0(cons_0(x0, inf_1(s_0(x0)))) and S_0(inf_1(z0)) -> U(z0) form no chain, because ECap^mu_R'(S_0(inf_1(z0))) = S_0(inf_1(z0)) does not unify with S_0(cons_0(x0, inf_1(s_0(x0)))). 
4.41/1.78	R' = 
4.41/1.78	   ( big_0, cons_1(x, cons_1(y, z)))
4.41/1.78	   ( big_0, cons_1(x, cons_0(y, z)))
4.41/1.78	   ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x)))
4.41/1.78	   ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x)))
4.41/1.78	   ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0))
4.41/1.78	   ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x)))
4.41/1.78	
4.41/1.78	
4.41/1.78	The rules CONS_0(inf_1(x0), x1) -> CONS_0(cons_0(x0, inf_1(s_0(x0))), x1) and CONS_0(inf_1(z0), z1) -> CONS_0(cons_0(z0, inf_1(s_0(z0))), z1) form no chain, because ECap^mu_R'(CONS_0(inf_1(z0), z1)) = CONS_0(inf_1(z0), x_1) does not unify with CONS_0(cons_0(x0, inf_1(s_0(x0))), x1). 
4.41/1.78	R' = 
4.41/1.78	   ( big_0, cons_1(x, cons_1(y, z)))
4.41/1.78	   ( big_0, cons_1(x, cons_0(y, z)))
4.41/1.78	   ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x)))
4.41/1.78	   ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x)))
4.41/1.78	   ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0))
4.41/1.78	   ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x)))
4.41/1.78	
4.41/1.78	
4.41/1.78	The rules CONS_0(inf_1(x0), x1) -> CONS_0(cons_0(x0, inf_1(s_0(x0))), x1) and CONS_0(inf_1(z0), z1) -> CONS_0(z0, inf_1(s_0(z0))) form no chain, because ECap^mu_R'(CONS_0(inf_1(z0), z1)) = CONS_0(inf_1(z0), x_1) does not unify with CONS_0(cons_0(x0, inf_1(s_0(x0))), x1). 
4.41/1.78	R' = 
4.41/1.78	   ( big_0, cons_1(x, cons_1(y, z)))
4.41/1.78	   ( big_0, cons_1(x, cons_0(y, z)))
4.41/1.78	   ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x)))
4.41/1.78	   ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x)))
4.41/1.78	   ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0))
4.41/1.78	   ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x)))
4.41/1.78	
4.41/1.78	
4.41/1.78	The rules CONS_0(inf_1(x0), x1) -> CONS_0(cons_0(x0, inf_1(s_0(x0))), x1) and CONS_0(inf_1(z0), z1) -> U(z0) form no chain, because ECap^mu_R'(CONS_0(inf_1(z0), z1)) = CONS_0(inf_1(z0), x_1) does not unify with CONS_0(cons_0(x0, inf_1(s_0(x0))), x1). 
4.41/1.78	R' = 
4.41/1.78	   ( big_0, cons_1(x, cons_1(y, z)))
4.41/1.78	   ( big_0, cons_1(x, cons_0(y, z)))
4.41/1.78	   ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x)))
4.41/1.78	   ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x)))
4.41/1.78	   ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0))
4.41/1.78	   ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x)))
4.41/1.78	
4.41/1.78	
4.41/1.78	
4.41/1.78	----------------------------------------
4.41/1.78	
4.41/1.78	(4)
4.41/1.78	Obligation:
4.41/1.78	Q-restricted context-sensitive dependency pair problem:
4.41/1.78	The symbols in {cons_0_2, *top*_0_1, s_0_1, CONS_0_2, S_0_1} are replacing on all positions.
4.41/1.78	The symbols in {cons_1_2, inf_1_1, U_1} are not replacing on any position.
4.41/1.78	
4.41/1.78	The TRS P consists of the following rules:
4.41/1.78	
4.41/1.78	   CONS_0(inf_1(x), x0) -> CONS_0(x, inf_1(s_0(x)))
4.41/1.78	   CONS_0(inf_1(x), x0) -> U(x)
4.41/1.78	   U(s_0(x_0)) -> U(x_0)
4.41/1.78	   U(cons_0(x_0, x_1)) -> U(x_0)
4.41/1.78	   U(s_0(x0)) -> S_0(x0)
4.41/1.78	   S_0(inf_1(x)) -> CONS_0(x, inf_1(s_0(x)))
4.41/1.78	   S_0(inf_1(x)) -> U(x)
4.41/1.78	   U(cons_0(x0, inf_1(s_0(x0)))) -> CONS_0(x0, inf_1(s_0(x0)))
4.41/1.78	
4.41/1.78	The TRS R consists of the following rules:
4.41/1.78	
4.41/1.78	   cons_1(x, cons_1(y, z)) -> big_0
4.41/1.78	   cons_1(x, cons_0(y, z)) -> big_0
4.41/1.78	   *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0)
4.41/1.78	   cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x))))
4.41/1.78	
4.41/1.78	Q is empty.
4.41/1.78	
4.41/1.78	----------------------------------------
4.41/1.78	
4.41/1.78	(5) QCSDPReductionPairProof (EQUIVALENT)
4.41/1.78	Using the order
4.41/1.78	
4.41/1.78	Polynomial interpretation with max and min functions [POLO,MAXPOLO]:
4.41/1.78	
4.41/1.78	   POL(CONS_0(x_1, x_2)) = x_1
4.41/1.78	   POL(S_0(x_1)) = x_1
4.41/1.78	   POL(U(x_1)) = x_1
4.41/1.78	   POL(big_0) = 0
4.41/1.78	   POL(cons_0(x_1, x_2)) = x_1
4.41/1.78	   POL(cons_1(x_1, x_2)) = x_1
4.41/1.78	   POL(inf_1(x_1)) = 1 + x_1
4.41/1.78	   POL(s_0(x_1)) = x_1
4.41/1.78	
4.41/1.78	
4.41/1.78	the following usable rules
4.41/1.78	
4.41/1.78	
4.41/1.78	   s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0)
4.41/1.78	   cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x))))
4.41/1.78	   cons_1(x, cons_1(y, z)) -> big_0
4.41/1.78	   cons_1(x, cons_0(y, z)) -> big_0
4.41/1.78	
4.41/1.78	
4.41/1.78	could all be oriented weakly.
4.41/1.78	
4.41/1.78	Furthermore, the pairs
4.41/1.78	
4.41/1.78	
4.41/1.78	   CONS_0(inf_1(x), x0) -> CONS_0(x, inf_1(s_0(x)))
4.41/1.78	   CONS_0(inf_1(x), x0) -> U(x)
4.41/1.78	   S_0(inf_1(x)) -> CONS_0(x, inf_1(s_0(x)))
4.41/1.78	   S_0(inf_1(x)) -> U(x)
4.41/1.78	
4.41/1.78	
4.41/1.78	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
4.41/1.78	
4.41/1.78	
4.41/1.78	----------------------------------------
4.41/1.78	
4.41/1.78	(6)
4.41/1.78	Obligation:
4.41/1.78	Q-restricted context-sensitive dependency pair problem:
4.41/1.78	The symbols in {cons_0_2, *top*_0_1, s_0_1, S_0_1, CONS_0_2} are replacing on all positions.
4.41/1.78	The symbols in {cons_1_2, inf_1_1, U_1} are not replacing on any position.
4.41/1.78	
4.41/1.78	The TRS P consists of the following rules:
4.41/1.78	
4.41/1.78	   U(s_0(x_0)) -> U(x_0)
4.41/1.78	   U(cons_0(x_0, x_1)) -> U(x_0)
4.41/1.78	   U(s_0(x0)) -> S_0(x0)
4.41/1.78	   U(cons_0(x0, inf_1(s_0(x0)))) -> CONS_0(x0, inf_1(s_0(x0)))
4.41/1.78	
4.41/1.78	The TRS R consists of the following rules:
4.41/1.78	
4.41/1.78	   cons_1(x, cons_1(y, z)) -> big_0
4.41/1.78	   cons_1(x, cons_0(y, z)) -> big_0
4.41/1.78	   *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0)
4.41/1.78	   cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x))))
4.41/1.78	
4.41/1.78	Q is empty.
4.41/1.78	
4.41/1.78	----------------------------------------
4.41/1.78	
4.41/1.78	(7) QCSDependencyGraphProof (EQUIVALENT)
4.41/1.78	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 2 less nodes.
4.41/1.78	
4.41/1.78	----------------------------------------
4.41/1.78	
4.41/1.78	(8)
4.41/1.78	Obligation:
4.41/1.78	Q-restricted context-sensitive dependency pair problem:
4.41/1.78	The symbols in {cons_0_2, *top*_0_1, s_0_1} are replacing on all positions.
4.41/1.78	The symbols in {cons_1_2, inf_1_1, U_1} are not replacing on any position.
4.41/1.78	
4.41/1.78	The TRS P consists of the following rules:
4.41/1.78	
4.41/1.78	   U(s_0(x_0)) -> U(x_0)
4.41/1.78	   U(cons_0(x_0, x_1)) -> U(x_0)
4.41/1.78	
4.41/1.78	The TRS R consists of the following rules:
4.41/1.78	
4.41/1.78	   cons_1(x, cons_1(y, z)) -> big_0
4.41/1.78	   cons_1(x, cons_0(y, z)) -> big_0
4.41/1.78	   *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x))))
4.41/1.78	   cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0)
4.41/1.78	   cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x))))
4.41/1.78	
4.41/1.78	Q is empty.
4.41/1.78	
4.41/1.78	----------------------------------------
4.41/1.78	
4.41/1.78	(9) QCSDPSubtermProof (EQUIVALENT)
4.41/1.78	We use the subterm processor [DA_EMMES].
4.41/1.78	
4.41/1.78	
4.41/1.78	The following pairs can be oriented strictly and are deleted.
4.41/1.78	
4.41/1.78	   U(s_0(x_0)) -> U(x_0)
4.41/1.78	   U(cons_0(x_0, x_1)) -> U(x_0)
4.50/1.78	The remaining pairs can at least be oriented weakly.
4.50/1.78	none
4.50/1.78	Used ordering:  Combined order from the following AFS and order.
4.50/1.78	U(x1)  =  x1
4.50/1.78	
4.50/1.78	
4.50/1.78	Subterm Order
4.50/1.78	
4.50/1.78	----------------------------------------
4.50/1.78	
4.50/1.78	(10)
4.50/1.78	Obligation:
4.50/1.78	Q-restricted context-sensitive dependency pair problem:
4.50/1.78	The symbols in {cons_0_2, *top*_0_1, s_0_1} are replacing on all positions.
4.50/1.78	The symbols in {cons_1_2, inf_1_1} are not replacing on any position.
4.50/1.78	
4.50/1.78	The TRS P consists of the following rules:
4.50/1.78	none
4.50/1.78	
4.50/1.78	The TRS R consists of the following rules:
4.50/1.78	
4.50/1.78	   cons_1(x, cons_1(y, z)) -> big_0
4.50/1.78	   cons_1(x, cons_0(y, z)) -> big_0
4.50/1.78	   *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x))))
4.50/1.78	   s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x))))
4.50/1.78	   cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0)
4.50/1.78	   cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x))))
4.50/1.78	
4.50/1.78	Q is empty.
4.50/1.78	
4.50/1.78	----------------------------------------
4.50/1.78	
4.50/1.78	(11) PIsEmptyProof (EQUIVALENT)
4.50/1.78	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
4.50/1.78	----------------------------------------
4.50/1.78	
4.50/1.78	(12)
4.50/1.78	YES
4.50/1.80	EOF