10.96/3.71	YES
10.96/3.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
10.96/3.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.96/3.77	
10.96/3.77	
10.96/3.77	Termination w.r.t. Q of the given QTRS could be proven:
10.96/3.77	
10.96/3.77	(0) QTRS
10.96/3.77	(1) QTRS Reverse [EQUIVALENT, 0 ms]
10.96/3.77	(2) QTRS
10.96/3.77	(3) DependencyPairsProof [EQUIVALENT, 25 ms]
10.96/3.77	(4) QDP
10.96/3.77	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
10.96/3.77	(6) AND
10.96/3.77	    (7) QDP
10.96/3.77	        (8) UsableRulesProof [EQUIVALENT, 2 ms]
10.96/3.77	        (9) QDP
10.96/3.77	        (10) QDPSizeChangeProof [EQUIVALENT, 1 ms]
10.96/3.77	        (11) YES
10.96/3.77	    (12) QDP
10.96/3.77	        (13) QDPOrderProof [EQUIVALENT, 44 ms]
10.96/3.77	        (14) QDP
10.96/3.77	        (15) PisEmptyProof [EQUIVALENT, 0 ms]
10.96/3.77	        (16) YES
10.96/3.77	
10.96/3.77	
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(0)
10.96/3.77	Obligation:
10.96/3.77	Q restricted rewrite system:
10.96/3.77	The TRS R consists of the following rules:
10.96/3.77	
10.96/3.77	   f(x1) -> n(c(c(x1)))
10.96/3.77	   c(f(x1)) -> f(c(c(x1)))
10.96/3.77	   c(c(x1)) -> c(x1)
10.96/3.77	   n(s(x1)) -> f(s(s(x1)))
10.96/3.77	   n(f(x1)) -> f(n(x1))
10.96/3.77	
10.96/3.77	Q is empty.
10.96/3.77	
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(1) QTRS Reverse (EQUIVALENT)
10.96/3.77	We applied the QTRS Reverse Processor [REVERSE].
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(2)
10.96/3.77	Obligation:
10.96/3.77	Q restricted rewrite system:
10.96/3.77	The TRS R consists of the following rules:
10.96/3.77	
10.96/3.77	   f(x1) -> c(c(n(x1)))
10.96/3.77	   f(c(x1)) -> c(c(f(x1)))
10.96/3.77	   c(c(x1)) -> c(x1)
10.96/3.77	   s(n(x1)) -> s(s(f(x1)))
10.96/3.77	   f(n(x1)) -> n(f(x1))
10.96/3.77	
10.96/3.77	Q is empty.
10.96/3.77	
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(3) DependencyPairsProof (EQUIVALENT)
10.96/3.77	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(4)
10.96/3.77	Obligation:
10.96/3.77	Q DP problem:
10.96/3.77	The TRS P consists of the following rules:
10.96/3.77	
10.96/3.77	   F(x1) -> C(c(n(x1)))
10.96/3.77	   F(x1) -> C(n(x1))
10.96/3.77	   F(c(x1)) -> C(c(f(x1)))
10.96/3.77	   F(c(x1)) -> C(f(x1))
10.96/3.77	   F(c(x1)) -> F(x1)
10.96/3.77	   S(n(x1)) -> S(s(f(x1)))
10.96/3.77	   S(n(x1)) -> S(f(x1))
10.96/3.77	   S(n(x1)) -> F(x1)
10.96/3.77	   F(n(x1)) -> F(x1)
10.96/3.77	
10.96/3.77	The TRS R consists of the following rules:
10.96/3.77	
10.96/3.77	   f(x1) -> c(c(n(x1)))
10.96/3.77	   f(c(x1)) -> c(c(f(x1)))
10.96/3.77	   c(c(x1)) -> c(x1)
10.96/3.77	   s(n(x1)) -> s(s(f(x1)))
10.96/3.77	   f(n(x1)) -> n(f(x1))
10.96/3.77	
10.96/3.77	Q is empty.
10.96/3.77	We have to consider all minimal (P,Q,R)-chains.
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(5) DependencyGraphProof (EQUIVALENT)
10.96/3.77	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes.
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(6)
10.96/3.77	Complex Obligation (AND)
10.96/3.77	
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(7)
10.96/3.77	Obligation:
10.96/3.77	Q DP problem:
10.96/3.77	The TRS P consists of the following rules:
10.96/3.77	
10.96/3.77	   F(n(x1)) -> F(x1)
10.96/3.77	   F(c(x1)) -> F(x1)
10.96/3.77	
10.96/3.77	The TRS R consists of the following rules:
10.96/3.77	
10.96/3.77	   f(x1) -> c(c(n(x1)))
10.96/3.77	   f(c(x1)) -> c(c(f(x1)))
10.96/3.77	   c(c(x1)) -> c(x1)
10.96/3.77	   s(n(x1)) -> s(s(f(x1)))
10.96/3.77	   f(n(x1)) -> n(f(x1))
10.96/3.77	
10.96/3.77	Q is empty.
10.96/3.77	We have to consider all minimal (P,Q,R)-chains.
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(8) UsableRulesProof (EQUIVALENT)
10.96/3.77	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(9)
10.96/3.77	Obligation:
10.96/3.77	Q DP problem:
10.96/3.77	The TRS P consists of the following rules:
10.96/3.77	
10.96/3.77	   F(n(x1)) -> F(x1)
10.96/3.77	   F(c(x1)) -> F(x1)
10.96/3.77	
10.96/3.77	R is empty.
10.96/3.77	Q is empty.
10.96/3.77	We have to consider all minimal (P,Q,R)-chains.
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(10) QDPSizeChangeProof (EQUIVALENT)
10.96/3.77	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
10.96/3.77	
10.96/3.77	From the DPs we obtained the following set of size-change graphs:
10.96/3.77	*F(n(x1)) -> F(x1)
10.96/3.77	The graph contains the following edges 1 > 1
10.96/3.77	
10.96/3.77	
10.96/3.77	*F(c(x1)) -> F(x1)
10.96/3.77	The graph contains the following edges 1 > 1
10.96/3.77	
10.96/3.77	
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(11)
10.96/3.77	YES
10.96/3.77	
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(12)
10.96/3.77	Obligation:
10.96/3.77	Q DP problem:
10.96/3.77	The TRS P consists of the following rules:
10.96/3.77	
10.96/3.77	   S(n(x1)) -> S(f(x1))
10.96/3.77	   S(n(x1)) -> S(s(f(x1)))
10.96/3.77	
10.96/3.77	The TRS R consists of the following rules:
10.96/3.77	
10.96/3.77	   f(x1) -> c(c(n(x1)))
10.96/3.77	   f(c(x1)) -> c(c(f(x1)))
10.96/3.77	   c(c(x1)) -> c(x1)
10.96/3.77	   s(n(x1)) -> s(s(f(x1)))
10.96/3.77	   f(n(x1)) -> n(f(x1))
10.96/3.77	
10.96/3.77	Q is empty.
10.96/3.77	We have to consider all minimal (P,Q,R)-chains.
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(13) QDPOrderProof (EQUIVALENT)
10.96/3.77	We use the reduction pair processor [LPAR04,JAR06].
10.96/3.77	
10.96/3.77	
10.96/3.77	The following pairs can be oriented strictly and are deleted.
10.96/3.77	
10.96/3.77	   S(n(x1)) -> S(f(x1))
10.96/3.77	   S(n(x1)) -> S(s(f(x1)))
10.96/3.77	The remaining pairs can at least be oriented weakly.
10.96/3.77	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
10.96/3.77	
10.96/3.77	POL( S_1(x_1) ) = max{0, 2x_1 - 1}
10.96/3.77	POL( f_1(x_1) ) = 2x_1
10.96/3.77	POL( c_1(x_1) ) = 0
10.96/3.77	POL( n_1(x_1) ) = 2x_1 + 2
10.96/3.77	POL( s_1(x_1) ) = max{0, -2}
10.96/3.77	
10.96/3.77	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
10.96/3.77	
10.96/3.77	   f(x1) -> c(c(n(x1)))
10.96/3.77	   f(c(x1)) -> c(c(f(x1)))
10.96/3.77	   f(n(x1)) -> n(f(x1))
10.96/3.77	   s(n(x1)) -> s(s(f(x1)))
10.96/3.77	   c(c(x1)) -> c(x1)
10.96/3.77	
10.96/3.77	
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(14)
10.96/3.77	Obligation:
10.96/3.77	Q DP problem:
10.96/3.77	P is empty.
10.96/3.77	The TRS R consists of the following rules:
10.96/3.77	
10.96/3.77	   f(x1) -> c(c(n(x1)))
10.96/3.77	   f(c(x1)) -> c(c(f(x1)))
10.96/3.77	   c(c(x1)) -> c(x1)
10.96/3.77	   s(n(x1)) -> s(s(f(x1)))
10.96/3.77	   f(n(x1)) -> n(f(x1))
10.96/3.77	
10.96/3.77	Q is empty.
10.96/3.77	We have to consider all minimal (P,Q,R)-chains.
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(15) PisEmptyProof (EQUIVALENT)
10.96/3.77	The TRS P is empty. Hence, there is no (P,Q,R) chain.
10.96/3.77	----------------------------------------
10.96/3.77	
10.96/3.77	(16)
10.96/3.77	YES
11.48/3.97	EOF