8.27/3.04	YES
8.27/3.08	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
8.27/3.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.27/3.08	
8.27/3.08	
8.27/3.08	Termination w.r.t. Q of the given QTRS could be proven:
8.27/3.08	
8.27/3.08	(0) QTRS
8.27/3.08	(1) QTRS Reverse [EQUIVALENT, 0 ms]
8.27/3.08	(2) QTRS
8.27/3.08	(3) DependencyPairsProof [EQUIVALENT, 7 ms]
8.27/3.08	(4) QDP
8.27/3.08	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
8.27/3.08	(6) AND
8.27/3.08	    (7) QDP
8.27/3.08	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
8.27/3.08	        (9) QDP
8.27/3.08	        (10) QDPSizeChangeProof [EQUIVALENT, 1 ms]
8.27/3.08	        (11) YES
8.27/3.08	    (12) QDP
8.27/3.08	        (13) QDPOrderProof [EQUIVALENT, 22 ms]
8.27/3.08	        (14) QDP
8.27/3.08	        (15) PisEmptyProof [EQUIVALENT, 0 ms]
8.27/3.08	        (16) YES
8.27/3.08	
8.27/3.08	
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(0)
8.27/3.08	Obligation:
8.27/3.08	Q restricted rewrite system:
8.27/3.08	The TRS R consists of the following rules:
8.27/3.08	
8.27/3.08	   f(x1) -> n(c(n(a(x1))))
8.27/3.08	   c(f(x1)) -> f(n(a(c(x1))))
8.27/3.08	   n(a(x1)) -> c(x1)
8.27/3.08	   c(c(x1)) -> c(x1)
8.27/3.08	   n(s(x1)) -> f(s(s(x1)))
8.27/3.08	   n(f(x1)) -> f(n(x1))
8.27/3.08	
8.27/3.08	Q is empty.
8.27/3.08	
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(1) QTRS Reverse (EQUIVALENT)
8.27/3.08	We applied the QTRS Reverse Processor [REVERSE].
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(2)
8.27/3.08	Obligation:
8.27/3.08	Q restricted rewrite system:
8.27/3.08	The TRS R consists of the following rules:
8.27/3.08	
8.27/3.08	   f(x1) -> a(n(c(n(x1))))
8.27/3.08	   f(c(x1)) -> c(a(n(f(x1))))
8.27/3.08	   a(n(x1)) -> c(x1)
8.27/3.08	   c(c(x1)) -> c(x1)
8.27/3.08	   s(n(x1)) -> s(s(f(x1)))
8.27/3.08	   f(n(x1)) -> n(f(x1))
8.27/3.08	
8.27/3.08	Q is empty.
8.27/3.08	
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(3) DependencyPairsProof (EQUIVALENT)
8.27/3.08	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(4)
8.27/3.08	Obligation:
8.27/3.08	Q DP problem:
8.27/3.08	The TRS P consists of the following rules:
8.27/3.08	
8.27/3.08	   F(x1) -> A(n(c(n(x1))))
8.27/3.08	   F(x1) -> C(n(x1))
8.27/3.08	   F(c(x1)) -> C(a(n(f(x1))))
8.27/3.08	   F(c(x1)) -> A(n(f(x1)))
8.27/3.08	   F(c(x1)) -> F(x1)
8.27/3.08	   A(n(x1)) -> C(x1)
8.27/3.08	   S(n(x1)) -> S(s(f(x1)))
8.27/3.08	   S(n(x1)) -> S(f(x1))
8.27/3.08	   S(n(x1)) -> F(x1)
8.27/3.08	   F(n(x1)) -> F(x1)
8.27/3.08	
8.27/3.08	The TRS R consists of the following rules:
8.27/3.08	
8.27/3.08	   f(x1) -> a(n(c(n(x1))))
8.27/3.08	   f(c(x1)) -> c(a(n(f(x1))))
8.27/3.08	   a(n(x1)) -> c(x1)
8.27/3.08	   c(c(x1)) -> c(x1)
8.27/3.08	   s(n(x1)) -> s(s(f(x1)))
8.27/3.08	   f(n(x1)) -> n(f(x1))
8.27/3.08	
8.27/3.08	Q is empty.
8.27/3.08	We have to consider all minimal (P,Q,R)-chains.
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(5) DependencyGraphProof (EQUIVALENT)
8.27/3.08	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes.
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(6)
8.27/3.08	Complex Obligation (AND)
8.27/3.08	
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(7)
8.27/3.08	Obligation:
8.27/3.08	Q DP problem:
8.27/3.08	The TRS P consists of the following rules:
8.27/3.08	
8.27/3.08	   F(n(x1)) -> F(x1)
8.27/3.08	   F(c(x1)) -> F(x1)
8.27/3.08	
8.27/3.08	The TRS R consists of the following rules:
8.27/3.08	
8.27/3.08	   f(x1) -> a(n(c(n(x1))))
8.27/3.08	   f(c(x1)) -> c(a(n(f(x1))))
8.27/3.08	   a(n(x1)) -> c(x1)
8.27/3.08	   c(c(x1)) -> c(x1)
8.27/3.08	   s(n(x1)) -> s(s(f(x1)))
8.27/3.08	   f(n(x1)) -> n(f(x1))
8.27/3.08	
8.27/3.08	Q is empty.
8.27/3.08	We have to consider all minimal (P,Q,R)-chains.
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(8) UsableRulesProof (EQUIVALENT)
8.27/3.08	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.
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(9)
8.27/3.08	Obligation:
8.27/3.08	Q DP problem:
8.27/3.08	The TRS P consists of the following rules:
8.27/3.08	
8.27/3.08	   F(n(x1)) -> F(x1)
8.27/3.08	   F(c(x1)) -> F(x1)
8.27/3.08	
8.27/3.08	R is empty.
8.27/3.08	Q is empty.
8.27/3.08	We have to consider all minimal (P,Q,R)-chains.
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(10) QDPSizeChangeProof (EQUIVALENT)
8.27/3.08	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. 
8.27/3.08	
8.27/3.08	From the DPs we obtained the following set of size-change graphs:
8.27/3.08	*F(n(x1)) -> F(x1)
8.27/3.08	The graph contains the following edges 1 > 1
8.27/3.08	
8.27/3.08	
8.27/3.08	*F(c(x1)) -> F(x1)
8.27/3.08	The graph contains the following edges 1 > 1
8.27/3.08	
8.27/3.08	
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(11)
8.27/3.08	YES
8.27/3.08	
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(12)
8.27/3.08	Obligation:
8.27/3.08	Q DP problem:
8.27/3.08	The TRS P consists of the following rules:
8.27/3.08	
8.27/3.08	   S(n(x1)) -> S(f(x1))
8.27/3.08	   S(n(x1)) -> S(s(f(x1)))
8.27/3.08	
8.27/3.08	The TRS R consists of the following rules:
8.27/3.08	
8.27/3.08	   f(x1) -> a(n(c(n(x1))))
8.27/3.08	   f(c(x1)) -> c(a(n(f(x1))))
8.27/3.08	   a(n(x1)) -> c(x1)
8.27/3.08	   c(c(x1)) -> c(x1)
8.27/3.08	   s(n(x1)) -> s(s(f(x1)))
8.27/3.08	   f(n(x1)) -> n(f(x1))
8.27/3.08	
8.27/3.08	Q is empty.
8.27/3.08	We have to consider all minimal (P,Q,R)-chains.
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(13) QDPOrderProof (EQUIVALENT)
8.27/3.08	We use the reduction pair processor [LPAR04,JAR06].
8.27/3.08	
8.27/3.08	
8.27/3.08	The following pairs can be oriented strictly and are deleted.
8.27/3.08	
8.27/3.08	   S(n(x1)) -> S(f(x1))
8.27/3.08	   S(n(x1)) -> S(s(f(x1)))
8.27/3.08	The remaining pairs can at least be oriented weakly.
8.27/3.08	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
8.27/3.08	
8.27/3.08	POL( S_1(x_1) ) = max{0, x_1 - 1}
8.27/3.08	POL( f_1(x_1) ) = 2x_1
8.27/3.08	POL( a_1(x_1) ) = max{0, -2}
8.27/3.08	POL( n_1(x_1) ) = 2x_1 + 2
8.27/3.08	POL( c_1(x_1) ) = max{0, -2}
8.27/3.08	POL( s_1(x_1) ) = max{0, -2}
8.27/3.08	
8.27/3.08	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
8.27/3.08	
8.27/3.08	   f(x1) -> a(n(c(n(x1))))
8.27/3.08	   f(c(x1)) -> c(a(n(f(x1))))
8.27/3.08	   f(n(x1)) -> n(f(x1))
8.27/3.08	   s(n(x1)) -> s(s(f(x1)))
8.27/3.08	   a(n(x1)) -> c(x1)
8.27/3.08	   c(c(x1)) -> c(x1)
8.27/3.08	
8.27/3.08	
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(14)
8.27/3.08	Obligation:
8.27/3.08	Q DP problem:
8.27/3.08	P is empty.
8.27/3.08	The TRS R consists of the following rules:
8.27/3.08	
8.27/3.08	   f(x1) -> a(n(c(n(x1))))
8.27/3.08	   f(c(x1)) -> c(a(n(f(x1))))
8.27/3.08	   a(n(x1)) -> c(x1)
8.27/3.08	   c(c(x1)) -> c(x1)
8.27/3.08	   s(n(x1)) -> s(s(f(x1)))
8.27/3.08	   f(n(x1)) -> n(f(x1))
8.27/3.08	
8.27/3.08	Q is empty.
8.27/3.08	We have to consider all minimal (P,Q,R)-chains.
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(15) PisEmptyProof (EQUIVALENT)
8.27/3.08	The TRS P is empty. Hence, there is no (P,Q,R) chain.
8.27/3.08	----------------------------------------
8.27/3.08	
8.27/3.08	(16)
8.27/3.08	YES
8.70/3.12	EOF