12.93/4.09	YES
12.93/4.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
12.93/4.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.93/4.13	
12.93/4.13	
12.93/4.13	Termination w.r.t. Q of the given QTRS could be proven:
12.93/4.13	
12.93/4.13	(0) QTRS
12.93/4.13	(1) QTRSRRRProof [EQUIVALENT, 50 ms]
12.93/4.13	(2) QTRS
12.93/4.13	(3) DependencyPairsProof [EQUIVALENT, 25 ms]
12.93/4.13	(4) QDP
12.93/4.13	(5) MRRProof [EQUIVALENT, 34 ms]
12.93/4.13	(6) QDP
12.93/4.13	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
12.93/4.13	(8) AND
12.93/4.13	    (9) QDP
12.93/4.13	        (10) QDPOrderProof [EQUIVALENT, 6 ms]
12.93/4.13	        (11) QDP
12.93/4.13	        (12) PisEmptyProof [EQUIVALENT, 0 ms]
12.93/4.13	        (13) YES
12.93/4.13	    (14) QDP
12.93/4.13	        (15) QDPOrderProof [EQUIVALENT, 36 ms]
12.93/4.13	        (16) QDP
12.93/4.13	        (17) DependencyGraphProof [EQUIVALENT, 0 ms]
12.93/4.13	        (18) TRUE
12.93/4.13	
12.93/4.13	
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(0)
12.93/4.13	Obligation:
12.93/4.13	Q restricted rewrite system:
12.93/4.13	The TRS R consists of the following rules:
12.93/4.13	
12.93/4.13	   a(a(x1)) -> b(b(b(x1)))
12.93/4.13	   a(x1) -> d(c(d(x1)))
12.93/4.13	   b(b(b(x1))) -> a(f(x1))
12.93/4.13	   b(b(x1)) -> c(c(c(x1)))
12.93/4.13	   c(c(x1)) -> d(d(d(x1)))
12.93/4.13	   c(d(d(x1))) -> f(x1)
12.93/4.13	   f(f(x1)) -> f(a(x1))
12.93/4.13	
12.93/4.13	Q is empty.
12.93/4.13	
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(1) QTRSRRRProof (EQUIVALENT)
12.93/4.13	Used ordering:
12.93/4.13	Polynomial interpretation [POLO]:
12.93/4.13	
12.93/4.13	   POL(a(x_1)) = 39 + x_1
12.93/4.13	   POL(b(x_1)) = 26 + x_1
12.93/4.13	   POL(c(x_1)) = 17 + x_1
12.93/4.13	   POL(d(x_1)) = 11 + x_1
12.93/4.13	   POL(f(x_1)) = 39 + x_1
12.93/4.13	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
12.93/4.13	
12.93/4.13	   b(b(x1)) -> c(c(c(x1)))
12.93/4.13	   c(c(x1)) -> d(d(d(x1)))
12.93/4.13	
12.93/4.13	
12.93/4.13	
12.93/4.13	
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(2)
12.93/4.13	Obligation:
12.93/4.13	Q restricted rewrite system:
12.93/4.13	The TRS R consists of the following rules:
12.93/4.13	
12.93/4.13	   a(a(x1)) -> b(b(b(x1)))
12.93/4.13	   a(x1) -> d(c(d(x1)))
12.93/4.13	   b(b(b(x1))) -> a(f(x1))
12.93/4.13	   c(d(d(x1))) -> f(x1)
12.93/4.13	   f(f(x1)) -> f(a(x1))
12.93/4.13	
12.93/4.13	Q is empty.
12.93/4.13	
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(3) DependencyPairsProof (EQUIVALENT)
12.93/4.13	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(4)
12.93/4.13	Obligation:
12.93/4.13	Q DP problem:
12.93/4.13	The TRS P consists of the following rules:
12.93/4.13	
12.93/4.13	   A(a(x1)) -> B(b(b(x1)))
12.93/4.13	   A(a(x1)) -> B(b(x1))
12.93/4.13	   A(a(x1)) -> B(x1)
12.93/4.13	   A(x1) -> C(d(x1))
12.93/4.13	   B(b(b(x1))) -> A(f(x1))
12.93/4.13	   B(b(b(x1))) -> F(x1)
12.93/4.13	   C(d(d(x1))) -> F(x1)
12.93/4.13	   F(f(x1)) -> F(a(x1))
12.93/4.13	   F(f(x1)) -> A(x1)
12.93/4.13	
12.93/4.13	The TRS R consists of the following rules:
12.93/4.13	
12.93/4.13	   a(a(x1)) -> b(b(b(x1)))
12.93/4.13	   a(x1) -> d(c(d(x1)))
12.93/4.13	   b(b(b(x1))) -> a(f(x1))
12.93/4.13	   c(d(d(x1))) -> f(x1)
12.93/4.13	   f(f(x1)) -> f(a(x1))
12.93/4.13	
12.93/4.13	Q is empty.
12.93/4.13	We have to consider all minimal (P,Q,R)-chains.
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(5) MRRProof (EQUIVALENT)
12.93/4.13	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
12.93/4.13	
12.93/4.13	Strictly oriented dependency pairs:
12.93/4.13	
12.93/4.13	   A(a(x1)) -> B(b(x1))
12.93/4.13	   A(a(x1)) -> B(x1)
12.93/4.13	   A(x1) -> C(d(x1))
12.93/4.13	   B(b(b(x1))) -> F(x1)
12.93/4.13	   C(d(d(x1))) -> F(x1)
12.93/4.13	   F(f(x1)) -> A(x1)
12.93/4.13	
12.93/4.13	
12.93/4.13	Used ordering: Polynomial interpretation [POLO]:
12.93/4.13	
12.93/4.13	   POL(A(x_1)) = 2 + 2*x_1
12.93/4.13	   POL(B(x_1)) = 2*x_1
12.93/4.13	   POL(C(x_1)) = 1 + 2*x_1
12.93/4.13	   POL(F(x_1)) = 2*x_1
12.93/4.13	   POL(a(x_1)) = 3 + x_1
12.93/4.13	   POL(b(x_1)) = 2 + x_1
12.93/4.13	   POL(c(x_1)) = 3 + x_1
12.93/4.13	   POL(d(x_1)) = x_1
12.93/4.13	   POL(f(x_1)) = 3 + x_1
12.93/4.13	
12.93/4.13	
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(6)
12.93/4.13	Obligation:
12.93/4.13	Q DP problem:
12.93/4.13	The TRS P consists of the following rules:
12.93/4.13	
12.93/4.13	   A(a(x1)) -> B(b(b(x1)))
12.93/4.13	   B(b(b(x1))) -> A(f(x1))
12.93/4.13	   F(f(x1)) -> F(a(x1))
12.93/4.13	
12.93/4.13	The TRS R consists of the following rules:
12.93/4.13	
12.93/4.13	   a(a(x1)) -> b(b(b(x1)))
12.93/4.13	   a(x1) -> d(c(d(x1)))
12.93/4.13	   b(b(b(x1))) -> a(f(x1))
12.93/4.13	   c(d(d(x1))) -> f(x1)
12.93/4.13	   f(f(x1)) -> f(a(x1))
12.93/4.13	
12.93/4.13	Q is empty.
12.93/4.13	We have to consider all minimal (P,Q,R)-chains.
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(7) DependencyGraphProof (EQUIVALENT)
12.93/4.13	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(8)
12.93/4.13	Complex Obligation (AND)
12.93/4.13	
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(9)
12.93/4.13	Obligation:
12.93/4.13	Q DP problem:
12.93/4.13	The TRS P consists of the following rules:
12.93/4.13	
12.93/4.13	   F(f(x1)) -> F(a(x1))
12.93/4.13	
12.93/4.13	The TRS R consists of the following rules:
12.93/4.13	
12.93/4.13	   a(a(x1)) -> b(b(b(x1)))
12.93/4.13	   a(x1) -> d(c(d(x1)))
12.93/4.13	   b(b(b(x1))) -> a(f(x1))
12.93/4.13	   c(d(d(x1))) -> f(x1)
12.93/4.13	   f(f(x1)) -> f(a(x1))
12.93/4.13	
12.93/4.13	Q is empty.
12.93/4.13	We have to consider all minimal (P,Q,R)-chains.
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(10) QDPOrderProof (EQUIVALENT)
12.93/4.13	We use the reduction pair processor [LPAR04,JAR06].
12.93/4.13	
12.93/4.13	
12.93/4.13	The following pairs can be oriented strictly and are deleted.
12.93/4.13	
12.93/4.13	   F(f(x1)) -> F(a(x1))
12.93/4.13	The remaining pairs can at least be oriented weakly.
12.93/4.13	Used ordering:  Polynomial interpretation [POLO]:
12.93/4.13	
12.93/4.13	   POL(F(x_1)) = x_1
12.93/4.13	   POL(a(x_1)) = 0
12.93/4.13	   POL(b(x_1)) = 0
12.93/4.13	   POL(c(x_1)) = 1 + x_1
12.93/4.13	   POL(d(x_1)) = 0
12.93/4.13	   POL(f(x_1)) = 1
12.93/4.13	
12.93/4.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.93/4.13	
12.93/4.13	   b(b(b(x1))) -> a(f(x1))
12.93/4.13	   a(a(x1)) -> b(b(b(x1)))
12.93/4.13	   a(x1) -> d(c(d(x1)))
12.93/4.13	
12.93/4.13	
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(11)
12.93/4.13	Obligation:
12.93/4.13	Q DP problem:
12.93/4.13	P is empty.
12.93/4.13	The TRS R consists of the following rules:
12.93/4.13	
12.93/4.13	   a(a(x1)) -> b(b(b(x1)))
12.93/4.13	   a(x1) -> d(c(d(x1)))
12.93/4.13	   b(b(b(x1))) -> a(f(x1))
12.93/4.13	   c(d(d(x1))) -> f(x1)
12.93/4.13	   f(f(x1)) -> f(a(x1))
12.93/4.13	
12.93/4.13	Q is empty.
12.93/4.13	We have to consider all minimal (P,Q,R)-chains.
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(12) PisEmptyProof (EQUIVALENT)
12.93/4.13	The TRS P is empty. Hence, there is no (P,Q,R) chain.
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(13)
12.93/4.13	YES
12.93/4.13	
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(14)
12.93/4.13	Obligation:
12.93/4.13	Q DP problem:
12.93/4.13	The TRS P consists of the following rules:
12.93/4.13	
12.93/4.13	   B(b(b(x1))) -> A(f(x1))
12.93/4.13	   A(a(x1)) -> B(b(b(x1)))
12.93/4.13	
12.93/4.13	The TRS R consists of the following rules:
12.93/4.13	
12.93/4.13	   a(a(x1)) -> b(b(b(x1)))
12.93/4.13	   a(x1) -> d(c(d(x1)))
12.93/4.13	   b(b(b(x1))) -> a(f(x1))
12.93/4.13	   c(d(d(x1))) -> f(x1)
12.93/4.13	   f(f(x1)) -> f(a(x1))
12.93/4.13	
12.93/4.13	Q is empty.
12.93/4.13	We have to consider all minimal (P,Q,R)-chains.
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(15) QDPOrderProof (EQUIVALENT)
12.93/4.13	We use the reduction pair processor [LPAR04,JAR06].
12.93/4.13	
12.93/4.13	
12.93/4.13	The following pairs can be oriented strictly and are deleted.
12.93/4.13	
12.93/4.13	   B(b(b(x1))) -> A(f(x1))
12.93/4.13	The remaining pairs can at least be oriented weakly.
12.93/4.13	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
12.93/4.13	
12.93/4.13	POL( A_1(x_1) ) = x_1 + 2
12.93/4.13	POL( B_1(x_1) ) = x_1 + 1
12.93/4.13	POL( a_1(x_1) ) = 2x_1 + 1
12.93/4.13	POL( d_1(x_1) ) = max{0, x_1 - 2}
12.93/4.13	POL( f_1(x_1) ) = max{0, -2}
12.93/4.13	POL( b_1(x_1) ) = x_1 + 1
12.93/4.13	POL( c_1(x_1) ) = 0
12.93/4.13	
12.93/4.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.93/4.13	
12.93/4.13	   f(f(x1)) -> f(a(x1))
12.93/4.13	   b(b(b(x1))) -> a(f(x1))
12.93/4.13	   a(a(x1)) -> b(b(b(x1)))
12.93/4.13	   a(x1) -> d(c(d(x1)))
12.93/4.13	   c(d(d(x1))) -> f(x1)
12.93/4.13	
12.93/4.13	
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(16)
12.93/4.13	Obligation:
12.93/4.13	Q DP problem:
12.93/4.13	The TRS P consists of the following rules:
12.93/4.13	
12.93/4.13	   A(a(x1)) -> B(b(b(x1)))
12.93/4.13	
12.93/4.13	The TRS R consists of the following rules:
12.93/4.13	
12.93/4.13	   a(a(x1)) -> b(b(b(x1)))
12.93/4.13	   a(x1) -> d(c(d(x1)))
12.93/4.13	   b(b(b(x1))) -> a(f(x1))
12.93/4.13	   c(d(d(x1))) -> f(x1)
12.93/4.13	   f(f(x1)) -> f(a(x1))
12.93/4.13	
12.93/4.13	Q is empty.
12.93/4.13	We have to consider all minimal (P,Q,R)-chains.
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(17) DependencyGraphProof (EQUIVALENT)
12.93/4.13	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
12.93/4.13	----------------------------------------
12.93/4.13	
12.93/4.13	(18)
12.93/4.13	TRUE
13.25/4.22	EOF