3.42/1.47	YES
3.42/1.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.42/1.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.42/1.48	
3.42/1.48	
3.42/1.48	Termination of the given CSR could be proven:
3.42/1.48	
3.42/1.48	(0) CSR
3.42/1.48	(1) CSRRRRProof [EQUIVALENT, 59 ms]
3.42/1.48	(2) CSR
3.42/1.48	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.42/1.48	(4) CSR
3.42/1.48	(5) CSRRRRProof [EQUIVALENT, 6 ms]
3.42/1.48	(6) CSR
3.42/1.48	(7) CSRInnermostProof [EQUIVALENT, 0 ms]
3.42/1.48	(8) CSR
3.42/1.48	(9) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.42/1.48	(10) QCSDP
3.42/1.48	(11) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.42/1.48	(12) TRUE
3.42/1.48	
3.42/1.48	
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(0)
3.42/1.48	Obligation:
3.42/1.48	Context-sensitive rewrite system:
3.42/1.48	The TRS R consists of the following rules:
3.42/1.48	
3.42/1.48	   *top*_0(a_1) -> *top*_0(f_0(a_1))
3.42/1.48	   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
3.42/1.48	   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
3.42/1.48	   f_0(f_0(f_0(a_1))) -> f_1(f_0(f_0(f_0(a_1))))
3.42/1.48	   f_1(f_1(f_0(f_0(x)))) -> c_0
3.42/1.48	   f_1(f_0(f_0(f_0(x)))) -> c_0
3.42/1.48	   f_1(f_1(f_1(f_0(x)))) -> c_0
3.42/1.48	   f_1(f_1(f_1(f_1(x)))) -> c_0
3.42/1.48	
3.42/1.48	The replacement map contains the following entries:
3.42/1.48	
3.42/1.48	*top*_0: {1}
3.42/1.48	a_1: empty set
3.42/1.48	f_0: {1}
3.42/1.48	f_1: empty set
3.42/1.48	c_0: empty set
3.42/1.48	
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(1) CSRRRRProof (EQUIVALENT)
3.42/1.48	The following CSR is given: Context-sensitive rewrite system:
3.42/1.48	The TRS R consists of the following rules:
3.42/1.48	
3.42/1.48	   *top*_0(a_1) -> *top*_0(f_0(a_1))
3.42/1.48	   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
3.42/1.48	   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
3.42/1.48	   f_0(f_0(f_0(a_1))) -> f_1(f_0(f_0(f_0(a_1))))
3.42/1.48	   f_1(f_1(f_0(f_0(x)))) -> c_0
3.42/1.48	   f_1(f_0(f_0(f_0(x)))) -> c_0
3.42/1.48	   f_1(f_1(f_1(f_0(x)))) -> c_0
3.42/1.48	   f_1(f_1(f_1(f_1(x)))) -> c_0
3.42/1.48	
3.42/1.48	The replacement map contains the following entries:
3.42/1.48	
3.42/1.48	*top*_0: {1}
3.42/1.48	a_1: empty set
3.42/1.48	f_0: {1}
3.42/1.48	f_1: empty set
3.42/1.48	c_0: empty set
3.42/1.48	Used ordering:
3.42/1.48	Polynomial interpretation [POLO]:
3.42/1.48	
3.42/1.48	   POL(*top*_0(x_1)) = x_1
3.42/1.48	   POL(a_1) = 1
3.42/1.48	   POL(c_0) = 0
3.42/1.48	   POL(f_0(x_1)) = x_1
3.42/1.48	   POL(f_1(x_1)) = 1
3.42/1.48	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.42/1.48	
3.42/1.48	   f_1(f_1(f_0(f_0(x)))) -> c_0
3.42/1.48	   f_1(f_0(f_0(f_0(x)))) -> c_0
3.42/1.48	   f_1(f_1(f_1(f_0(x)))) -> c_0
3.42/1.48	   f_1(f_1(f_1(f_1(x)))) -> c_0
3.42/1.48	
3.42/1.48	
3.42/1.48	
3.42/1.48	
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(2)
3.42/1.48	Obligation:
3.42/1.48	Context-sensitive rewrite system:
3.42/1.48	The TRS R consists of the following rules:
3.42/1.48	
3.42/1.48	   *top*_0(a_1) -> *top*_0(f_0(a_1))
3.42/1.48	   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
3.42/1.48	   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
3.42/1.48	   f_0(f_0(f_0(a_1))) -> f_1(f_0(f_0(f_0(a_1))))
3.42/1.48	
3.42/1.48	The replacement map contains the following entries:
3.42/1.48	
3.42/1.48	*top*_0: {1}
3.42/1.48	a_1: empty set
3.42/1.48	f_0: {1}
3.42/1.48	f_1: empty set
3.42/1.48	
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(3) CSRRRRProof (EQUIVALENT)
3.42/1.48	The following CSR is given: Context-sensitive rewrite system:
3.42/1.48	The TRS R consists of the following rules:
3.42/1.48	
3.42/1.48	   *top*_0(a_1) -> *top*_0(f_0(a_1))
3.42/1.48	   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
3.42/1.48	   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
3.42/1.48	   f_0(f_0(f_0(a_1))) -> f_1(f_0(f_0(f_0(a_1))))
3.42/1.48	
3.42/1.48	The replacement map contains the following entries:
3.42/1.48	
3.42/1.48	*top*_0: {1}
3.42/1.48	a_1: empty set
3.42/1.48	f_0: {1}
3.42/1.48	f_1: empty set
3.42/1.48	Used ordering:
3.42/1.48	Polynomial interpretation [POLO]:
3.42/1.48	
3.42/1.48	   POL(*top*_0(x_1)) = x_1
3.42/1.48	   POL(a_1) = 1
3.42/1.48	   POL(f_0(x_1)) = x_1
3.42/1.48	   POL(f_1(x_1)) = 0
3.42/1.48	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.42/1.48	
3.42/1.48	   f_0(f_0(f_0(a_1))) -> f_1(f_0(f_0(f_0(a_1))))
3.42/1.48	
3.42/1.48	
3.42/1.48	
3.42/1.48	
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(4)
3.42/1.48	Obligation:
3.42/1.48	Context-sensitive rewrite system:
3.42/1.48	The TRS R consists of the following rules:
3.42/1.48	
3.42/1.48	   *top*_0(a_1) -> *top*_0(f_0(a_1))
3.42/1.48	   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
3.42/1.48	   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
3.42/1.48	
3.42/1.48	The replacement map contains the following entries:
3.42/1.48	
3.42/1.48	*top*_0: {1}
3.42/1.48	a_1: empty set
3.42/1.48	f_0: {1}
3.42/1.48	
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(5) CSRRRRProof (EQUIVALENT)
3.42/1.48	The following CSR is given: Context-sensitive rewrite system:
3.42/1.48	The TRS R consists of the following rules:
3.42/1.48	
3.42/1.48	   *top*_0(a_1) -> *top*_0(f_0(a_1))
3.42/1.48	   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
3.42/1.48	   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
3.42/1.48	
3.42/1.48	The replacement map contains the following entries:
3.42/1.48	
3.42/1.48	*top*_0: {1}
3.42/1.48	a_1: empty set
3.42/1.48	f_0: {1}
3.42/1.48	Used ordering:
3.42/1.48	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.42/1.48	
3.42/1.48	   <<<
3.42/1.48	 POL(*top*_0(x_1)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1
3.42/1.48	>>>
3.42/1.48	
3.42/1.48	   <<<
3.42/1.48	 POL(a_1) =  	[[0], [1]]
3.42/1.48	>>>
3.42/1.48	
3.42/1.48	   <<<
3.42/1.48	 POL(f_0(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
3.42/1.48	>>>
3.42/1.48	
3.42/1.48	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.42/1.48	
3.42/1.48	   *top*_0(a_1) -> *top*_0(f_0(a_1))
3.42/1.48	
3.42/1.48	
3.42/1.48	
3.42/1.48	
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(6)
3.42/1.48	Obligation:
3.42/1.48	Context-sensitive rewrite system:
3.42/1.48	The TRS R consists of the following rules:
3.42/1.48	
3.42/1.48	   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
3.42/1.48	   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
3.42/1.48	
3.42/1.48	The replacement map contains the following entries:
3.42/1.48	
3.42/1.48	*top*_0: {1}
3.42/1.48	a_1: empty set
3.42/1.48	f_0: {1}
3.42/1.48	
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(7) CSRInnermostProof (EQUIVALENT)
3.42/1.48	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(8)
3.42/1.48	Obligation:
3.42/1.48	Context-sensitive rewrite system:
3.42/1.48	The TRS R consists of the following rules:
3.42/1.48	
3.42/1.48	   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
3.42/1.48	   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
3.42/1.48	
3.42/1.48	The replacement map contains the following entries:
3.42/1.48	
3.42/1.48	*top*_0: {1}
3.42/1.48	a_1: empty set
3.42/1.48	f_0: {1}
3.42/1.48	
3.42/1.48	
3.42/1.48	Innermost Strategy.
3.42/1.48	
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(9) CSDependencyPairsProof (EQUIVALENT)
3.42/1.48	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(10)
3.42/1.48	Obligation:
3.42/1.48	Q-restricted context-sensitive dependency pair problem:
3.42/1.48	The symbols in {*top*_0_1, f_0_1, *TOP*_0_1} are replacing on all positions.
3.42/1.48	
3.42/1.48	The ordinary context-sensitive dependency pairs DP_o are:
3.42/1.48	   *TOP*_0(f_0(a_1)) -> *TOP*_0(f_0(f_0(a_1)))
3.42/1.48	   *TOP*_0(f_0(f_0(a_1))) -> *TOP*_0(f_0(f_0(f_0(a_1))))
3.42/1.48	
3.42/1.48	The TRS R consists of the following rules:
3.42/1.48	
3.42/1.48	   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
3.42/1.48	   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
3.42/1.48	
3.42/1.48	The set Q consists of the following terms:
3.42/1.48	
3.42/1.48	   *top*_0(f_0(a_1))
3.42/1.48	   *top*_0(f_0(f_0(a_1)))
3.42/1.48	
3.42/1.48	
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(11) QCSDependencyGraphProof (EQUIVALENT)
3.42/1.48	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
3.42/1.48	The rules *TOP*_0(f_0(a_1)) -> *TOP*_0(f_0(f_0(a_1))) and *TOP*_0(f_0(a_1)) -> *TOP*_0(f_0(f_0(a_1))) form no chain, because ECap^mu(*TOP*_0(f_0(f_0(a_1)))) = *TOP*_0(f_0(f_0(a_1))) does not unify with *TOP*_0(f_0(a_1)). The rules *TOP*_0(f_0(f_0(a_1))) -> *TOP*_0(f_0(f_0(f_0(a_1)))) and *TOP*_0(f_0(a_1)) -> *TOP*_0(f_0(f_0(a_1))) form no chain, because ECap^mu(*TOP*_0(f_0(f_0(f_0(a_1))))) = *TOP*_0(f_0(f_0(f_0(a_1)))) does not unify with *TOP*_0(f_0(a_1)). The rules *TOP*_0(f_0(f_0(a_1))) -> *TOP*_0(f_0(f_0(f_0(a_1)))) and *TOP*_0(f_0(f_0(a_1))) -> *TOP*_0(f_0(f_0(f_0(a_1)))) form no chain, because ECap^mu(*TOP*_0(f_0(f_0(f_0(a_1))))) = *TOP*_0(f_0(f_0(f_0(a_1)))) does not unify with *TOP*_0(f_0(f_0(a_1))). 
3.42/1.48	----------------------------------------
3.42/1.48	
3.42/1.48	(12)
3.42/1.48	TRUE
3.42/1.52	EOF