3.22/1.55	YES
3.22/1.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.22/1.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.22/1.56	
3.22/1.56	
3.22/1.56	Termination of the given CSR could be proven:
3.22/1.56	
3.22/1.56	(0) CSR
3.22/1.56	(1) CSRRRRProof [EQUIVALENT, 43 ms]
3.22/1.56	(2) CSR
3.22/1.56	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.22/1.56	(4) CSR
3.22/1.56	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.22/1.56	(6) CSR
3.22/1.56	(7) CSRRRRProof [EQUIVALENT, 3 ms]
3.22/1.56	(8) CSR
3.22/1.56	(9) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.22/1.56	(10) QCSDP
3.22/1.56	(11) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.22/1.56	(12) TRUE
3.22/1.56	
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(0)
3.22/1.56	Obligation:
3.22/1.56	Context-sensitive rewrite system:
3.22/1.56	The TRS R consists of the following rules:
3.22/1.56	
3.22/1.56	   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
3.22/1.56	   f_1(x, i_0(g_1(x))) -> a_0
3.22/1.56	   f_1(x, y) -> x
3.22/1.56	   f_1(x, i_0(x)) -> f_1(x, x)
3.22/1.56	   g_1(x) -> i_0(x)
3.22/1.56	
3.22/1.56	The replacement map contains the following entries:
3.22/1.56	
3.22/1.56	f_1: empty set
3.22/1.56	i_0: {1}
3.22/1.56	g_1: empty set
3.22/1.56	a_0: empty set
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(1) CSRRRRProof (EQUIVALENT)
3.22/1.56	The following CSR is given: Context-sensitive rewrite system:
3.22/1.56	The TRS R consists of the following rules:
3.22/1.56	
3.22/1.56	   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
3.22/1.56	   f_1(x, i_0(g_1(x))) -> a_0
3.22/1.56	   f_1(x, y) -> x
3.22/1.56	   f_1(x, i_0(x)) -> f_1(x, x)
3.22/1.56	   g_1(x) -> i_0(x)
3.22/1.56	
3.22/1.56	The replacement map contains the following entries:
3.22/1.56	
3.22/1.56	f_1: empty set
3.22/1.56	i_0: {1}
3.22/1.56	g_1: empty set
3.22/1.56	a_0: empty set
3.22/1.56	Used ordering:
3.22/1.56	Polynomial interpretation [POLO]:
3.22/1.56	
3.22/1.56	   POL(a_0) = 1
3.22/1.56	   POL(f_1(x_1, x_2)) = 1 + x_1 + x_2
3.22/1.56	   POL(g_1(x_1)) = x_1
3.22/1.56	   POL(i_0(x_1)) = x_1
3.22/1.56	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.22/1.56	
3.22/1.56	   f_1(x, y) -> x
3.22/1.56	
3.22/1.56	
3.22/1.56	
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(2)
3.22/1.56	Obligation:
3.22/1.56	Context-sensitive rewrite system:
3.22/1.56	The TRS R consists of the following rules:
3.22/1.56	
3.22/1.56	   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
3.22/1.56	   f_1(x, i_0(g_1(x))) -> a_0
3.22/1.56	   f_1(x, i_0(x)) -> f_1(x, x)
3.22/1.56	   g_1(x) -> i_0(x)
3.22/1.56	
3.22/1.56	The replacement map contains the following entries:
3.22/1.56	
3.22/1.56	f_1: empty set
3.22/1.56	i_0: {1}
3.22/1.56	g_1: empty set
3.22/1.56	a_0: empty set
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(3) CSRRRRProof (EQUIVALENT)
3.22/1.56	The following CSR is given: Context-sensitive rewrite system:
3.22/1.56	The TRS R consists of the following rules:
3.22/1.56	
3.22/1.56	   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
3.22/1.56	   f_1(x, i_0(g_1(x))) -> a_0
3.22/1.56	   f_1(x, i_0(x)) -> f_1(x, x)
3.22/1.56	   g_1(x) -> i_0(x)
3.22/1.56	
3.22/1.56	The replacement map contains the following entries:
3.22/1.56	
3.22/1.56	f_1: empty set
3.22/1.56	i_0: {1}
3.22/1.56	g_1: empty set
3.22/1.56	a_0: empty set
3.22/1.56	Used ordering:
3.22/1.56	Polynomial interpretation [POLO]:
3.22/1.56	
3.22/1.56	   POL(a_0) = 0
3.22/1.56	   POL(f_1(x_1, x_2)) = 2
3.22/1.56	   POL(g_1(x_1)) = 1 + x_1
3.22/1.56	   POL(i_0(x_1)) = 1 + x_1
3.22/1.56	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.22/1.56	
3.22/1.56	   f_1(x, i_0(g_1(x))) -> a_0
3.22/1.56	
3.22/1.56	
3.22/1.56	
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(4)
3.22/1.56	Obligation:
3.22/1.56	Context-sensitive rewrite system:
3.22/1.56	The TRS R consists of the following rules:
3.22/1.56	
3.22/1.56	   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
3.22/1.56	   f_1(x, i_0(x)) -> f_1(x, x)
3.22/1.56	   g_1(x) -> i_0(x)
3.22/1.56	
3.22/1.56	The replacement map contains the following entries:
3.22/1.56	
3.22/1.56	f_1: empty set
3.22/1.56	i_0: {1}
3.22/1.56	g_1: empty set
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(5) CSRRRRProof (EQUIVALENT)
3.22/1.56	The following CSR is given: Context-sensitive rewrite system:
3.22/1.56	The TRS R consists of the following rules:
3.22/1.56	
3.22/1.56	   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
3.22/1.56	   f_1(x, i_0(x)) -> f_1(x, x)
3.22/1.56	   g_1(x) -> i_0(x)
3.22/1.56	
3.22/1.56	The replacement map contains the following entries:
3.22/1.56	
3.22/1.56	f_1: empty set
3.22/1.56	i_0: {1}
3.22/1.56	g_1: empty set
3.22/1.56	Used ordering:
3.22/1.56	Polynomial interpretation [POLO]:
3.22/1.56	
3.22/1.56	   POL(f_1(x_1, x_2)) = x_1
3.22/1.56	   POL(g_1(x_1)) = 1 + x_1
3.22/1.56	   POL(i_0(x_1)) = x_1
3.22/1.56	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.22/1.56	
3.22/1.56	   g_1(x) -> i_0(x)
3.22/1.56	
3.22/1.56	
3.22/1.56	
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(6)
3.22/1.56	Obligation:
3.22/1.56	Context-sensitive rewrite system:
3.22/1.56	The TRS R consists of the following rules:
3.22/1.56	
3.22/1.56	   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
3.22/1.56	   f_1(x, i_0(x)) -> f_1(x, x)
3.22/1.56	
3.22/1.56	The replacement map contains the following entries:
3.22/1.56	
3.22/1.56	f_1: empty set
3.22/1.56	i_0: {1}
3.22/1.56	g_1: empty set
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(7) CSRRRRProof (EQUIVALENT)
3.22/1.56	The following CSR is given: Context-sensitive rewrite system:
3.22/1.56	The TRS R consists of the following rules:
3.22/1.56	
3.22/1.56	   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
3.22/1.56	   f_1(x, i_0(x)) -> f_1(x, x)
3.22/1.56	
3.22/1.56	The replacement map contains the following entries:
3.22/1.56	
3.22/1.56	f_1: empty set
3.22/1.56	i_0: {1}
3.22/1.56	g_1: empty set
3.22/1.56	Used ordering:
3.22/1.56	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.22/1.56	
3.22/1.56	   <<<
3.22/1.56	 POL(f_1(x_1, x_2)) =  	[[1]] 	 +  	[[1, 0]] 	* 	x_1 	 +  	[[1, 1]] 	* 	x_2
3.22/1.56	>>>
3.22/1.56	
3.22/1.56	   <<<
3.22/1.56	 POL(i_0(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
3.22/1.56	>>>
3.22/1.56	
3.22/1.56	   <<<
3.22/1.56	 POL(g_1(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
3.22/1.56	>>>
3.22/1.56	
3.22/1.56	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.22/1.56	
3.22/1.56	   f_1(x, i_0(x)) -> f_1(x, x)
3.22/1.56	
3.22/1.56	
3.22/1.56	
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(8)
3.22/1.56	Obligation:
3.22/1.56	Context-sensitive rewrite system:
3.22/1.56	The TRS R consists of the following rules:
3.22/1.56	
3.22/1.56	   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
3.22/1.56	
3.22/1.56	The replacement map contains the following entries:
3.22/1.56	
3.22/1.56	f_1: empty set
3.22/1.56	i_0: {1}
3.22/1.56	g_1: empty set
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(9) CSDependencyPairsProof (EQUIVALENT)
3.22/1.56	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(10)
3.22/1.56	Obligation:
3.22/1.56	Q-restricted context-sensitive dependency pair problem:
3.22/1.56	The symbols in {i_0_1} are replacing on all positions.
3.22/1.56	The symbols in {f_1_2, g_1_1, F_1_2} are not replacing on any position.
3.22/1.56	
3.22/1.56	The ordinary context-sensitive dependency pairs DP_o are:
3.22/1.56	   F_1(x, x) -> F_1(i_0(x), g_1(g_1(x)))
3.22/1.56	
3.22/1.56	The TRS R consists of the following rules:
3.22/1.56	
3.22/1.56	   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
3.22/1.56	
3.22/1.56	Q is empty.
3.22/1.56	
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(11) QCSDependencyGraphProof (EQUIVALENT)
3.22/1.56	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
3.22/1.56	The rules F_1(z0, z0) -> F_1(i_0(z0), g_1(g_1(z0))) and F_1(x0, x0) -> F_1(i_0(x0), g_1(g_1(x0))) form no chain, because ECap^mu(F_1(i_0(z0), g_1(g_1(z0)))) = F_1(i_0(z0), g_1(g_1(z0))) does not unify with F_1(x0, x0). 
3.22/1.56	----------------------------------------
3.22/1.56	
3.22/1.56	(12)
3.22/1.56	TRUE
3.22/1.58	EOF