3.35/1.59	YES
3.35/1.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.35/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.35/1.60	
3.35/1.60	
3.35/1.60	Termination of the given CSR could be proven:
3.35/1.60	
3.35/1.60	(0) CSR
3.35/1.60	(1) CSRRRRProof [EQUIVALENT, 56 ms]
3.35/1.60	(2) CSR
3.35/1.60	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.35/1.60	(4) CSR
3.35/1.60	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.35/1.60	(6) CSR
3.35/1.60	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.35/1.60	(8) CSR
3.35/1.60	(9) CSRRRRProof [EQUIVALENT, 0 ms]
3.35/1.60	(10) CSR
3.35/1.60	(11) RisEmptyProof [EQUIVALENT, 0 ms]
3.35/1.60	(12) YES
3.35/1.60	
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(0)
3.35/1.60	Obligation:
3.35/1.60	Context-sensitive rewrite system:
3.35/1.60	The TRS R consists of the following rules:
3.35/1.60	
3.35/1.60	   *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x)
3.35/1.60	   f_0(f_1(f_0(g_1(x)))) -> f_0(x)
3.35/1.60	   g_0(f_1(f_0(g_1(x)))) -> g_1(x)
3.35/1.60	   *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x))
3.35/1.60	   f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x))
3.35/1.60	   g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x))
3.35/1.60	   f_1(f_0(g_0(x))) -> x
3.35/1.60	   *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x)
3.35/1.60	   f_0(f_1(f_0(g_0(x)))) -> f_1(x)
3.35/1.60	   g_0(f_1(f_0(g_0(x)))) -> g_0(x)
3.35/1.60	   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	The replacement map contains the following entries:
3.35/1.60	
3.35/1.60	*top*_0: {1}
3.35/1.60	f_1: empty set
3.35/1.60	f_0: {1}
3.35/1.60	g_1: empty set
3.35/1.60	g_0: {1}
3.35/1.60	b_0: empty set
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(1) CSRRRRProof (EQUIVALENT)
3.35/1.60	The following CSR is given: Context-sensitive rewrite system:
3.35/1.60	The TRS R consists of the following rules:
3.35/1.60	
3.35/1.60	   *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x)
3.35/1.60	   f_0(f_1(f_0(g_1(x)))) -> f_0(x)
3.35/1.60	   g_0(f_1(f_0(g_1(x)))) -> g_1(x)
3.35/1.60	   *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x))
3.35/1.60	   f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x))
3.35/1.60	   g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x))
3.35/1.60	   f_1(f_0(g_0(x))) -> x
3.35/1.60	   *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x)
3.35/1.60	   f_0(f_1(f_0(g_0(x)))) -> f_1(x)
3.35/1.60	   g_0(f_1(f_0(g_0(x)))) -> g_0(x)
3.35/1.60	   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	The replacement map contains the following entries:
3.35/1.60	
3.35/1.60	*top*_0: {1}
3.35/1.60	f_1: empty set
3.35/1.60	f_0: {1}
3.35/1.60	g_1: empty set
3.35/1.60	g_0: {1}
3.35/1.60	b_0: empty set
3.35/1.60	Used ordering:
3.35/1.60	Polynomial interpretation [POLO]:
3.35/1.60	
3.35/1.60	   POL(*top*_0(x_1)) = x_1
3.35/1.60	   POL(b_0) = 0
3.35/1.60	   POL(f_0(x_1)) = x_1
3.35/1.60	   POL(f_1(x_1)) = x_1
3.35/1.60	   POL(g_0(x_1)) = 1 + x_1
3.35/1.60	   POL(g_1(x_1)) = x_1
3.35/1.60	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.35/1.60	
3.35/1.60	   g_0(f_1(f_0(g_1(x)))) -> g_1(x)
3.35/1.60	   *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x))
3.35/1.60	   f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x))
3.35/1.60	   g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x))
3.35/1.60	   f_1(f_0(g_0(x))) -> x
3.35/1.60	   *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x)
3.35/1.60	   f_0(f_1(f_0(g_0(x)))) -> f_1(x)
3.35/1.60	   g_0(f_1(f_0(g_0(x)))) -> g_0(x)
3.35/1.60	
3.35/1.60	
3.35/1.60	
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(2)
3.35/1.60	Obligation:
3.35/1.60	Context-sensitive rewrite system:
3.35/1.60	The TRS R consists of the following rules:
3.35/1.60	
3.35/1.60	   *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x)
3.35/1.60	   f_0(f_1(f_0(g_1(x)))) -> f_0(x)
3.35/1.60	   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	The replacement map contains the following entries:
3.35/1.60	
3.35/1.60	*top*_0: {1}
3.35/1.60	f_1: empty set
3.35/1.60	f_0: {1}
3.35/1.60	g_1: empty set
3.35/1.60	g_0: {1}
3.35/1.60	b_0: empty set
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(3) CSRRRRProof (EQUIVALENT)
3.35/1.60	The following CSR is given: Context-sensitive rewrite system:
3.35/1.60	The TRS R consists of the following rules:
3.35/1.60	
3.35/1.60	   *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x)
3.35/1.60	   f_0(f_1(f_0(g_1(x)))) -> f_0(x)
3.35/1.60	   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	The replacement map contains the following entries:
3.35/1.60	
3.35/1.60	*top*_0: {1}
3.35/1.60	f_1: empty set
3.35/1.60	f_0: {1}
3.35/1.60	g_1: empty set
3.35/1.60	g_0: {1}
3.35/1.60	b_0: empty set
3.35/1.60	Used ordering:
3.35/1.60	Polynomial interpretation [POLO]:
3.35/1.60	
3.35/1.60	   POL(*top*_0(x_1)) = x_1
3.35/1.60	   POL(b_0) = 0
3.35/1.60	   POL(f_0(x_1)) = x_1
3.35/1.60	   POL(f_1(x_1)) = x_1
3.35/1.60	   POL(g_0(x_1)) = x_1
3.35/1.60	   POL(g_1(x_1)) = 1 + x_1
3.35/1.60	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.35/1.60	
3.35/1.60	   *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x)
3.35/1.60	   f_0(f_1(f_0(g_1(x)))) -> f_0(x)
3.35/1.60	
3.35/1.60	
3.35/1.60	
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(4)
3.35/1.60	Obligation:
3.35/1.60	Context-sensitive rewrite system:
3.35/1.60	The TRS R consists of the following rules:
3.35/1.60	
3.35/1.60	   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	The replacement map contains the following entries:
3.35/1.60	
3.35/1.60	*top*_0: {1}
3.35/1.60	f_1: empty set
3.35/1.60	f_0: {1}
3.35/1.60	g_1: empty set
3.35/1.60	g_0: {1}
3.35/1.60	b_0: empty set
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(5) CSRRRRProof (EQUIVALENT)
3.35/1.60	The following CSR is given: Context-sensitive rewrite system:
3.35/1.60	The TRS R consists of the following rules:
3.35/1.60	
3.35/1.60	   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	The replacement map contains the following entries:
3.35/1.60	
3.35/1.60	*top*_0: {1}
3.35/1.60	f_1: empty set
3.35/1.60	f_0: {1}
3.35/1.60	g_1: empty set
3.35/1.60	g_0: {1}
3.35/1.60	b_0: empty set
3.35/1.60	Used ordering:
3.35/1.60	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(*top*_0(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(g_1(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(b_0) =  	[[0], [1]]
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(f_0(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(f_1(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(g_0(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.35/1.60	
3.35/1.60	   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	
3.35/1.60	
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(6)
3.35/1.60	Obligation:
3.35/1.60	Context-sensitive rewrite system:
3.35/1.60	The TRS R consists of the following rules:
3.35/1.60	
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	The replacement map contains the following entries:
3.35/1.60	
3.35/1.60	f_1: empty set
3.35/1.60	f_0: {1}
3.35/1.60	g_1: empty set
3.35/1.60	g_0: {1}
3.35/1.60	b_0: empty set
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(7) CSRRRRProof (EQUIVALENT)
3.35/1.60	The following CSR is given: Context-sensitive rewrite system:
3.35/1.60	The TRS R consists of the following rules:
3.35/1.60	
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	The replacement map contains the following entries:
3.35/1.60	
3.35/1.60	f_1: empty set
3.35/1.60	f_0: {1}
3.35/1.60	g_1: empty set
3.35/1.60	g_0: {1}
3.35/1.60	b_0: empty set
3.35/1.60	Used ordering:
3.35/1.60	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(f_0(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(g_1(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(b_0) =  	[[0], [1]]
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(f_1(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	   <<<
3.35/1.60	 POL(g_0(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
3.35/1.60	>>>
3.35/1.60	
3.35/1.60	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.35/1.60	
3.35/1.60	   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	
3.35/1.60	
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(8)
3.35/1.60	Obligation:
3.35/1.60	Context-sensitive rewrite system:
3.35/1.60	The TRS R consists of the following rules:
3.35/1.60	
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	The replacement map contains the following entries:
3.35/1.60	
3.35/1.60	f_1: empty set
3.35/1.60	f_0: {1}
3.35/1.60	g_1: empty set
3.35/1.60	b_0: empty set
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(9) CSRRRRProof (EQUIVALENT)
3.35/1.60	The following CSR is given: Context-sensitive rewrite system:
3.35/1.60	The TRS R consists of the following rules:
3.35/1.60	
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	The replacement map contains the following entries:
3.35/1.60	
3.35/1.60	f_1: empty set
3.35/1.60	f_0: {1}
3.35/1.60	g_1: empty set
3.35/1.60	b_0: empty set
3.35/1.60	Used ordering:
3.35/1.60	Polynomial interpretation [POLO]:
3.35/1.60	
3.35/1.60	   POL(b_0) = 1
3.35/1.60	   POL(f_0(x_1)) = 2 + x_1
3.35/1.60	   POL(f_1(x_1)) = 0
3.35/1.60	   POL(g_1(x_1)) = 2 + 2*x_1
3.35/1.60	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.35/1.60	
3.35/1.60	   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
3.35/1.60	
3.35/1.60	
3.35/1.60	
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(10)
3.35/1.60	Obligation:
3.35/1.60	Context-sensitive rewrite system:
3.35/1.60	R is empty.
3.35/1.60	
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(11) RisEmptyProof (EQUIVALENT)
3.35/1.60	The CSR R is empty. Hence, termination is trivially proven.
3.35/1.60	----------------------------------------
3.35/1.60	
3.35/1.60	(12)
3.35/1.60	YES
3.63/1.63	EOF