3.43/1.49	YES
3.43/1.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.43/1.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.43/1.50	
3.43/1.50	
3.43/1.50	Termination of the given CSR could be proven:
3.43/1.50	
3.43/1.50	(0) CSR
3.43/1.50	(1) CSRRRRProof [EQUIVALENT, 52 ms]
3.43/1.50	(2) CSR
3.43/1.50	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.43/1.50	(4) CSR
3.43/1.50	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.43/1.50	(6) CSR
3.43/1.50	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.43/1.50	(8) CSR
3.43/1.50	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.43/1.50	(10) YES
3.43/1.50	
3.43/1.50	
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(0)
3.43/1.50	Obligation:
3.43/1.50	Context-sensitive rewrite system:
3.43/1.50	The TRS R consists of the following rules:
3.43/1.50	
3.43/1.50	   f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
3.43/1.50	   f_1(f_0(x)) -> b_0
3.43/1.50	   f_1(f_1(x)) -> b_0
3.43/1.50	   g_1(x) -> f_1(g_1(x))
3.43/1.50	   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
3.43/1.50	   f_0(g_1(x)) -> f_1(f_0(g_1(x)))
3.43/1.50	   s_0(g_1(x)) -> s_0(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	The replacement map contains the following entries:
3.43/1.50	
3.43/1.50	f_1: empty set
3.43/1.50	g_1: empty set
3.43/1.50	a_0: empty set
3.43/1.50	f_0: {1}
3.43/1.50	s_0: {1}
3.43/1.50	b_0: empty set
3.43/1.50	*top*_0: {1}
3.43/1.50	
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(1) CSRRRRProof (EQUIVALENT)
3.43/1.50	The following CSR is given: Context-sensitive rewrite system:
3.43/1.50	The TRS R consists of the following rules:
3.43/1.50	
3.43/1.50	   f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
3.43/1.50	   f_1(f_0(x)) -> b_0
3.43/1.50	   f_1(f_1(x)) -> b_0
3.43/1.50	   g_1(x) -> f_1(g_1(x))
3.43/1.50	   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
3.43/1.50	   f_0(g_1(x)) -> f_1(f_0(g_1(x)))
3.43/1.50	   s_0(g_1(x)) -> s_0(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	The replacement map contains the following entries:
3.43/1.50	
3.43/1.50	f_1: empty set
3.43/1.50	g_1: empty set
3.43/1.50	a_0: empty set
3.43/1.50	f_0: {1}
3.43/1.50	s_0: {1}
3.43/1.50	b_0: empty set
3.43/1.50	*top*_0: {1}
3.43/1.50	Used ordering:
3.43/1.50	Polynomial interpretation [POLO]:
3.43/1.50	
3.43/1.50	   POL(*top*_0(x_1)) = x_1
3.43/1.50	   POL(a_0) = 1
3.43/1.50	   POL(b_0) = 0
3.43/1.50	   POL(f_0(x_1)) = x_1
3.43/1.50	   POL(f_1(x_1)) = x_1
3.43/1.50	   POL(g_1(x_1)) = 2*x_1
3.43/1.50	   POL(s_0(x_1)) = 1 + x_1
3.43/1.50	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/1.50	
3.43/1.50	   f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
3.43/1.50	
3.43/1.50	
3.43/1.50	
3.43/1.50	
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(2)
3.43/1.50	Obligation:
3.43/1.50	Context-sensitive rewrite system:
3.43/1.50	The TRS R consists of the following rules:
3.43/1.50	
3.43/1.50	   f_1(f_0(x)) -> b_0
3.43/1.50	   f_1(f_1(x)) -> b_0
3.43/1.50	   g_1(x) -> f_1(g_1(x))
3.43/1.50	   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
3.43/1.50	   f_0(g_1(x)) -> f_1(f_0(g_1(x)))
3.43/1.50	   s_0(g_1(x)) -> s_0(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	The replacement map contains the following entries:
3.43/1.50	
3.43/1.50	f_1: empty set
3.43/1.50	g_1: empty set
3.43/1.50	f_0: {1}
3.43/1.50	s_0: {1}
3.43/1.50	b_0: empty set
3.43/1.50	*top*_0: {1}
3.43/1.50	
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(3) CSRRRRProof (EQUIVALENT)
3.43/1.50	The following CSR is given: Context-sensitive rewrite system:
3.43/1.50	The TRS R consists of the following rules:
3.43/1.50	
3.43/1.50	   f_1(f_0(x)) -> b_0
3.43/1.50	   f_1(f_1(x)) -> b_0
3.43/1.50	   g_1(x) -> f_1(g_1(x))
3.43/1.50	   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
3.43/1.50	   f_0(g_1(x)) -> f_1(f_0(g_1(x)))
3.43/1.50	   s_0(g_1(x)) -> s_0(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	The replacement map contains the following entries:
3.43/1.50	
3.43/1.50	f_1: empty set
3.43/1.50	g_1: empty set
3.43/1.50	f_0: {1}
3.43/1.50	s_0: {1}
3.43/1.50	b_0: empty set
3.43/1.50	*top*_0: {1}
3.43/1.50	Used ordering:
3.43/1.50	Polynomial interpretation [POLO]:
3.43/1.50	
3.43/1.50	   POL(*top*_0(x_1)) = x_1
3.43/1.50	   POL(b_0) = 0
3.43/1.50	   POL(f_0(x_1)) = x_1
3.43/1.50	   POL(f_1(x_1)) = 1
3.43/1.50	   POL(g_1(x_1)) = 2
3.43/1.50	   POL(s_0(x_1)) = x_1
3.43/1.50	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/1.50	
3.43/1.50	   f_1(f_0(x)) -> b_0
3.43/1.50	   f_1(f_1(x)) -> b_0
3.43/1.50	   g_1(x) -> f_1(g_1(x))
3.43/1.50	   f_0(g_1(x)) -> f_1(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	
3.43/1.50	
3.43/1.50	
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(4)
3.43/1.50	Obligation:
3.43/1.50	Context-sensitive rewrite system:
3.43/1.50	The TRS R consists of the following rules:
3.43/1.50	
3.43/1.50	   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
3.43/1.50	   s_0(g_1(x)) -> s_0(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	The replacement map contains the following entries:
3.43/1.50	
3.43/1.50	g_1: empty set
3.43/1.50	f_0: {1}
3.43/1.50	s_0: {1}
3.43/1.50	*top*_0: {1}
3.43/1.50	
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(5) CSRRRRProof (EQUIVALENT)
3.43/1.50	The following CSR is given: Context-sensitive rewrite system:
3.43/1.50	The TRS R consists of the following rules:
3.43/1.50	
3.43/1.50	   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
3.43/1.50	   s_0(g_1(x)) -> s_0(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	The replacement map contains the following entries:
3.43/1.50	
3.43/1.50	g_1: empty set
3.43/1.50	f_0: {1}
3.43/1.50	s_0: {1}
3.43/1.50	*top*_0: {1}
3.43/1.50	Used ordering:
3.43/1.50	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.43/1.50	
3.43/1.50	   <<<
3.43/1.50	 POL(*top*_0(x_1)) =  	[[1]] 	 +  	[[1, 1]] 	* 	x_1
3.43/1.50	>>>
3.43/1.50	
3.43/1.50	   <<<
3.43/1.50	 POL(g_1(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
3.43/1.50	>>>
3.43/1.50	
3.43/1.50	   <<<
3.43/1.50	 POL(f_0(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
3.43/1.50	>>>
3.43/1.50	
3.43/1.50	   <<<
3.43/1.50	 POL(s_0(x_1)) =  	[[0]] 	 +  	[[1, 0]] 	* 	x_1
3.43/1.50	>>>
3.43/1.50	
3.43/1.50	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/1.50	
3.43/1.50	   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	
3.43/1.50	
3.43/1.50	
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(6)
3.43/1.50	Obligation:
3.43/1.50	Context-sensitive rewrite system:
3.43/1.50	The TRS R consists of the following rules:
3.43/1.50	
3.43/1.50	   s_0(g_1(x)) -> s_0(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	The replacement map contains the following entries:
3.43/1.50	
3.43/1.50	g_1: empty set
3.43/1.50	f_0: {1}
3.43/1.50	s_0: {1}
3.43/1.50	
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(7) CSRRRRProof (EQUIVALENT)
3.43/1.50	The following CSR is given: Context-sensitive rewrite system:
3.43/1.50	The TRS R consists of the following rules:
3.43/1.50	
3.43/1.50	   s_0(g_1(x)) -> s_0(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	The replacement map contains the following entries:
3.43/1.50	
3.43/1.50	g_1: empty set
3.43/1.50	f_0: {1}
3.43/1.50	s_0: {1}
3.43/1.50	Used ordering:
3.43/1.50	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.43/1.50	
3.43/1.50	   <<<
3.43/1.50	 POL(s_0(x_1)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1
3.43/1.50	>>>
3.43/1.50	
3.43/1.50	   <<<
3.43/1.50	 POL(g_1(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
3.43/1.50	>>>
3.43/1.50	
3.43/1.50	   <<<
3.43/1.50	 POL(f_0(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
3.43/1.50	>>>
3.43/1.50	
3.43/1.50	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/1.50	
3.43/1.50	   s_0(g_1(x)) -> s_0(f_0(g_1(x)))
3.43/1.50	
3.43/1.50	
3.43/1.50	
3.43/1.50	
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(8)
3.43/1.50	Obligation:
3.43/1.50	Context-sensitive rewrite system:
3.43/1.50	R is empty.
3.43/1.50	
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(9) RisEmptyProof (EQUIVALENT)
3.43/1.50	The CSR R is empty. Hence, termination is trivially proven.
3.43/1.50	----------------------------------------
3.43/1.50	
3.43/1.50	(10)
3.43/1.50	YES
3.43/1.52	EOF