3.23/1.47	YES
3.23/1.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.23/1.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.23/1.48	
3.23/1.48	
3.23/1.48	Termination of the given CSR could be proven:
3.23/1.48	
3.23/1.48	(0) CSR
3.23/1.48	(1) CSRRRRProof [EQUIVALENT, 53 ms]
3.23/1.48	(2) CSR
3.23/1.48	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.23/1.48	(4) CSR
3.23/1.48	(5) CSRRRRProof [EQUIVALENT, 5 ms]
3.23/1.48	(6) CSR
3.23/1.48	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.23/1.48	(8) CSR
3.23/1.48	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.23/1.48	(10) YES
3.23/1.48	
3.23/1.48	
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(0)
3.23/1.48	Obligation:
3.23/1.48	Context-sensitive rewrite system:
3.23/1.48	The TRS R consists of the following rules:
3.23/1.48	
3.23/1.48	   f_1(h_1(x)) -> f_1(i_1(x))
3.23/1.48	   h_1(x) -> f_1(h_1(x))
3.23/1.48	   *top*_0(f_1(i_1(x))) -> *top*_0(x)
3.23/1.48	   f_0(f_1(i_1(x))) -> f_1(x)
3.23/1.48	   f_1(i_1(x)) -> x
3.23/1.48	   i_1(x) -> h_1(x)
3.23/1.48	
3.23/1.48	The replacement map contains the following entries:
3.23/1.48	
3.23/1.48	f_1: empty set
3.23/1.48	h_1: empty set
3.23/1.48	i_1: empty set
3.23/1.48	*top*_0: {1}
3.23/1.48	f_0: {1}
3.23/1.48	
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(1) CSRRRRProof (EQUIVALENT)
3.23/1.48	The following CSR is given: Context-sensitive rewrite system:
3.23/1.48	The TRS R consists of the following rules:
3.23/1.48	
3.23/1.48	   f_1(h_1(x)) -> f_1(i_1(x))
3.23/1.48	   h_1(x) -> f_1(h_1(x))
3.23/1.48	   *top*_0(f_1(i_1(x))) -> *top*_0(x)
3.23/1.48	   f_0(f_1(i_1(x))) -> f_1(x)
3.23/1.48	   f_1(i_1(x)) -> x
3.23/1.48	   i_1(x) -> h_1(x)
3.23/1.48	
3.23/1.48	The replacement map contains the following entries:
3.23/1.48	
3.23/1.48	f_1: empty set
3.23/1.48	h_1: empty set
3.23/1.48	i_1: empty set
3.23/1.48	*top*_0: {1}
3.23/1.48	f_0: {1}
3.23/1.48	Used ordering:
3.23/1.48	Polynomial interpretation [POLO]:
3.23/1.48	
3.23/1.48	   POL(*top*_0(x_1)) = x_1
3.23/1.48	   POL(f_0(x_1)) = 2 + x_1
3.23/1.48	   POL(f_1(x_1)) = x_1
3.23/1.48	   POL(h_1(x_1)) = 2*x_1
3.23/1.48	   POL(i_1(x_1)) = 2*x_1
3.23/1.48	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.23/1.48	
3.23/1.48	   f_0(f_1(i_1(x))) -> f_1(x)
3.23/1.48	
3.23/1.48	
3.23/1.48	
3.23/1.48	
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(2)
3.23/1.48	Obligation:
3.23/1.48	Context-sensitive rewrite system:
3.23/1.48	The TRS R consists of the following rules:
3.23/1.48	
3.23/1.48	   f_1(h_1(x)) -> f_1(i_1(x))
3.23/1.48	   h_1(x) -> f_1(h_1(x))
3.23/1.48	   *top*_0(f_1(i_1(x))) -> *top*_0(x)
3.23/1.48	   f_1(i_1(x)) -> x
3.23/1.48	   i_1(x) -> h_1(x)
3.23/1.48	
3.23/1.48	The replacement map contains the following entries:
3.23/1.48	
3.23/1.48	f_1: empty set
3.23/1.48	h_1: empty set
3.23/1.48	i_1: empty set
3.23/1.48	*top*_0: {1}
3.23/1.48	
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(3) CSRRRRProof (EQUIVALENT)
3.23/1.48	The following CSR is given: Context-sensitive rewrite system:
3.23/1.48	The TRS R consists of the following rules:
3.23/1.48	
3.23/1.48	   f_1(h_1(x)) -> f_1(i_1(x))
3.23/1.48	   h_1(x) -> f_1(h_1(x))
3.23/1.48	   *top*_0(f_1(i_1(x))) -> *top*_0(x)
3.23/1.48	   f_1(i_1(x)) -> x
3.23/1.48	   i_1(x) -> h_1(x)
3.23/1.48	
3.23/1.48	The replacement map contains the following entries:
3.23/1.48	
3.23/1.48	f_1: empty set
3.23/1.48	h_1: empty set
3.23/1.48	i_1: empty set
3.23/1.48	*top*_0: {1}
3.23/1.48	Used ordering:
3.23/1.48	Polynomial interpretation [POLO]:
3.23/1.48	
3.23/1.48	   POL(*top*_0(x_1)) = x_1
3.23/1.48	   POL(f_1(x_1)) = x_1
3.23/1.48	   POL(h_1(x_1)) = 1 + x_1
3.23/1.48	   POL(i_1(x_1)) = 1 + x_1
3.23/1.48	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.23/1.48	
3.23/1.48	   *top*_0(f_1(i_1(x))) -> *top*_0(x)
3.23/1.48	   f_1(i_1(x)) -> x
3.23/1.48	
3.23/1.48	
3.23/1.48	
3.23/1.48	
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(4)
3.23/1.48	Obligation:
3.23/1.48	Context-sensitive rewrite system:
3.23/1.48	The TRS R consists of the following rules:
3.23/1.48	
3.23/1.48	   f_1(h_1(x)) -> f_1(i_1(x))
3.23/1.48	   h_1(x) -> f_1(h_1(x))
3.23/1.48	   i_1(x) -> h_1(x)
3.23/1.48	
3.23/1.48	The replacement map contains the following entries:
3.23/1.48	
3.23/1.48	f_1: empty set
3.23/1.48	h_1: empty set
3.23/1.48	i_1: empty set
3.23/1.48	
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(5) CSRRRRProof (EQUIVALENT)
3.23/1.48	The following CSR is given: Context-sensitive rewrite system:
3.23/1.48	The TRS R consists of the following rules:
3.23/1.48	
3.23/1.48	   f_1(h_1(x)) -> f_1(i_1(x))
3.23/1.48	   h_1(x) -> f_1(h_1(x))
3.23/1.48	   i_1(x) -> h_1(x)
3.23/1.48	
3.23/1.48	The replacement map contains the following entries:
3.23/1.48	
3.23/1.48	f_1: empty set
3.23/1.48	h_1: empty set
3.23/1.48	i_1: empty set
3.23/1.48	Used ordering:
3.23/1.48	Polynomial interpretation [POLO]:
3.23/1.48	
3.23/1.48	   POL(f_1(x_1)) = 0
3.23/1.48	   POL(h_1(x_1)) = 1 + 2*x_1
3.23/1.48	   POL(i_1(x_1)) = 2 + 2*x_1
3.23/1.48	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.23/1.48	
3.23/1.48	   h_1(x) -> f_1(h_1(x))
3.23/1.48	   i_1(x) -> h_1(x)
3.23/1.48	
3.23/1.48	
3.23/1.48	
3.23/1.48	
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(6)
3.23/1.48	Obligation:
3.23/1.48	Context-sensitive rewrite system:
3.23/1.48	The TRS R consists of the following rules:
3.23/1.48	
3.23/1.48	   f_1(h_1(x)) -> f_1(i_1(x))
3.23/1.48	
3.23/1.48	The replacement map contains the following entries:
3.23/1.48	
3.23/1.48	f_1: empty set
3.23/1.48	h_1: empty set
3.23/1.48	i_1: empty set
3.23/1.48	
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(7) CSRRRRProof (EQUIVALENT)
3.23/1.48	The following CSR is given: Context-sensitive rewrite system:
3.23/1.48	The TRS R consists of the following rules:
3.23/1.48	
3.23/1.48	   f_1(h_1(x)) -> f_1(i_1(x))
3.23/1.48	
3.23/1.48	The replacement map contains the following entries:
3.23/1.48	
3.23/1.48	f_1: empty set
3.23/1.48	h_1: empty set
3.23/1.48	i_1: empty set
3.23/1.48	Used ordering:
3.23/1.48	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.23/1.48	
3.23/1.48	   <<<
3.23/1.48	 POL(f_1(x_1)) =  	[[1]] 	 +  	[[1, 1]] 	* 	x_1
3.23/1.48	>>>
3.23/1.48	
3.23/1.48	   <<<
3.23/1.48	 POL(h_1(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
3.23/1.48	>>>
3.23/1.48	
3.23/1.48	   <<<
3.23/1.48	 POL(i_1(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
3.23/1.48	>>>
3.23/1.48	
3.23/1.48	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.23/1.48	
3.23/1.48	   f_1(h_1(x)) -> f_1(i_1(x))
3.23/1.48	
3.23/1.48	
3.23/1.48	
3.23/1.48	
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(8)
3.23/1.48	Obligation:
3.23/1.48	Context-sensitive rewrite system:
3.23/1.48	R is empty.
3.23/1.48	
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(9) RisEmptyProof (EQUIVALENT)
3.23/1.48	The CSR R is empty. Hence, termination is trivially proven.
3.23/1.48	----------------------------------------
3.23/1.48	
3.23/1.48	(10)
3.23/1.48	YES
3.23/1.49	EOF