3.20/1.46	YES
3.24/1.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.24/1.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.24/1.47	
3.24/1.47	
3.24/1.47	Termination of the given CSR could be proven:
3.24/1.47	
3.24/1.47	(0) CSR
3.24/1.47	(1) CSRRRRProof [EQUIVALENT, 74 ms]
3.24/1.47	(2) CSR
3.24/1.47	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.24/1.47	(4) CSR
3.24/1.47	(5) CSRRRRProof [EQUIVALENT, 6 ms]
3.24/1.47	(6) CSR
3.24/1.47	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.24/1.47	(8) YES
3.24/1.47	
3.24/1.47	
3.24/1.47	----------------------------------------
3.24/1.47	
3.24/1.47	(0)
3.24/1.47	Obligation:
3.24/1.47	Context-sensitive rewrite system:
3.24/1.47	The TRS R consists of the following rules:
3.24/1.47	
3.24/1.47	   f(0) -> cons(0, f(s(0)))
3.24/1.47	   f(s(0)) -> f(p(s(0)))
3.24/1.47	   p(s(X)) -> X
3.24/1.47	
3.24/1.47	The replacement map contains the following entries:
3.24/1.47	
3.24/1.47	f: {1}
3.24/1.47	0: empty set
3.24/1.47	cons: {1}
3.24/1.47	s: {1}
3.24/1.47	p: {1}
3.24/1.47	
3.24/1.47	----------------------------------------
3.24/1.47	
3.24/1.47	(1) CSRRRRProof (EQUIVALENT)
3.24/1.47	The following CSR is given: Context-sensitive rewrite system:
3.24/1.47	The TRS R consists of the following rules:
3.24/1.47	
3.24/1.47	   f(0) -> cons(0, f(s(0)))
3.24/1.47	   f(s(0)) -> f(p(s(0)))
3.24/1.47	   p(s(X)) -> X
3.24/1.47	
3.24/1.47	The replacement map contains the following entries:
3.24/1.47	
3.24/1.47	f: {1}
3.24/1.47	0: empty set
3.24/1.47	cons: {1}
3.24/1.47	s: {1}
3.24/1.47	p: {1}
3.24/1.47	Used ordering:
3.24/1.47	Polynomial interpretation [POLO]:
3.24/1.47	
3.24/1.47	   POL(0) = 0
3.24/1.47	   POL(cons(x_1, x_2)) = x_1
3.24/1.47	   POL(f(x_1)) = 1 + x_1
3.24/1.47	   POL(p(x_1)) = 2*x_1
3.24/1.47	   POL(s(x_1)) = 2*x_1
3.24/1.47	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.24/1.47	
3.24/1.47	   f(0) -> cons(0, f(s(0)))
3.24/1.47	
3.24/1.47	
3.24/1.47	
3.24/1.47	
3.24/1.47	----------------------------------------
3.24/1.47	
3.24/1.47	(2)
3.24/1.47	Obligation:
3.24/1.47	Context-sensitive rewrite system:
3.24/1.47	The TRS R consists of the following rules:
3.24/1.47	
3.24/1.47	   f(s(0)) -> f(p(s(0)))
3.24/1.47	   p(s(X)) -> X
3.24/1.47	
3.24/1.47	The replacement map contains the following entries:
3.24/1.47	
3.24/1.47	f: {1}
3.24/1.47	0: empty set
3.24/1.47	s: {1}
3.24/1.47	p: {1}
3.24/1.47	
3.24/1.47	----------------------------------------
3.24/1.47	
3.24/1.47	(3) CSRRRRProof (EQUIVALENT)
3.24/1.47	The following CSR is given: Context-sensitive rewrite system:
3.24/1.47	The TRS R consists of the following rules:
3.24/1.47	
3.24/1.47	   f(s(0)) -> f(p(s(0)))
3.24/1.47	   p(s(X)) -> X
3.24/1.47	
3.24/1.47	The replacement map contains the following entries:
3.24/1.47	
3.24/1.47	f: {1}
3.24/1.47	0: empty set
3.24/1.47	s: {1}
3.24/1.47	p: {1}
3.24/1.47	Used ordering:
3.24/1.47	Polynomial interpretation [POLO]:
3.24/1.47	
3.24/1.47	   POL(0) = 0
3.24/1.47	   POL(f(x_1)) = 2*x_1
3.24/1.47	   POL(p(x_1)) = x_1
3.24/1.47	   POL(s(x_1)) = 1 + x_1
3.24/1.47	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.24/1.47	
3.24/1.47	   p(s(X)) -> X
3.24/1.47	
3.24/1.47	
3.24/1.47	
3.24/1.47	
3.24/1.47	----------------------------------------
3.24/1.47	
3.24/1.47	(4)
3.24/1.47	Obligation:
3.24/1.47	Context-sensitive rewrite system:
3.24/1.47	The TRS R consists of the following rules:
3.24/1.47	
3.24/1.47	   f(s(0)) -> f(p(s(0)))
3.24/1.47	
3.24/1.47	The replacement map contains the following entries:
3.24/1.47	
3.24/1.47	f: {1}
3.24/1.47	0: empty set
3.24/1.47	s: {1}
3.24/1.47	p: {1}
3.24/1.47	
3.24/1.47	----------------------------------------
3.24/1.47	
3.24/1.47	(5) CSRRRRProof (EQUIVALENT)
3.24/1.47	The following CSR is given: Context-sensitive rewrite system:
3.24/1.47	The TRS R consists of the following rules:
3.24/1.47	
3.24/1.47	   f(s(0)) -> f(p(s(0)))
3.24/1.47	
3.24/1.47	The replacement map contains the following entries:
3.24/1.47	
3.24/1.47	f: {1}
3.24/1.47	0: empty set
3.24/1.47	s: {1}
3.24/1.47	p: {1}
3.24/1.47	Used ordering:
3.24/1.47	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.24/1.47	
3.24/1.47	   <<<
3.24/1.47	 POL(f(x_1)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1
3.24/1.47	>>>
3.24/1.47	
3.24/1.47	   <<<
3.24/1.47	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
3.24/1.47	>>>
3.24/1.47	
3.24/1.47	   <<<
3.24/1.47	 POL(0) =  	[[1], [1]]
3.24/1.47	>>>
3.24/1.47	
3.24/1.47	   <<<
3.24/1.47	 POL(p(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
3.24/1.47	>>>
3.24/1.47	
3.24/1.47	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.24/1.47	
3.24/1.47	   f(s(0)) -> f(p(s(0)))
3.24/1.47	
3.24/1.47	
3.24/1.47	
3.24/1.47	
3.24/1.47	----------------------------------------
3.24/1.47	
3.24/1.47	(6)
3.24/1.47	Obligation:
3.24/1.47	Context-sensitive rewrite system:
3.24/1.47	R is empty.
3.24/1.47	
3.24/1.47	----------------------------------------
3.24/1.47	
3.24/1.47	(7) RisEmptyProof (EQUIVALENT)
3.24/1.47	The CSR R is empty. Hence, termination is trivially proven.
3.24/1.47	----------------------------------------
3.24/1.47	
3.24/1.47	(8)
3.24/1.47	YES
3.24/1.50	EOF