3.32/2.29	YES
3.34/2.30	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.34/2.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.34/2.30	
3.34/2.30	
3.34/2.30	Termination of the given CSR could be proven:
3.34/2.30	
3.34/2.30	(0) CSR
3.34/2.30	(1) CSRRRRProof [EQUIVALENT, 48 ms]
3.34/2.30	(2) CSR
3.34/2.30	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.34/2.30	(4) CSR
3.34/2.30	(5) CSRRRRProof [EQUIVALENT, 7 ms]
3.34/2.30	(6) CSR
3.34/2.30	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.34/2.30	(8) YES
3.34/2.30	
3.34/2.30	
3.34/2.30	----------------------------------------
3.34/2.30	
3.34/2.30	(0)
3.34/2.30	Obligation:
3.34/2.30	Context-sensitive rewrite system:
3.34/2.30	The TRS R consists of the following rules:
3.34/2.30	
3.34/2.30	   f(0) -> cons(0, f(s(0)))
3.34/2.30	   f(s(0)) -> f(p(s(0)))
3.34/2.30	   p(s(0)) -> 0
3.34/2.30	
3.34/2.30	The replacement map contains the following entries:
3.34/2.30	
3.34/2.30	f: {1}
3.34/2.30	0: empty set
3.34/2.30	cons: {1}
3.34/2.30	s: {1}
3.34/2.30	p: {1}
3.34/2.30	
3.34/2.30	----------------------------------------
3.34/2.30	
3.34/2.30	(1) CSRRRRProof (EQUIVALENT)
3.34/2.30	The following CSR is given: Context-sensitive rewrite system:
3.34/2.30	The TRS R consists of the following rules:
3.34/2.30	
3.34/2.30	   f(0) -> cons(0, f(s(0)))
3.34/2.30	   f(s(0)) -> f(p(s(0)))
3.34/2.30	   p(s(0)) -> 0
3.34/2.30	
3.34/2.30	The replacement map contains the following entries:
3.34/2.30	
3.34/2.30	f: {1}
3.34/2.30	0: empty set
3.34/2.30	cons: {1}
3.34/2.30	s: {1}
3.34/2.30	p: {1}
3.34/2.30	Used ordering:
3.34/2.30	Polynomial interpretation [POLO]:
3.34/2.30	
3.34/2.30	   POL(0) = 1
3.34/2.30	   POL(cons(x_1, x_2)) = 1 + x_1
3.34/2.30	   POL(f(x_1)) = 1 + x_1
3.34/2.30	   POL(p(x_1)) = x_1
3.34/2.30	   POL(s(x_1)) = 1 + x_1
3.34/2.30	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.34/2.30	
3.34/2.30	   p(s(0)) -> 0
3.34/2.30	
3.34/2.30	
3.34/2.30	
3.34/2.30	
3.34/2.30	----------------------------------------
3.34/2.30	
3.34/2.30	(2)
3.34/2.30	Obligation:
3.34/2.30	Context-sensitive rewrite system:
3.34/2.30	The TRS R consists of the following rules:
3.34/2.30	
3.34/2.30	   f(0) -> cons(0, f(s(0)))
3.34/2.30	   f(s(0)) -> f(p(s(0)))
3.34/2.30	
3.34/2.30	The replacement map contains the following entries:
3.34/2.30	
3.34/2.30	f: {1}
3.34/2.30	0: empty set
3.34/2.30	cons: {1}
3.34/2.30	s: {1}
3.34/2.30	p: {1}
3.34/2.30	
3.34/2.30	----------------------------------------
3.34/2.30	
3.34/2.30	(3) CSRRRRProof (EQUIVALENT)
3.34/2.30	The following CSR is given: Context-sensitive rewrite system:
3.34/2.30	The TRS R consists of the following rules:
3.34/2.30	
3.34/2.30	   f(0) -> cons(0, f(s(0)))
3.34/2.30	   f(s(0)) -> f(p(s(0)))
3.34/2.30	
3.34/2.30	The replacement map contains the following entries:
3.34/2.30	
3.34/2.30	f: {1}
3.34/2.30	0: empty set
3.34/2.30	cons: {1}
3.34/2.30	s: {1}
3.34/2.30	p: {1}
3.34/2.30	Used ordering:
3.34/2.30	Polynomial interpretation [POLO]:
3.34/2.30	
3.34/2.30	   POL(0) = 1
3.34/2.30	   POL(cons(x_1, x_2)) = x_1
3.34/2.30	   POL(f(x_1)) = 1 + x_1
3.34/2.30	   POL(p(x_1)) = x_1
3.34/2.30	   POL(s(x_1)) = 1 + x_1
3.34/2.30	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.34/2.30	
3.34/2.30	   f(0) -> cons(0, f(s(0)))
3.34/2.30	
3.34/2.30	
3.34/2.30	
3.34/2.30	
3.34/2.30	----------------------------------------
3.34/2.30	
3.34/2.30	(4)
3.34/2.30	Obligation:
3.34/2.30	Context-sensitive rewrite system:
3.34/2.30	The TRS R consists of the following rules:
3.34/2.30	
3.34/2.30	   f(s(0)) -> f(p(s(0)))
3.34/2.30	
3.34/2.30	The replacement map contains the following entries:
3.34/2.30	
3.34/2.30	f: {1}
3.34/2.30	0: empty set
3.34/2.30	s: {1}
3.34/2.30	p: {1}
3.34/2.30	
3.34/2.30	----------------------------------------
3.34/2.30	
3.34/2.30	(5) CSRRRRProof (EQUIVALENT)
3.34/2.30	The following CSR is given: Context-sensitive rewrite system:
3.34/2.30	The TRS R consists of the following rules:
3.34/2.30	
3.34/2.30	   f(s(0)) -> f(p(s(0)))
3.34/2.30	
3.34/2.30	The replacement map contains the following entries:
3.34/2.30	
3.34/2.30	f: {1}
3.34/2.30	0: empty set
3.34/2.30	s: {1}
3.34/2.30	p: {1}
3.34/2.30	Used ordering:
3.34/2.30	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.34/2.30	
3.34/2.30	   <<<
3.34/2.30	 POL(f(x_1)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1
3.34/2.30	>>>
3.34/2.30	
3.34/2.30	   <<<
3.34/2.30	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
3.34/2.30	>>>
3.34/2.30	
3.34/2.30	   <<<
3.34/2.30	 POL(0) =  	[[1], [1]]
3.34/2.30	>>>
3.34/2.30	
3.34/2.30	   <<<
3.34/2.30	 POL(p(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
3.34/2.30	>>>
3.34/2.30	
3.34/2.30	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.34/2.30	
3.34/2.30	   f(s(0)) -> f(p(s(0)))
3.34/2.30	
3.34/2.30	
3.34/2.30	
3.34/2.30	
3.34/2.30	----------------------------------------
3.34/2.30	
3.34/2.30	(6)
3.34/2.30	Obligation:
3.34/2.30	Context-sensitive rewrite system:
3.34/2.30	R is empty.
3.34/2.30	
3.34/2.30	----------------------------------------
3.34/2.30	
3.34/2.30	(7) RisEmptyProof (EQUIVALENT)
3.34/2.30	The CSR R is empty. Hence, termination is trivially proven.
3.34/2.30	----------------------------------------
3.34/2.30	
3.34/2.30	(8)
3.34/2.30	YES
3.34/2.37	EOF