3.00/1.45	YES
3.00/1.46	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.00/1.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.00/1.46	
3.00/1.46	
3.00/1.46	Termination of the given CSR could be proven:
3.00/1.46	
3.00/1.46	(0) CSR
3.00/1.46	(1) CSRRRRProof [EQUIVALENT, 76 ms]
3.00/1.46	(2) CSR
3.00/1.46	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.00/1.46	(4) CSR
3.00/1.46	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.00/1.46	(6) CSR
3.00/1.46	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.00/1.46	(8) YES
3.00/1.46	
3.00/1.46	
3.00/1.46	----------------------------------------
3.00/1.46	
3.00/1.46	(0)
3.00/1.46	Obligation:
3.00/1.46	Context-sensitive rewrite system:
3.00/1.46	The TRS R consists of the following rules:
3.00/1.46	
3.00/1.46	   U11(tt, M, N) -> U12(tt, M, N)
3.00/1.46	   U12(tt, M, N) -> s(plus(N, M))
3.00/1.46	   plus(N, 0) -> N
3.00/1.46	   plus(N, s(M)) -> U11(tt, M, N)
3.00/1.46	
3.00/1.46	The replacement map contains the following entries:
3.00/1.46	
3.00/1.46	U11: {1}
3.00/1.46	tt: empty set
3.00/1.46	U12: {1}
3.00/1.46	s: {1}
3.00/1.46	plus: {1, 2}
3.00/1.46	0: empty set
3.00/1.46	
3.00/1.46	----------------------------------------
3.00/1.46	
3.00/1.46	(1) CSRRRRProof (EQUIVALENT)
3.00/1.46	The following CSR is given: Context-sensitive rewrite system:
3.00/1.46	The TRS R consists of the following rules:
3.00/1.46	
3.00/1.46	   U11(tt, M, N) -> U12(tt, M, N)
3.00/1.46	   U12(tt, M, N) -> s(plus(N, M))
3.00/1.46	   plus(N, 0) -> N
3.00/1.46	   plus(N, s(M)) -> U11(tt, M, N)
3.00/1.46	
3.00/1.46	The replacement map contains the following entries:
3.00/1.46	
3.00/1.46	U11: {1}
3.00/1.46	tt: empty set
3.00/1.46	U12: {1}
3.00/1.46	s: {1}
3.00/1.46	plus: {1, 2}
3.00/1.46	0: empty set
3.00/1.46	Used ordering:
3.00/1.46	Polynomial interpretation [POLO]:
3.00/1.46	
3.00/1.46	   POL(0) = 2
3.00/1.46	   POL(U11(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
3.00/1.46	   POL(U12(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
3.00/1.46	   POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2
3.00/1.46	   POL(s(x_1)) = x_1
3.00/1.46	   POL(tt) = 0
3.00/1.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.00/1.46	
3.00/1.46	   plus(N, 0) -> N
3.00/1.46	
3.00/1.46	
3.00/1.46	
3.00/1.46	
3.00/1.46	----------------------------------------
3.00/1.46	
3.00/1.46	(2)
3.00/1.46	Obligation:
3.00/1.46	Context-sensitive rewrite system:
3.00/1.46	The TRS R consists of the following rules:
3.00/1.46	
3.00/1.46	   U11(tt, M, N) -> U12(tt, M, N)
3.00/1.46	   U12(tt, M, N) -> s(plus(N, M))
3.00/1.46	   plus(N, s(M)) -> U11(tt, M, N)
3.00/1.46	
3.00/1.46	The replacement map contains the following entries:
3.00/1.46	
3.00/1.46	U11: {1}
3.00/1.46	tt: empty set
3.00/1.46	U12: {1}
3.00/1.46	s: {1}
3.00/1.46	plus: {1, 2}
3.00/1.46	
3.00/1.46	----------------------------------------
3.00/1.46	
3.00/1.46	(3) CSRRRRProof (EQUIVALENT)
3.00/1.46	The following CSR is given: Context-sensitive rewrite system:
3.00/1.46	The TRS R consists of the following rules:
3.00/1.46	
3.00/1.46	   U11(tt, M, N) -> U12(tt, M, N)
3.00/1.46	   U12(tt, M, N) -> s(plus(N, M))
3.00/1.46	   plus(N, s(M)) -> U11(tt, M, N)
3.00/1.46	
3.00/1.46	The replacement map contains the following entries:
3.00/1.46	
3.00/1.46	U11: {1}
3.00/1.46	tt: empty set
3.00/1.46	U12: {1}
3.00/1.46	s: {1}
3.00/1.46	plus: {1, 2}
3.00/1.46	Used ordering:
3.00/1.46	Polynomial interpretation [POLO]:
3.00/1.46	
3.00/1.46	   POL(U11(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
3.00/1.46	   POL(U12(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
3.00/1.46	   POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2
3.00/1.46	   POL(s(x_1)) = 2 + x_1
3.00/1.46	   POL(tt) = 1
3.00/1.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.00/1.46	
3.00/1.46	   U12(tt, M, N) -> s(plus(N, M))
3.00/1.46	   plus(N, s(M)) -> U11(tt, M, N)
3.00/1.46	
3.00/1.46	
3.00/1.46	
3.00/1.46	
3.00/1.46	----------------------------------------
3.00/1.46	
3.00/1.46	(4)
3.00/1.46	Obligation:
3.00/1.46	Context-sensitive rewrite system:
3.00/1.46	The TRS R consists of the following rules:
3.00/1.46	
3.00/1.46	   U11(tt, M, N) -> U12(tt, M, N)
3.00/1.46	
3.00/1.46	The replacement map contains the following entries:
3.00/1.46	
3.00/1.46	U11: {1}
3.00/1.46	tt: empty set
3.00/1.46	U12: {1}
3.00/1.46	
3.00/1.46	----------------------------------------
3.00/1.46	
3.00/1.46	(5) CSRRRRProof (EQUIVALENT)
3.00/1.46	The following CSR is given: Context-sensitive rewrite system:
3.00/1.46	The TRS R consists of the following rules:
3.00/1.46	
3.00/1.46	   U11(tt, M, N) -> U12(tt, M, N)
3.00/1.46	
3.00/1.46	The replacement map contains the following entries:
3.00/1.46	
3.00/1.46	U11: {1}
3.00/1.46	tt: empty set
3.00/1.46	U12: {1}
3.00/1.46	Used ordering:
3.00/1.46	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.00/1.46	
3.00/1.46	   <<<
3.00/1.46	 POL(U11(x_1, x_2, x_3)) =  	[[1]] 	 +  	[[1, 1]] 	* 	x_1 	 +  	[[1, 1]] 	* 	x_2 	 +  	[[1, 1]] 	* 	x_3
3.00/1.46	>>>
3.00/1.46	
3.00/1.46	   <<<
3.00/1.46	 POL(tt) =  	[[1], [1]]
3.00/1.46	>>>
3.00/1.46	
3.00/1.46	   <<<
3.00/1.46	 POL(U12(x_1, x_2, x_3)) =  	[[1]] 	 +  	[[1, 0]] 	* 	x_1 	 +  	[[1, 1]] 	* 	x_2 	 +  	[[1, 1]] 	* 	x_3
3.00/1.46	>>>
3.00/1.46	
3.00/1.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.00/1.46	
3.00/1.46	   U11(tt, M, N) -> U12(tt, M, N)
3.00/1.46	
3.00/1.46	
3.00/1.46	
3.00/1.46	
3.00/1.46	----------------------------------------
3.00/1.46	
3.00/1.46	(6)
3.00/1.46	Obligation:
3.00/1.46	Context-sensitive rewrite system:
3.00/1.46	R is empty.
3.00/1.46	
3.00/1.46	----------------------------------------
3.00/1.46	
3.00/1.46	(7) RisEmptyProof (EQUIVALENT)
3.00/1.46	The CSR R is empty. Hence, termination is trivially proven.
3.00/1.46	----------------------------------------
3.00/1.46	
3.00/1.46	(8)
3.00/1.46	YES
3.00/1.47	EOF