3.75/1.74	YES
3.75/1.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.75/1.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.75/1.75	
3.75/1.75	
3.75/1.75	Termination w.r.t. Q of the given QTRS could be proven:
3.75/1.75	
3.75/1.75	(0) QTRS
3.75/1.75	(1) QTRSToCSRProof [SOUND, 0 ms]
3.75/1.75	(2) CSR
3.75/1.75	(3) CSRRRRProof [EQUIVALENT, 56 ms]
3.75/1.75	(4) CSR
3.75/1.75	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.75/1.75	(6) CSR
3.75/1.75	(7) CSRRRRProof [EQUIVALENT, 7 ms]
3.75/1.75	(8) CSR
3.75/1.75	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.75/1.75	(10) YES
3.75/1.75	
3.75/1.75	
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(0)
3.75/1.75	Obligation:
3.75/1.75	Q restricted rewrite system:
3.75/1.75	The TRS R consists of the following rules:
3.75/1.75	
3.75/1.75	   active(U11(tt, M, N)) -> mark(U12(tt, M, N))
3.75/1.75	   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
3.75/1.75	   active(plus(N, 0)) -> mark(N)
3.75/1.75	   active(plus(N, s(M))) -> mark(U11(tt, M, N))
3.75/1.75	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
3.75/1.75	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
3.75/1.75	   active(s(X)) -> s(active(X))
3.75/1.75	   active(plus(X1, X2)) -> plus(active(X1), X2)
3.75/1.75	   active(plus(X1, X2)) -> plus(X1, active(X2))
3.75/1.75	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
3.75/1.75	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
3.75/1.75	   s(mark(X)) -> mark(s(X))
3.75/1.75	   plus(mark(X1), X2) -> mark(plus(X1, X2))
3.75/1.75	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
3.75/1.75	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
3.75/1.75	   proper(tt) -> ok(tt)
3.75/1.75	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
3.75/1.75	   proper(s(X)) -> s(proper(X))
3.75/1.75	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
3.75/1.75	   proper(0) -> ok(0)
3.75/1.75	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
3.75/1.75	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
3.75/1.75	   s(ok(X)) -> ok(s(X))
3.75/1.75	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
3.75/1.75	   top(mark(X)) -> top(proper(X))
3.75/1.75	   top(ok(X)) -> top(active(X))
3.75/1.75	
3.75/1.75	The set Q consists of the following terms:
3.75/1.75	
3.75/1.75	   active(U11(x0, x1, x2))
3.75/1.75	   active(U12(x0, x1, x2))
3.75/1.75	   active(s(x0))
3.75/1.75	   active(plus(x0, x1))
3.75/1.75	   U11(mark(x0), x1, x2)
3.75/1.75	   U12(mark(x0), x1, x2)
3.75/1.75	   s(mark(x0))
3.75/1.75	   plus(mark(x0), x1)
3.75/1.75	   plus(x0, mark(x1))
3.75/1.75	   proper(U11(x0, x1, x2))
3.75/1.75	   proper(tt)
3.75/1.75	   proper(U12(x0, x1, x2))
3.75/1.75	   proper(s(x0))
3.75/1.75	   proper(plus(x0, x1))
3.75/1.75	   proper(0)
3.75/1.75	   U11(ok(x0), ok(x1), ok(x2))
3.75/1.75	   U12(ok(x0), ok(x1), ok(x2))
3.75/1.75	   s(ok(x0))
3.75/1.75	   plus(ok(x0), ok(x1))
3.75/1.75	   top(mark(x0))
3.75/1.75	   top(ok(x0))
3.75/1.75	
3.75/1.75	
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(1) QTRSToCSRProof (SOUND)
3.75/1.75	The following Q TRS is given: Q restricted rewrite system:
3.75/1.75	The TRS R consists of the following rules:
3.75/1.75	
3.75/1.75	   active(U11(tt, M, N)) -> mark(U12(tt, M, N))
3.75/1.75	   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
3.75/1.75	   active(plus(N, 0)) -> mark(N)
3.75/1.75	   active(plus(N, s(M))) -> mark(U11(tt, M, N))
3.75/1.75	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
3.75/1.75	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
3.75/1.75	   active(s(X)) -> s(active(X))
3.75/1.75	   active(plus(X1, X2)) -> plus(active(X1), X2)
3.75/1.75	   active(plus(X1, X2)) -> plus(X1, active(X2))
3.75/1.75	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
3.75/1.75	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
3.75/1.75	   s(mark(X)) -> mark(s(X))
3.75/1.75	   plus(mark(X1), X2) -> mark(plus(X1, X2))
3.75/1.75	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
3.75/1.75	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
3.75/1.75	   proper(tt) -> ok(tt)
3.75/1.75	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
3.75/1.75	   proper(s(X)) -> s(proper(X))
3.75/1.75	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
3.75/1.75	   proper(0) -> ok(0)
3.75/1.75	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
3.75/1.75	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
3.75/1.75	   s(ok(X)) -> ok(s(X))
3.75/1.75	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
3.75/1.75	   top(mark(X)) -> top(proper(X))
3.75/1.75	   top(ok(X)) -> top(active(X))
3.75/1.75	
3.75/1.75	The set Q consists of the following terms:
3.75/1.75	
3.75/1.75	   active(U11(x0, x1, x2))
3.75/1.75	   active(U12(x0, x1, x2))
3.75/1.75	   active(s(x0))
3.75/1.75	   active(plus(x0, x1))
3.75/1.75	   U11(mark(x0), x1, x2)
3.75/1.75	   U12(mark(x0), x1, x2)
3.75/1.75	   s(mark(x0))
3.75/1.75	   plus(mark(x0), x1)
3.75/1.75	   plus(x0, mark(x1))
3.75/1.75	   proper(U11(x0, x1, x2))
3.75/1.75	   proper(tt)
3.75/1.75	   proper(U12(x0, x1, x2))
3.75/1.75	   proper(s(x0))
3.75/1.75	   proper(plus(x0, x1))
3.75/1.75	   proper(0)
3.75/1.75	   U11(ok(x0), ok(x1), ok(x2))
3.75/1.75	   U12(ok(x0), ok(x1), ok(x2))
3.75/1.75	   s(ok(x0))
3.75/1.75	   plus(ok(x0), ok(x1))
3.75/1.75	   top(mark(x0))
3.75/1.75	   top(ok(x0))
3.75/1.75	
3.75/1.75	Special symbols used for the transformation (see [GM04]):
3.75/1.75	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.75/1.75	The replacement map contains the following entries:
3.75/1.75	
3.75/1.75	U11: {1}
3.75/1.75	tt: empty set
3.75/1.75	U12: {1}
3.75/1.75	s: {1}
3.75/1.75	plus: {1, 2}
3.75/1.75	0: empty set
3.75/1.75	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(2)
3.75/1.75	Obligation:
3.75/1.75	Context-sensitive rewrite system:
3.75/1.75	The TRS R consists of the following rules:
3.75/1.75	
3.75/1.75	   U11(tt, M, N) -> U12(tt, M, N)
3.75/1.75	   U12(tt, M, N) -> s(plus(N, M))
3.75/1.75	   plus(N, 0) -> N
3.75/1.75	   plus(N, s(M)) -> U11(tt, M, N)
3.75/1.75	
3.75/1.75	The replacement map contains the following entries:
3.75/1.75	
3.75/1.75	U11: {1}
3.75/1.75	tt: empty set
3.75/1.75	U12: {1}
3.75/1.75	s: {1}
3.75/1.75	plus: {1, 2}
3.75/1.75	0: empty set
3.75/1.75	
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(3) CSRRRRProof (EQUIVALENT)
3.75/1.75	The following CSR is given: Context-sensitive rewrite system:
3.75/1.75	The TRS R consists of the following rules:
3.75/1.75	
3.75/1.75	   U11(tt, M, N) -> U12(tt, M, N)
3.75/1.75	   U12(tt, M, N) -> s(plus(N, M))
3.75/1.75	   plus(N, 0) -> N
3.75/1.75	   plus(N, s(M)) -> U11(tt, M, N)
3.75/1.75	
3.75/1.75	The replacement map contains the following entries:
3.75/1.75	
3.75/1.75	U11: {1}
3.75/1.75	tt: empty set
3.75/1.75	U12: {1}
3.75/1.75	s: {1}
3.75/1.75	plus: {1, 2}
3.75/1.75	0: empty set
3.75/1.75	Used ordering:
3.75/1.75	Polynomial interpretation [POLO]:
3.75/1.75	
3.75/1.75	   POL(0) = 2
3.75/1.75	   POL(U11(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
3.75/1.75	   POL(U12(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
3.75/1.75	   POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2
3.75/1.75	   POL(s(x_1)) = x_1
3.75/1.75	   POL(tt) = 0
3.75/1.75	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.75/1.75	
3.75/1.75	   plus(N, 0) -> N
3.75/1.75	
3.75/1.75	
3.75/1.75	
3.75/1.75	
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(4)
3.75/1.75	Obligation:
3.75/1.75	Context-sensitive rewrite system:
3.75/1.75	The TRS R consists of the following rules:
3.75/1.75	
3.75/1.75	   U11(tt, M, N) -> U12(tt, M, N)
3.75/1.75	   U12(tt, M, N) -> s(plus(N, M))
3.75/1.75	   plus(N, s(M)) -> U11(tt, M, N)
3.75/1.75	
3.75/1.75	The replacement map contains the following entries:
3.75/1.75	
3.75/1.75	U11: {1}
3.75/1.75	tt: empty set
3.75/1.75	U12: {1}
3.75/1.75	s: {1}
3.75/1.75	plus: {1, 2}
3.75/1.75	
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(5) CSRRRRProof (EQUIVALENT)
3.75/1.75	The following CSR is given: Context-sensitive rewrite system:
3.75/1.75	The TRS R consists of the following rules:
3.75/1.75	
3.75/1.75	   U11(tt, M, N) -> U12(tt, M, N)
3.75/1.75	   U12(tt, M, N) -> s(plus(N, M))
3.75/1.75	   plus(N, s(M)) -> U11(tt, M, N)
3.75/1.75	
3.75/1.75	The replacement map contains the following entries:
3.75/1.75	
3.75/1.75	U11: {1}
3.75/1.75	tt: empty set
3.75/1.75	U12: {1}
3.75/1.75	s: {1}
3.75/1.75	plus: {1, 2}
3.75/1.75	Used ordering:
3.75/1.75	Polynomial interpretation [POLO]:
3.75/1.75	
3.75/1.75	   POL(U11(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
3.75/1.75	   POL(U12(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
3.75/1.75	   POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2
3.75/1.75	   POL(s(x_1)) = 2 + x_1
3.75/1.75	   POL(tt) = 1
3.75/1.75	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.75/1.75	
3.75/1.75	   U12(tt, M, N) -> s(plus(N, M))
3.75/1.75	   plus(N, s(M)) -> U11(tt, M, N)
3.75/1.75	
3.75/1.75	
3.75/1.75	
3.75/1.75	
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(6)
3.75/1.75	Obligation:
3.75/1.75	Context-sensitive rewrite system:
3.75/1.75	The TRS R consists of the following rules:
3.75/1.75	
3.75/1.75	   U11(tt, M, N) -> U12(tt, M, N)
3.75/1.75	
3.75/1.75	The replacement map contains the following entries:
3.75/1.75	
3.75/1.75	U11: {1}
3.75/1.75	tt: empty set
3.75/1.75	U12: {1}
3.75/1.75	
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(7) CSRRRRProof (EQUIVALENT)
3.75/1.75	The following CSR is given: Context-sensitive rewrite system:
3.75/1.75	The TRS R consists of the following rules:
3.75/1.75	
3.75/1.75	   U11(tt, M, N) -> U12(tt, M, N)
3.75/1.75	
3.75/1.75	The replacement map contains the following entries:
3.75/1.75	
3.75/1.75	U11: {1}
3.75/1.75	tt: empty set
3.75/1.75	U12: {1}
3.75/1.75	Used ordering:
3.75/1.75	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.75/1.75	
3.75/1.75	   <<<
3.75/1.75	 POL(U11(x_1, x_2, x_3)) =  	[[1]] 	 +  	[[1, 1]] 	* 	x_1 	 +  	[[1, 1]] 	* 	x_2 	 +  	[[1, 1]] 	* 	x_3
3.75/1.75	>>>
3.75/1.75	
3.75/1.75	   <<<
3.75/1.75	 POL(tt) =  	[[1], [1]]
3.75/1.75	>>>
3.75/1.75	
3.75/1.75	   <<<
3.75/1.75	 POL(U12(x_1, x_2, x_3)) =  	[[1]] 	 +  	[[1, 0]] 	* 	x_1 	 +  	[[1, 1]] 	* 	x_2 	 +  	[[1, 1]] 	* 	x_3
3.75/1.75	>>>
3.75/1.75	
3.75/1.75	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.75/1.75	
3.75/1.75	   U11(tt, M, N) -> U12(tt, M, N)
3.75/1.75	
3.75/1.75	
3.75/1.75	
3.75/1.75	
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(8)
3.75/1.75	Obligation:
3.75/1.75	Context-sensitive rewrite system:
3.75/1.75	R is empty.
3.75/1.75	
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(9) RisEmptyProof (EQUIVALENT)
3.75/1.75	The CSR R is empty. Hence, termination is trivially proven.
3.75/1.75	----------------------------------------
3.75/1.75	
3.75/1.75	(10)
3.75/1.75	YES
3.75/1.78	EOF