3.32/1.67	YES
3.32/1.68	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.32/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.32/1.68	
3.32/1.68	
3.32/1.68	Termination w.r.t. Q of the given QTRS could be proven:
3.32/1.68	
3.32/1.68	(0) QTRS
3.32/1.68	(1) QTRSToCSRProof [SOUND, 0 ms]
3.32/1.68	(2) CSR
3.32/1.68	(3) CSRRRRProof [EQUIVALENT, 66 ms]
3.32/1.68	(4) CSR
3.32/1.68	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.32/1.68	(6) YES
3.32/1.68	
3.32/1.68	
3.32/1.68	----------------------------------------
3.32/1.68	
3.32/1.68	(0)
3.32/1.68	Obligation:
3.32/1.68	Q restricted rewrite system:
3.32/1.68	The TRS R consists of the following rules:
3.32/1.68	
3.32/1.68	   active(and(tt, X)) -> mark(X)
3.32/1.68	   active(plus(N, 0)) -> mark(N)
3.32/1.68	   active(plus(N, s(M))) -> mark(s(plus(N, M)))
3.32/1.68	   active(and(X1, X2)) -> and(active(X1), X2)
3.32/1.68	   active(plus(X1, X2)) -> plus(active(X1), X2)
3.32/1.68	   active(plus(X1, X2)) -> plus(X1, active(X2))
3.32/1.68	   active(s(X)) -> s(active(X))
3.32/1.68	   and(mark(X1), X2) -> mark(and(X1, X2))
3.32/1.68	   plus(mark(X1), X2) -> mark(plus(X1, X2))
3.32/1.68	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
3.32/1.68	   s(mark(X)) -> mark(s(X))
3.32/1.68	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
3.32/1.68	   proper(tt) -> ok(tt)
3.32/1.68	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
3.32/1.68	   proper(0) -> ok(0)
3.32/1.68	   proper(s(X)) -> s(proper(X))
3.32/1.68	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
3.32/1.68	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
3.32/1.68	   s(ok(X)) -> ok(s(X))
3.32/1.68	   top(mark(X)) -> top(proper(X))
3.32/1.68	   top(ok(X)) -> top(active(X))
3.32/1.68	
3.32/1.68	The set Q consists of the following terms:
3.32/1.68	
3.32/1.68	   active(and(x0, x1))
3.32/1.68	   active(plus(x0, x1))
3.32/1.68	   active(s(x0))
3.32/1.68	   and(mark(x0), x1)
3.32/1.68	   plus(mark(x0), x1)
3.32/1.68	   plus(x0, mark(x1))
3.32/1.68	   s(mark(x0))
3.32/1.68	   proper(and(x0, x1))
3.32/1.68	   proper(tt)
3.32/1.68	   proper(plus(x0, x1))
3.32/1.68	   proper(0)
3.32/1.68	   proper(s(x0))
3.32/1.68	   and(ok(x0), ok(x1))
3.32/1.68	   plus(ok(x0), ok(x1))
3.32/1.68	   s(ok(x0))
3.32/1.68	   top(mark(x0))
3.32/1.68	   top(ok(x0))
3.32/1.68	
3.32/1.68	
3.32/1.68	----------------------------------------
3.32/1.68	
3.32/1.68	(1) QTRSToCSRProof (SOUND)
3.32/1.68	The following Q TRS is given: Q restricted rewrite system:
3.32/1.68	The TRS R consists of the following rules:
3.32/1.68	
3.32/1.68	   active(and(tt, X)) -> mark(X)
3.32/1.68	   active(plus(N, 0)) -> mark(N)
3.32/1.68	   active(plus(N, s(M))) -> mark(s(plus(N, M)))
3.32/1.68	   active(and(X1, X2)) -> and(active(X1), X2)
3.32/1.68	   active(plus(X1, X2)) -> plus(active(X1), X2)
3.32/1.68	   active(plus(X1, X2)) -> plus(X1, active(X2))
3.32/1.68	   active(s(X)) -> s(active(X))
3.32/1.68	   and(mark(X1), X2) -> mark(and(X1, X2))
3.32/1.68	   plus(mark(X1), X2) -> mark(plus(X1, X2))
3.32/1.68	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
3.32/1.68	   s(mark(X)) -> mark(s(X))
3.32/1.68	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
3.32/1.68	   proper(tt) -> ok(tt)
3.32/1.68	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
3.32/1.68	   proper(0) -> ok(0)
3.32/1.68	   proper(s(X)) -> s(proper(X))
3.32/1.68	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
3.32/1.68	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
3.32/1.68	   s(ok(X)) -> ok(s(X))
3.32/1.68	   top(mark(X)) -> top(proper(X))
3.32/1.68	   top(ok(X)) -> top(active(X))
3.32/1.68	
3.32/1.68	The set Q consists of the following terms:
3.32/1.68	
3.32/1.68	   active(and(x0, x1))
3.32/1.68	   active(plus(x0, x1))
3.32/1.68	   active(s(x0))
3.32/1.68	   and(mark(x0), x1)
3.32/1.68	   plus(mark(x0), x1)
3.32/1.68	   plus(x0, mark(x1))
3.32/1.68	   s(mark(x0))
3.32/1.68	   proper(and(x0, x1))
3.32/1.68	   proper(tt)
3.32/1.68	   proper(plus(x0, x1))
3.32/1.68	   proper(0)
3.32/1.68	   proper(s(x0))
3.32/1.68	   and(ok(x0), ok(x1))
3.32/1.68	   plus(ok(x0), ok(x1))
3.32/1.68	   s(ok(x0))
3.32/1.68	   top(mark(x0))
3.32/1.68	   top(ok(x0))
3.32/1.68	
3.32/1.68	Special symbols used for the transformation (see [GM04]):
3.32/1.68	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.32/1.68	The replacement map contains the following entries:
3.32/1.68	
3.32/1.68	and: {1}
3.32/1.68	tt: empty set
3.32/1.68	plus: {1, 2}
3.32/1.68	0: empty set
3.32/1.68	s: {1}
3.32/1.68	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.
3.32/1.68	----------------------------------------
3.32/1.68	
3.32/1.68	(2)
3.32/1.68	Obligation:
3.32/1.68	Context-sensitive rewrite system:
3.32/1.68	The TRS R consists of the following rules:
3.32/1.68	
3.32/1.68	   and(tt, X) -> X
3.32/1.68	   plus(N, 0) -> N
3.32/1.68	   plus(N, s(M)) -> s(plus(N, M))
3.32/1.68	
3.32/1.68	The replacement map contains the following entries:
3.32/1.68	
3.32/1.68	and: {1}
3.32/1.68	tt: empty set
3.32/1.68	plus: {1, 2}
3.32/1.68	0: empty set
3.32/1.68	s: {1}
3.32/1.68	
3.32/1.68	----------------------------------------
3.32/1.68	
3.32/1.68	(3) CSRRRRProof (EQUIVALENT)
3.32/1.68	The following CSR is given: Context-sensitive rewrite system:
3.32/1.68	The TRS R consists of the following rules:
3.32/1.68	
3.32/1.68	   and(tt, X) -> X
3.32/1.68	   plus(N, 0) -> N
3.32/1.68	   plus(N, s(M)) -> s(plus(N, M))
3.32/1.68	
3.32/1.68	The replacement map contains the following entries:
3.32/1.68	
3.32/1.68	and: {1}
3.32/1.68	tt: empty set
3.32/1.68	plus: {1, 2}
3.32/1.68	0: empty set
3.32/1.68	s: {1}
3.32/1.68	Used ordering:
3.32/1.68	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.32/1.68	
3.32/1.68	   <<<
3.32/1.68	 POL(and(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 1]] 	* 	x_2
3.32/1.68	>>>
3.32/1.68	
3.32/1.68	   <<<
3.32/1.68	 POL(tt) =  	[[0], [1]]
3.32/1.68	>>>
3.32/1.68	
3.32/1.68	   <<<
3.32/1.68	 POL(plus(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 0]] 	* 	x_2
3.32/1.68	>>>
3.32/1.68	
3.32/1.68	   <<<
3.32/1.68	 POL(0) =  	[[0], [1]]
3.32/1.68	>>>
3.32/1.68	
3.32/1.68	   <<<
3.32/1.68	 POL(s(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
3.32/1.68	>>>
3.32/1.68	
3.32/1.68	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.32/1.68	
3.32/1.68	   and(tt, X) -> X
3.32/1.68	   plus(N, 0) -> N
3.32/1.68	   plus(N, s(M)) -> s(plus(N, M))
3.32/1.68	
3.32/1.68	
3.32/1.68	
3.32/1.68	
3.32/1.68	----------------------------------------
3.32/1.68	
3.32/1.68	(4)
3.32/1.68	Obligation:
3.32/1.68	Context-sensitive rewrite system:
3.32/1.68	R is empty.
3.32/1.68	
3.32/1.68	----------------------------------------
3.32/1.68	
3.32/1.68	(5) RisEmptyProof (EQUIVALENT)
3.32/1.68	The CSR R is empty. Hence, termination is trivially proven.
3.32/1.68	----------------------------------------
3.32/1.68	
3.32/1.68	(6)
3.32/1.68	YES
3.36/1.70	EOF