3.46/1.69	YES
3.46/1.69	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.46/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.46/1.69	
3.46/1.69	
3.46/1.69	Termination w.r.t. Q of the given QTRS could be proven:
3.46/1.69	
3.46/1.69	(0) QTRS
3.46/1.69	(1) QTRSToCSRProof [SOUND, 0 ms]
3.46/1.69	(2) CSR
3.46/1.69	(3) CSRRRRProof [EQUIVALENT, 64 ms]
3.46/1.69	(4) CSR
3.46/1.69	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.46/1.69	(6) CSR
3.46/1.69	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.46/1.69	(8) CSR
3.46/1.69	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.46/1.69	(10) YES
3.46/1.69	
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(0)
3.46/1.69	Obligation:
3.46/1.69	Q restricted rewrite system:
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   active(f(0)) -> mark(cons(0, f(s(0))))
3.46/1.69	   active(f(s(0))) -> mark(f(p(s(0))))
3.46/1.69	   active(p(s(X))) -> mark(X)
3.46/1.69	   active(f(X)) -> f(active(X))
3.46/1.69	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.46/1.69	   active(s(X)) -> s(active(X))
3.46/1.69	   active(p(X)) -> p(active(X))
3.46/1.69	   f(mark(X)) -> mark(f(X))
3.46/1.69	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.46/1.69	   s(mark(X)) -> mark(s(X))
3.46/1.69	   p(mark(X)) -> mark(p(X))
3.46/1.69	   proper(f(X)) -> f(proper(X))
3.46/1.69	   proper(0) -> ok(0)
3.46/1.69	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.46/1.69	   proper(s(X)) -> s(proper(X))
3.46/1.69	   proper(p(X)) -> p(proper(X))
3.46/1.69	   f(ok(X)) -> ok(f(X))
3.46/1.69	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.46/1.69	   s(ok(X)) -> ok(s(X))
3.46/1.69	   p(ok(X)) -> ok(p(X))
3.46/1.69	   top(mark(X)) -> top(proper(X))
3.46/1.69	   top(ok(X)) -> top(active(X))
3.46/1.69	
3.46/1.69	The set Q consists of the following terms:
3.46/1.69	
3.46/1.69	   active(f(x0))
3.46/1.69	   active(cons(x0, x1))
3.46/1.69	   active(s(x0))
3.46/1.69	   active(p(x0))
3.46/1.69	   f(mark(x0))
3.46/1.69	   cons(mark(x0), x1)
3.46/1.69	   s(mark(x0))
3.46/1.69	   p(mark(x0))
3.46/1.69	   proper(f(x0))
3.46/1.69	   proper(0)
3.46/1.69	   proper(cons(x0, x1))
3.46/1.69	   proper(s(x0))
3.46/1.69	   proper(p(x0))
3.46/1.69	   f(ok(x0))
3.46/1.69	   cons(ok(x0), ok(x1))
3.46/1.69	   s(ok(x0))
3.46/1.69	   p(ok(x0))
3.46/1.69	   top(mark(x0))
3.46/1.69	   top(ok(x0))
3.46/1.69	
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(1) QTRSToCSRProof (SOUND)
3.46/1.69	The following Q TRS is given: Q restricted rewrite system:
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   active(f(0)) -> mark(cons(0, f(s(0))))
3.46/1.69	   active(f(s(0))) -> mark(f(p(s(0))))
3.46/1.69	   active(p(s(X))) -> mark(X)
3.46/1.69	   active(f(X)) -> f(active(X))
3.46/1.69	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.46/1.69	   active(s(X)) -> s(active(X))
3.46/1.69	   active(p(X)) -> p(active(X))
3.46/1.69	   f(mark(X)) -> mark(f(X))
3.46/1.69	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.46/1.69	   s(mark(X)) -> mark(s(X))
3.46/1.69	   p(mark(X)) -> mark(p(X))
3.46/1.69	   proper(f(X)) -> f(proper(X))
3.46/1.69	   proper(0) -> ok(0)
3.46/1.69	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.46/1.69	   proper(s(X)) -> s(proper(X))
3.46/1.69	   proper(p(X)) -> p(proper(X))
3.46/1.69	   f(ok(X)) -> ok(f(X))
3.46/1.69	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.46/1.69	   s(ok(X)) -> ok(s(X))
3.46/1.69	   p(ok(X)) -> ok(p(X))
3.46/1.69	   top(mark(X)) -> top(proper(X))
3.46/1.69	   top(ok(X)) -> top(active(X))
3.46/1.69	
3.46/1.69	The set Q consists of the following terms:
3.46/1.69	
3.46/1.69	   active(f(x0))
3.46/1.69	   active(cons(x0, x1))
3.46/1.69	   active(s(x0))
3.46/1.69	   active(p(x0))
3.46/1.69	   f(mark(x0))
3.46/1.69	   cons(mark(x0), x1)
3.46/1.69	   s(mark(x0))
3.46/1.69	   p(mark(x0))
3.46/1.69	   proper(f(x0))
3.46/1.69	   proper(0)
3.46/1.69	   proper(cons(x0, x1))
3.46/1.69	   proper(s(x0))
3.46/1.69	   proper(p(x0))
3.46/1.69	   f(ok(x0))
3.46/1.69	   cons(ok(x0), ok(x1))
3.46/1.69	   s(ok(x0))
3.46/1.69	   p(ok(x0))
3.46/1.69	   top(mark(x0))
3.46/1.69	   top(ok(x0))
3.46/1.69	
3.46/1.69	Special symbols used for the transformation (see [GM04]):
3.46/1.69	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.46/1.69	The replacement map contains the following entries:
3.46/1.69	
3.46/1.69	f: {1}
3.46/1.69	0: empty set
3.46/1.69	cons: {1}
3.46/1.69	s: {1}
3.46/1.69	p: {1}
3.46/1.69	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.
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(2)
3.46/1.69	Obligation:
3.46/1.69	Context-sensitive rewrite system:
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   f(0) -> cons(0, f(s(0)))
3.46/1.69	   f(s(0)) -> f(p(s(0)))
3.46/1.69	   p(s(X)) -> X
3.46/1.69	
3.46/1.69	The replacement map contains the following entries:
3.46/1.69	
3.46/1.69	f: {1}
3.46/1.69	0: empty set
3.46/1.69	cons: {1}
3.46/1.69	s: {1}
3.46/1.69	p: {1}
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(3) CSRRRRProof (EQUIVALENT)
3.46/1.69	The following CSR is given: Context-sensitive rewrite system:
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   f(0) -> cons(0, f(s(0)))
3.46/1.69	   f(s(0)) -> f(p(s(0)))
3.46/1.69	   p(s(X)) -> X
3.46/1.69	
3.46/1.69	The replacement map contains the following entries:
3.46/1.69	
3.46/1.69	f: {1}
3.46/1.69	0: empty set
3.46/1.69	cons: {1}
3.46/1.69	s: {1}
3.46/1.69	p: {1}
3.46/1.69	Used ordering:
3.46/1.69	Polynomial interpretation [POLO]:
3.46/1.69	
3.46/1.69	   POL(0) = 1
3.46/1.69	   POL(cons(x_1, x_2)) = 1 + x_1
3.46/1.69	   POL(f(x_1)) = 1 + x_1
3.46/1.69	   POL(p(x_1)) = x_1
3.46/1.69	   POL(s(x_1)) = 1 + x_1
3.46/1.69	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.46/1.69	
3.46/1.69	   p(s(X)) -> X
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(4)
3.46/1.69	Obligation:
3.46/1.69	Context-sensitive rewrite system:
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   f(0) -> cons(0, f(s(0)))
3.46/1.69	   f(s(0)) -> f(p(s(0)))
3.46/1.69	
3.46/1.69	The replacement map contains the following entries:
3.46/1.69	
3.46/1.69	f: {1}
3.46/1.69	0: empty set
3.46/1.69	cons: {1}
3.46/1.69	s: {1}
3.46/1.69	p: {1}
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(5) CSRRRRProof (EQUIVALENT)
3.46/1.69	The following CSR is given: Context-sensitive rewrite system:
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   f(0) -> cons(0, f(s(0)))
3.46/1.69	   f(s(0)) -> f(p(s(0)))
3.46/1.69	
3.46/1.69	The replacement map contains the following entries:
3.46/1.69	
3.46/1.69	f: {1}
3.46/1.69	0: empty set
3.46/1.69	cons: {1}
3.46/1.69	s: {1}
3.46/1.69	p: {1}
3.46/1.69	Used ordering:
3.46/1.69	Polynomial interpretation [POLO]:
3.46/1.69	
3.46/1.69	   POL(0) = 1
3.46/1.69	   POL(cons(x_1, x_2)) = x_1
3.46/1.69	   POL(f(x_1)) = 1 + x_1
3.46/1.69	   POL(p(x_1)) = x_1
3.46/1.69	   POL(s(x_1)) = 1 + x_1
3.46/1.69	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.46/1.69	
3.46/1.69	   f(0) -> cons(0, f(s(0)))
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(6)
3.46/1.69	Obligation:
3.46/1.69	Context-sensitive rewrite system:
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   f(s(0)) -> f(p(s(0)))
3.46/1.69	
3.46/1.69	The replacement map contains the following entries:
3.46/1.69	
3.46/1.69	f: {1}
3.46/1.69	0: empty set
3.46/1.69	s: {1}
3.46/1.69	p: {1}
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(7) CSRRRRProof (EQUIVALENT)
3.46/1.69	The following CSR is given: Context-sensitive rewrite system:
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   f(s(0)) -> f(p(s(0)))
3.46/1.69	
3.46/1.69	The replacement map contains the following entries:
3.46/1.69	
3.46/1.69	f: {1}
3.46/1.69	0: empty set
3.46/1.69	s: {1}
3.46/1.69	p: {1}
3.46/1.69	Used ordering:
3.46/1.69	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.46/1.69	
3.46/1.69	   <<<
3.46/1.69	 POL(f(x_1)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1
3.46/1.69	>>>
3.46/1.69	
3.46/1.69	   <<<
3.46/1.69	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
3.46/1.69	>>>
3.46/1.69	
3.46/1.69	   <<<
3.46/1.69	 POL(0) =  	[[1], [1]]
3.46/1.69	>>>
3.46/1.69	
3.46/1.69	   <<<
3.46/1.69	 POL(p(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
3.46/1.69	>>>
3.46/1.69	
3.46/1.69	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.46/1.69	
3.46/1.69	   f(s(0)) -> f(p(s(0)))
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(8)
3.46/1.69	Obligation:
3.46/1.69	Context-sensitive rewrite system:
3.46/1.69	R is empty.
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(9) RisEmptyProof (EQUIVALENT)
3.46/1.69	The CSR R is empty. Hence, termination is trivially proven.
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(10)
3.46/1.69	YES
3.46/1.71	EOF