3.88/3.05	YES
3.88/3.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.88/3.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.88/3.06	
3.88/3.06	
3.88/3.06	Termination w.r.t. Q of the given QTRS could be proven:
3.88/3.06	
3.88/3.06	(0) QTRS
3.88/3.06	(1) QTRSToCSRProof [SOUND, 0 ms]
3.88/3.06	(2) CSR
3.88/3.06	(3) CSRRRRProof [EQUIVALENT, 55 ms]
3.88/3.06	(4) CSR
3.88/3.06	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.88/3.06	(6) CSR
3.88/3.06	(7) CSRRRRProof [EQUIVALENT, 8 ms]
3.88/3.06	(8) CSR
3.88/3.06	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.88/3.06	(10) YES
3.88/3.06	
3.88/3.06	
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(0)
3.88/3.06	Obligation:
3.88/3.06	Q restricted rewrite system:
3.88/3.06	The TRS R consists of the following rules:
3.88/3.06	
3.88/3.06	   active(app(nil, YS)) -> mark(YS)
3.88/3.06	   active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
3.88/3.06	   active(from(X)) -> mark(cons(X, from(s(X))))
3.88/3.06	   active(zWadr(nil, YS)) -> mark(nil)
3.88/3.06	   active(zWadr(XS, nil)) -> mark(nil)
3.88/3.06	   active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
3.88/3.06	   active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
3.88/3.06	   active(app(X1, X2)) -> app(active(X1), X2)
3.88/3.06	   active(app(X1, X2)) -> app(X1, active(X2))
3.88/3.06	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.88/3.06	   active(from(X)) -> from(active(X))
3.88/3.06	   active(s(X)) -> s(active(X))
3.88/3.06	   active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
3.88/3.06	   active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
3.88/3.06	   active(prefix(X)) -> prefix(active(X))
3.88/3.06	   app(mark(X1), X2) -> mark(app(X1, X2))
3.88/3.06	   app(X1, mark(X2)) -> mark(app(X1, X2))
3.88/3.06	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.88/3.06	   from(mark(X)) -> mark(from(X))
3.88/3.06	   s(mark(X)) -> mark(s(X))
3.88/3.06	   zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
3.88/3.06	   zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
3.88/3.06	   prefix(mark(X)) -> mark(prefix(X))
3.88/3.06	   proper(app(X1, X2)) -> app(proper(X1), proper(X2))
3.88/3.06	   proper(nil) -> ok(nil)
3.88/3.06	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.88/3.06	   proper(from(X)) -> from(proper(X))
3.88/3.06	   proper(s(X)) -> s(proper(X))
3.88/3.06	   proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
3.88/3.06	   proper(prefix(X)) -> prefix(proper(X))
3.88/3.06	   app(ok(X1), ok(X2)) -> ok(app(X1, X2))
3.88/3.06	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.88/3.06	   from(ok(X)) -> ok(from(X))
3.88/3.06	   s(ok(X)) -> ok(s(X))
3.88/3.06	   zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
3.88/3.06	   prefix(ok(X)) -> ok(prefix(X))
3.88/3.06	   top(mark(X)) -> top(proper(X))
3.88/3.06	   top(ok(X)) -> top(active(X))
3.88/3.06	
3.88/3.06	The set Q consists of the following terms:
3.88/3.06	
3.88/3.06	   active(from(x0))
3.88/3.06	   active(prefix(x0))
3.88/3.06	   active(app(x0, x1))
3.88/3.06	   active(cons(x0, x1))
3.88/3.06	   active(s(x0))
3.88/3.06	   active(zWadr(x0, x1))
3.88/3.06	   app(mark(x0), x1)
3.88/3.06	   app(x0, mark(x1))
3.88/3.06	   cons(mark(x0), x1)
3.88/3.06	   from(mark(x0))
3.88/3.06	   s(mark(x0))
3.88/3.06	   zWadr(mark(x0), x1)
3.88/3.06	   zWadr(x0, mark(x1))
3.88/3.06	   prefix(mark(x0))
3.88/3.06	   proper(app(x0, x1))
3.88/3.06	   proper(nil)
3.88/3.06	   proper(cons(x0, x1))
3.88/3.06	   proper(from(x0))
3.88/3.06	   proper(s(x0))
3.88/3.06	   proper(zWadr(x0, x1))
3.88/3.06	   proper(prefix(x0))
3.88/3.06	   app(ok(x0), ok(x1))
3.88/3.06	   cons(ok(x0), ok(x1))
3.88/3.06	   from(ok(x0))
3.88/3.06	   s(ok(x0))
3.88/3.06	   zWadr(ok(x0), ok(x1))
3.88/3.06	   prefix(ok(x0))
3.88/3.06	   top(mark(x0))
3.88/3.06	   top(ok(x0))
3.88/3.06	
3.88/3.06	
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(1) QTRSToCSRProof (SOUND)
3.88/3.06	The following Q TRS is given: Q restricted rewrite system:
3.88/3.06	The TRS R consists of the following rules:
3.88/3.06	
3.88/3.06	   active(app(nil, YS)) -> mark(YS)
3.88/3.06	   active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
3.88/3.06	   active(from(X)) -> mark(cons(X, from(s(X))))
3.88/3.06	   active(zWadr(nil, YS)) -> mark(nil)
3.88/3.06	   active(zWadr(XS, nil)) -> mark(nil)
3.88/3.06	   active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
3.88/3.06	   active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
3.88/3.06	   active(app(X1, X2)) -> app(active(X1), X2)
3.88/3.06	   active(app(X1, X2)) -> app(X1, active(X2))
3.88/3.06	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.88/3.06	   active(from(X)) -> from(active(X))
3.88/3.06	   active(s(X)) -> s(active(X))
3.88/3.06	   active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
3.88/3.06	   active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
3.88/3.06	   active(prefix(X)) -> prefix(active(X))
3.88/3.06	   app(mark(X1), X2) -> mark(app(X1, X2))
3.88/3.06	   app(X1, mark(X2)) -> mark(app(X1, X2))
3.88/3.06	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.88/3.06	   from(mark(X)) -> mark(from(X))
3.88/3.06	   s(mark(X)) -> mark(s(X))
3.88/3.06	   zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
3.88/3.06	   zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
3.88/3.06	   prefix(mark(X)) -> mark(prefix(X))
3.88/3.06	   proper(app(X1, X2)) -> app(proper(X1), proper(X2))
3.88/3.06	   proper(nil) -> ok(nil)
3.88/3.06	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.88/3.06	   proper(from(X)) -> from(proper(X))
3.88/3.06	   proper(s(X)) -> s(proper(X))
3.88/3.06	   proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
3.88/3.06	   proper(prefix(X)) -> prefix(proper(X))
3.88/3.06	   app(ok(X1), ok(X2)) -> ok(app(X1, X2))
3.88/3.06	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.88/3.06	   from(ok(X)) -> ok(from(X))
3.88/3.06	   s(ok(X)) -> ok(s(X))
3.88/3.06	   zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
3.88/3.06	   prefix(ok(X)) -> ok(prefix(X))
3.88/3.06	   top(mark(X)) -> top(proper(X))
3.88/3.06	   top(ok(X)) -> top(active(X))
3.88/3.06	
3.88/3.06	The set Q consists of the following terms:
3.88/3.06	
3.88/3.06	   active(from(x0))
3.88/3.06	   active(prefix(x0))
3.88/3.06	   active(app(x0, x1))
3.88/3.06	   active(cons(x0, x1))
3.88/3.06	   active(s(x0))
3.88/3.06	   active(zWadr(x0, x1))
3.88/3.06	   app(mark(x0), x1)
3.88/3.06	   app(x0, mark(x1))
3.88/3.06	   cons(mark(x0), x1)
3.88/3.06	   from(mark(x0))
3.88/3.06	   s(mark(x0))
3.88/3.06	   zWadr(mark(x0), x1)
3.88/3.06	   zWadr(x0, mark(x1))
3.88/3.06	   prefix(mark(x0))
3.88/3.06	   proper(app(x0, x1))
3.88/3.06	   proper(nil)
3.88/3.06	   proper(cons(x0, x1))
3.88/3.06	   proper(from(x0))
3.88/3.06	   proper(s(x0))
3.88/3.06	   proper(zWadr(x0, x1))
3.88/3.06	   proper(prefix(x0))
3.88/3.06	   app(ok(x0), ok(x1))
3.88/3.06	   cons(ok(x0), ok(x1))
3.88/3.06	   from(ok(x0))
3.88/3.06	   s(ok(x0))
3.88/3.06	   zWadr(ok(x0), ok(x1))
3.88/3.06	   prefix(ok(x0))
3.88/3.06	   top(mark(x0))
3.88/3.06	   top(ok(x0))
3.88/3.06	
3.88/3.06	Special symbols used for the transformation (see [GM04]):
3.88/3.06	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.88/3.06	The replacement map contains the following entries:
3.88/3.06	
3.88/3.06	app: {1, 2}
3.88/3.06	nil: empty set
3.88/3.06	cons: {1}
3.88/3.06	from: {1}
3.88/3.06	s: {1}
3.88/3.06	zWadr: {1, 2}
3.88/3.06	prefix: {1}
3.88/3.06	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.
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(2)
3.88/3.06	Obligation:
3.88/3.06	Context-sensitive rewrite system:
3.88/3.06	The TRS R consists of the following rules:
3.88/3.06	
3.88/3.06	   app(nil, YS) -> YS
3.88/3.06	   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
3.88/3.06	   from(X) -> cons(X, from(s(X)))
3.88/3.06	   zWadr(nil, YS) -> nil
3.88/3.06	   zWadr(XS, nil) -> nil
3.88/3.06	   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
3.88/3.06	   prefix(L) -> cons(nil, zWadr(L, prefix(L)))
3.88/3.06	
3.88/3.06	The replacement map contains the following entries:
3.88/3.06	
3.88/3.06	app: {1, 2}
3.88/3.06	nil: empty set
3.88/3.06	cons: {1}
3.88/3.06	from: {1}
3.88/3.06	s: {1}
3.88/3.06	zWadr: {1, 2}
3.88/3.06	prefix: {1}
3.88/3.06	
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(3) CSRRRRProof (EQUIVALENT)
3.88/3.06	The following CSR is given: Context-sensitive rewrite system:
3.88/3.06	The TRS R consists of the following rules:
3.88/3.06	
3.88/3.06	   app(nil, YS) -> YS
3.88/3.06	   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
3.88/3.06	   from(X) -> cons(X, from(s(X)))
3.88/3.06	   zWadr(nil, YS) -> nil
3.88/3.06	   zWadr(XS, nil) -> nil
3.88/3.06	   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
3.88/3.06	   prefix(L) -> cons(nil, zWadr(L, prefix(L)))
3.88/3.06	
3.88/3.06	The replacement map contains the following entries:
3.88/3.06	
3.88/3.06	app: {1, 2}
3.88/3.06	nil: empty set
3.88/3.06	cons: {1}
3.88/3.06	from: {1}
3.88/3.06	s: {1}
3.88/3.06	zWadr: {1, 2}
3.88/3.06	prefix: {1}
3.88/3.06	Used ordering:
3.88/3.06	Polynomial interpretation [POLO]:
3.88/3.06	
3.88/3.06	   POL(app(x_1, x_2)) = x_1 + x_2
3.88/3.06	   POL(cons(x_1, x_2)) = 1 + x_1
3.88/3.06	   POL(from(x_1)) = 1 + x_1
3.88/3.06	   POL(nil) = 0
3.88/3.06	   POL(prefix(x_1)) = 1 + x_1
3.88/3.06	   POL(s(x_1)) = x_1
3.88/3.06	   POL(zWadr(x_1, x_2)) = 1 + x_1 + x_2
3.88/3.06	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.88/3.06	
3.88/3.06	   zWadr(nil, YS) -> nil
3.88/3.06	   zWadr(XS, nil) -> nil
3.88/3.06	   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
3.88/3.06	
3.88/3.06	
3.88/3.06	
3.88/3.06	
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(4)
3.88/3.06	Obligation:
3.88/3.06	Context-sensitive rewrite system:
3.88/3.06	The TRS R consists of the following rules:
3.88/3.06	
3.88/3.06	   app(nil, YS) -> YS
3.88/3.06	   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
3.88/3.06	   from(X) -> cons(X, from(s(X)))
3.88/3.06	   prefix(L) -> cons(nil, zWadr(L, prefix(L)))
3.88/3.06	
3.88/3.06	The replacement map contains the following entries:
3.88/3.06	
3.88/3.06	app: {1, 2}
3.88/3.06	nil: empty set
3.88/3.06	cons: {1}
3.88/3.06	from: {1}
3.88/3.06	s: {1}
3.88/3.06	zWadr: {1, 2}
3.88/3.06	prefix: {1}
3.88/3.06	
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(5) CSRRRRProof (EQUIVALENT)
3.88/3.06	The following CSR is given: Context-sensitive rewrite system:
3.88/3.06	The TRS R consists of the following rules:
3.88/3.06	
3.88/3.06	   app(nil, YS) -> YS
3.88/3.06	   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
3.88/3.06	   from(X) -> cons(X, from(s(X)))
3.88/3.06	   prefix(L) -> cons(nil, zWadr(L, prefix(L)))
3.88/3.06	
3.88/3.06	The replacement map contains the following entries:
3.88/3.06	
3.88/3.06	app: {1, 2}
3.88/3.06	nil: empty set
3.88/3.06	cons: {1}
3.88/3.06	from: {1}
3.88/3.06	s: {1}
3.88/3.06	zWadr: {1, 2}
3.88/3.06	prefix: {1}
3.88/3.06	Used ordering:
3.88/3.06	Polynomial interpretation [POLO]:
3.88/3.06	
3.88/3.06	   POL(app(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
3.88/3.06	   POL(cons(x_1, x_2)) = 2*x_1
3.88/3.06	   POL(from(x_1)) = 2*x_1
3.88/3.06	   POL(nil) = 1
3.88/3.06	   POL(prefix(x_1)) = 2 + 2*x_1
3.88/3.06	   POL(s(x_1)) = 1 + x_1
3.88/3.06	   POL(zWadr(x_1, x_2)) = 1 + x_1 + x_2
3.88/3.06	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.88/3.06	
3.88/3.06	   app(nil, YS) -> YS
3.88/3.06	   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
3.88/3.06	
3.88/3.06	
3.88/3.06	
3.88/3.06	
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(6)
3.88/3.06	Obligation:
3.88/3.06	Context-sensitive rewrite system:
3.88/3.06	The TRS R consists of the following rules:
3.88/3.06	
3.88/3.06	   from(X) -> cons(X, from(s(X)))
3.88/3.06	   prefix(L) -> cons(nil, zWadr(L, prefix(L)))
3.88/3.06	
3.88/3.06	The replacement map contains the following entries:
3.88/3.06	
3.88/3.06	nil: empty set
3.88/3.06	cons: {1}
3.88/3.06	from: {1}
3.88/3.06	s: {1}
3.88/3.06	zWadr: {1, 2}
3.88/3.06	prefix: {1}
3.88/3.06	
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(7) CSRRRRProof (EQUIVALENT)
3.88/3.06	The following CSR is given: Context-sensitive rewrite system:
3.88/3.06	The TRS R consists of the following rules:
3.88/3.06	
3.88/3.06	   from(X) -> cons(X, from(s(X)))
3.88/3.06	   prefix(L) -> cons(nil, zWadr(L, prefix(L)))
3.88/3.06	
3.88/3.06	The replacement map contains the following entries:
3.88/3.06	
3.88/3.06	nil: empty set
3.88/3.06	cons: {1}
3.88/3.06	from: {1}
3.88/3.06	s: {1}
3.88/3.06	zWadr: {1, 2}
3.88/3.06	prefix: {1}
3.88/3.06	Used ordering:
3.88/3.06	Polynomial interpretation [POLO]:
3.88/3.06	
3.88/3.06	   POL(cons(x_1, x_2)) = x_1
3.88/3.06	   POL(from(x_1)) = 1 + 2*x_1
3.88/3.06	   POL(nil) = 0
3.88/3.06	   POL(prefix(x_1)) = 1 + x_1
3.88/3.06	   POL(s(x_1)) = x_1
3.88/3.06	   POL(zWadr(x_1, x_2)) = 2 + x_1 + x_2
3.88/3.06	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.88/3.06	
3.88/3.06	   from(X) -> cons(X, from(s(X)))
3.88/3.06	   prefix(L) -> cons(nil, zWadr(L, prefix(L)))
3.88/3.06	
3.88/3.06	
3.88/3.06	
3.88/3.06	
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(8)
3.88/3.06	Obligation:
3.88/3.06	Context-sensitive rewrite system:
3.88/3.06	R is empty.
3.88/3.06	
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(9) RisEmptyProof (EQUIVALENT)
3.88/3.06	The CSR R is empty. Hence, termination is trivially proven.
3.88/3.06	----------------------------------------
3.88/3.06	
3.88/3.06	(10)
3.88/3.06	YES
3.88/3.10	EOF