3.56/1.70	YES
3.56/1.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.56/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.56/1.71	
3.56/1.71	
3.56/1.71	Termination w.r.t. Q of the given QTRS could be proven:
3.56/1.71	
3.56/1.71	(0) QTRS
3.56/1.71	(1) QTRSToCSRProof [SOUND, 0 ms]
3.56/1.71	(2) CSR
3.56/1.71	(3) CSRRRRProof [EQUIVALENT, 51 ms]
3.56/1.71	(4) CSR
3.56/1.71	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.56/1.71	(6) CSR
3.56/1.71	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.56/1.71	(8) CSR
3.56/1.71	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.56/1.71	(10) YES
3.56/1.71	
3.56/1.71	
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(0)
3.56/1.71	Obligation:
3.56/1.71	Q restricted rewrite system:
3.56/1.71	The TRS R consists of the following rules:
3.56/1.71	
3.56/1.71	   active(f(X)) -> mark(if(X, c, f(true)))
3.56/1.71	   active(if(true, X, Y)) -> mark(X)
3.56/1.71	   active(if(false, X, Y)) -> mark(Y)
3.56/1.71	   active(f(X)) -> f(active(X))
3.56/1.71	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
3.56/1.71	   active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
3.56/1.71	   f(mark(X)) -> mark(f(X))
3.56/1.71	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
3.56/1.71	   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
3.56/1.71	   proper(f(X)) -> f(proper(X))
3.56/1.71	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
3.56/1.71	   proper(c) -> ok(c)
3.56/1.71	   proper(true) -> ok(true)
3.56/1.71	   proper(false) -> ok(false)
3.56/1.71	   f(ok(X)) -> ok(f(X))
3.56/1.71	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
3.56/1.71	   top(mark(X)) -> top(proper(X))
3.56/1.71	   top(ok(X)) -> top(active(X))
3.56/1.71	
3.56/1.71	The set Q consists of the following terms:
3.56/1.71	
3.56/1.71	   active(f(x0))
3.56/1.71	   active(if(x0, x1, x2))
3.56/1.71	   f(mark(x0))
3.56/1.71	   if(mark(x0), x1, x2)
3.56/1.71	   if(x0, mark(x1), x2)
3.56/1.71	   proper(f(x0))
3.56/1.71	   proper(if(x0, x1, x2))
3.56/1.71	   proper(c)
3.56/1.71	   proper(true)
3.56/1.71	   proper(false)
3.56/1.71	   f(ok(x0))
3.56/1.71	   if(ok(x0), ok(x1), ok(x2))
3.56/1.71	   top(mark(x0))
3.56/1.71	   top(ok(x0))
3.56/1.71	
3.56/1.71	
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(1) QTRSToCSRProof (SOUND)
3.56/1.71	The following Q TRS is given: Q restricted rewrite system:
3.56/1.71	The TRS R consists of the following rules:
3.56/1.71	
3.56/1.71	   active(f(X)) -> mark(if(X, c, f(true)))
3.56/1.71	   active(if(true, X, Y)) -> mark(X)
3.56/1.71	   active(if(false, X, Y)) -> mark(Y)
3.56/1.71	   active(f(X)) -> f(active(X))
3.56/1.71	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
3.56/1.71	   active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
3.56/1.71	   f(mark(X)) -> mark(f(X))
3.56/1.71	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
3.56/1.71	   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
3.56/1.71	   proper(f(X)) -> f(proper(X))
3.56/1.71	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
3.56/1.71	   proper(c) -> ok(c)
3.56/1.71	   proper(true) -> ok(true)
3.56/1.71	   proper(false) -> ok(false)
3.56/1.71	   f(ok(X)) -> ok(f(X))
3.56/1.71	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
3.56/1.71	   top(mark(X)) -> top(proper(X))
3.56/1.71	   top(ok(X)) -> top(active(X))
3.56/1.71	
3.56/1.71	The set Q consists of the following terms:
3.56/1.71	
3.56/1.71	   active(f(x0))
3.56/1.71	   active(if(x0, x1, x2))
3.56/1.71	   f(mark(x0))
3.56/1.71	   if(mark(x0), x1, x2)
3.56/1.71	   if(x0, mark(x1), x2)
3.56/1.71	   proper(f(x0))
3.56/1.71	   proper(if(x0, x1, x2))
3.56/1.71	   proper(c)
3.56/1.71	   proper(true)
3.56/1.71	   proper(false)
3.56/1.71	   f(ok(x0))
3.56/1.71	   if(ok(x0), ok(x1), ok(x2))
3.56/1.71	   top(mark(x0))
3.56/1.71	   top(ok(x0))
3.56/1.71	
3.56/1.71	Special symbols used for the transformation (see [GM04]):
3.56/1.71	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.56/1.71	The replacement map contains the following entries:
3.56/1.71	
3.56/1.71	f: {1}
3.56/1.71	if: {1, 2}
3.56/1.71	c: empty set
3.56/1.71	true: empty set
3.56/1.71	false: empty set
3.56/1.71	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.
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(2)
3.56/1.71	Obligation:
3.56/1.71	Context-sensitive rewrite system:
3.56/1.71	The TRS R consists of the following rules:
3.56/1.71	
3.56/1.71	   f(X) -> if(X, c, f(true))
3.56/1.71	   if(true, X, Y) -> X
3.56/1.71	   if(false, X, Y) -> Y
3.56/1.71	
3.56/1.71	The replacement map contains the following entries:
3.56/1.71	
3.56/1.71	f: {1}
3.56/1.71	if: {1, 2}
3.56/1.71	c: empty set
3.56/1.71	true: empty set
3.56/1.71	false: empty set
3.56/1.71	
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(3) CSRRRRProof (EQUIVALENT)
3.56/1.71	The following CSR is given: Context-sensitive rewrite system:
3.56/1.71	The TRS R consists of the following rules:
3.56/1.71	
3.56/1.71	   f(X) -> if(X, c, f(true))
3.56/1.71	   if(true, X, Y) -> X
3.56/1.71	   if(false, X, Y) -> Y
3.56/1.71	
3.56/1.71	The replacement map contains the following entries:
3.56/1.71	
3.56/1.71	f: {1}
3.56/1.71	if: {1, 2}
3.56/1.71	c: empty set
3.56/1.71	true: empty set
3.56/1.71	false: empty set
3.56/1.71	Used ordering:
3.56/1.71	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.56/1.71	
3.56/1.71	   <<<
3.56/1.71	 POL(f(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
3.56/1.71	>>>
3.56/1.71	
3.56/1.71	   <<<
3.56/1.71	 POL(if(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2 	 +  	[[1, 0], [0, 1]] 	* 	x_3
3.56/1.71	>>>
3.56/1.71	
3.56/1.71	   <<<
3.56/1.71	 POL(c) =  	[[0], [0]]
3.56/1.71	>>>
3.56/1.71	
3.56/1.71	   <<<
3.56/1.71	 POL(true) =  	[[0], [1]]
3.56/1.71	>>>
3.56/1.71	
3.56/1.71	   <<<
3.56/1.71	 POL(false) =  	[[1], [1]]
3.56/1.71	>>>
3.56/1.71	
3.56/1.71	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.56/1.71	
3.56/1.71	   if(false, X, Y) -> Y
3.56/1.71	
3.56/1.71	
3.56/1.71	
3.56/1.71	
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(4)
3.56/1.71	Obligation:
3.56/1.71	Context-sensitive rewrite system:
3.56/1.71	The TRS R consists of the following rules:
3.56/1.71	
3.56/1.71	   f(X) -> if(X, c, f(true))
3.56/1.71	   if(true, X, Y) -> X
3.56/1.71	
3.56/1.71	The replacement map contains the following entries:
3.56/1.71	
3.56/1.71	f: {1}
3.56/1.71	if: {1, 2}
3.56/1.71	c: empty set
3.56/1.71	true: empty set
3.56/1.71	
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(5) CSRRRRProof (EQUIVALENT)
3.56/1.71	The following CSR is given: Context-sensitive rewrite system:
3.56/1.71	The TRS R consists of the following rules:
3.56/1.71	
3.56/1.71	   f(X) -> if(X, c, f(true))
3.56/1.71	   if(true, X, Y) -> X
3.56/1.71	
3.56/1.71	The replacement map contains the following entries:
3.56/1.71	
3.56/1.71	f: {1}
3.56/1.71	if: {1, 2}
3.56/1.71	c: empty set
3.56/1.71	true: empty set
3.56/1.71	Used ordering:
3.56/1.71	Polynomial interpretation [POLO]:
3.56/1.71	
3.56/1.71	   POL(c) = 0
3.56/1.71	   POL(f(x_1)) = 1 + 2*x_1
3.56/1.71	   POL(if(x_1, x_2, x_3)) = x_1 + 2*x_2
3.56/1.71	   POL(true) = 0
3.56/1.71	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.56/1.71	
3.56/1.71	   f(X) -> if(X, c, f(true))
3.56/1.71	
3.56/1.71	
3.56/1.71	
3.56/1.71	
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(6)
3.56/1.71	Obligation:
3.56/1.71	Context-sensitive rewrite system:
3.56/1.71	The TRS R consists of the following rules:
3.56/1.71	
3.56/1.71	   if(true, X, Y) -> X
3.56/1.71	
3.56/1.71	The replacement map contains the following entries:
3.56/1.71	
3.56/1.71	if: {1, 2}
3.56/1.71	true: empty set
3.56/1.71	
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(7) CSRRRRProof (EQUIVALENT)
3.56/1.71	The following CSR is given: Context-sensitive rewrite system:
3.56/1.71	The TRS R consists of the following rules:
3.56/1.71	
3.56/1.71	   if(true, X, Y) -> X
3.56/1.71	
3.56/1.71	The replacement map contains the following entries:
3.56/1.71	
3.56/1.71	if: {1, 2}
3.56/1.71	true: empty set
3.56/1.71	Used ordering:
3.56/1.71	Polynomial interpretation [POLO]:
3.56/1.71	
3.56/1.71	   POL(if(x_1, x_2, x_3)) = 2 + x_1 + x_2
3.56/1.71	   POL(true) = 1
3.56/1.71	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.56/1.71	
3.56/1.71	   if(true, X, Y) -> X
3.56/1.71	
3.56/1.71	
3.56/1.71	
3.56/1.71	
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(8)
3.56/1.71	Obligation:
3.56/1.71	Context-sensitive rewrite system:
3.56/1.71	R is empty.
3.56/1.71	
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(9) RisEmptyProof (EQUIVALENT)
3.56/1.71	The CSR R is empty. Hence, termination is trivially proven.
3.56/1.71	----------------------------------------
3.56/1.71	
3.56/1.71	(10)
3.56/1.71	YES
3.56/1.74	EOF