2.81/1.41	YES
2.81/1.42	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
2.81/1.42	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.81/1.42	
2.81/1.42	
2.81/1.42	Termination of the given CSR could be proven:
2.81/1.42	
2.81/1.42	(0) CSR
2.81/1.42	(1) CSRRRRProof [EQUIVALENT, 43 ms]
2.81/1.42	(2) CSR
2.81/1.42	(3) CSRRRRProof [EQUIVALENT, 0 ms]
2.81/1.42	(4) CSR
2.81/1.42	(5) CSRRRRProof [EQUIVALENT, 3 ms]
2.81/1.42	(6) CSR
2.81/1.42	(7) RisEmptyProof [EQUIVALENT, 0 ms]
2.81/1.42	(8) YES
2.81/1.42	
2.81/1.42	
2.81/1.42	----------------------------------------
2.81/1.42	
2.81/1.42	(0)
2.81/1.42	Obligation:
2.81/1.42	Context-sensitive rewrite system:
2.81/1.42	The TRS R consists of the following rules:
2.81/1.42	
2.81/1.42	   f(X) -> if(X, c, f(true))
2.81/1.42	   if(true, X, Y) -> X
2.81/1.42	   if(false, X, Y) -> Y
2.81/1.42	
2.81/1.42	The replacement map contains the following entries:
2.81/1.42	
2.81/1.42	f: {1}
2.81/1.42	if: {1, 2}
2.81/1.42	c: empty set
2.81/1.42	true: empty set
2.81/1.42	false: empty set
2.81/1.42	
2.81/1.42	----------------------------------------
2.81/1.42	
2.81/1.42	(1) CSRRRRProof (EQUIVALENT)
2.81/1.42	The following CSR is given: Context-sensitive rewrite system:
2.81/1.42	The TRS R consists of the following rules:
2.81/1.42	
2.81/1.42	   f(X) -> if(X, c, f(true))
2.81/1.42	   if(true, X, Y) -> X
2.81/1.42	   if(false, X, Y) -> Y
2.81/1.42	
2.81/1.42	The replacement map contains the following entries:
2.81/1.42	
2.81/1.42	f: {1}
2.81/1.42	if: {1, 2}
2.81/1.42	c: empty set
2.81/1.42	true: empty set
2.81/1.42	false: empty set
2.81/1.42	Used ordering:
2.81/1.42	Polynomial interpretation [POLO]:
2.81/1.42	
2.81/1.42	   POL(c) = 0
2.81/1.42	   POL(f(x_1)) = x_1
2.81/1.42	   POL(false) = 1
2.81/1.42	   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
2.81/1.42	   POL(true) = 0
2.81/1.42	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.81/1.42	
2.81/1.42	   if(false, X, Y) -> Y
2.81/1.42	
2.81/1.42	
2.81/1.42	
2.81/1.42	
2.81/1.42	----------------------------------------
2.81/1.42	
2.81/1.42	(2)
2.81/1.42	Obligation:
2.81/1.42	Context-sensitive rewrite system:
2.81/1.42	The TRS R consists of the following rules:
2.81/1.42	
2.81/1.42	   f(X) -> if(X, c, f(true))
2.81/1.42	   if(true, X, Y) -> X
2.81/1.42	
2.81/1.42	The replacement map contains the following entries:
2.81/1.42	
2.81/1.42	f: {1}
2.81/1.42	if: {1, 2}
2.81/1.42	c: empty set
2.81/1.42	true: empty set
2.81/1.42	
2.81/1.42	----------------------------------------
2.81/1.42	
2.81/1.42	(3) CSRRRRProof (EQUIVALENT)
2.81/1.42	The following CSR is given: Context-sensitive rewrite system:
2.81/1.42	The TRS R consists of the following rules:
2.81/1.42	
2.81/1.42	   f(X) -> if(X, c, f(true))
2.81/1.42	   if(true, X, Y) -> X
2.81/1.42	
2.81/1.42	The replacement map contains the following entries:
2.81/1.42	
2.81/1.42	f: {1}
2.81/1.42	if: {1, 2}
2.81/1.42	c: empty set
2.81/1.42	true: empty set
2.81/1.42	Used ordering:
2.81/1.42	Polynomial interpretation [POLO]:
2.81/1.42	
2.81/1.42	   POL(c) = 0
2.81/1.42	   POL(f(x_1)) = 1 + 2*x_1
2.81/1.42	   POL(if(x_1, x_2, x_3)) = x_1 + 2*x_2
2.81/1.42	   POL(true) = 0
2.81/1.42	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.81/1.42	
2.81/1.42	   f(X) -> if(X, c, f(true))
2.81/1.42	
2.81/1.42	
2.81/1.42	
2.81/1.42	
2.81/1.42	----------------------------------------
2.81/1.42	
2.81/1.42	(4)
2.81/1.42	Obligation:
2.81/1.42	Context-sensitive rewrite system:
2.81/1.42	The TRS R consists of the following rules:
2.81/1.42	
2.81/1.42	   if(true, X, Y) -> X
2.81/1.42	
2.81/1.42	The replacement map contains the following entries:
2.81/1.42	
2.81/1.42	if: {1, 2}
2.81/1.42	true: empty set
2.81/1.42	
2.81/1.42	----------------------------------------
2.81/1.42	
2.81/1.42	(5) CSRRRRProof (EQUIVALENT)
2.81/1.42	The following CSR is given: Context-sensitive rewrite system:
2.81/1.42	The TRS R consists of the following rules:
2.81/1.42	
2.81/1.42	   if(true, X, Y) -> X
2.81/1.42	
2.81/1.42	The replacement map contains the following entries:
2.81/1.42	
2.81/1.42	if: {1, 2}
2.81/1.42	true: empty set
2.81/1.42	Used ordering:
2.81/1.42	Polynomial interpretation [POLO]:
2.81/1.42	
2.81/1.42	   POL(if(x_1, x_2, x_3)) = x_1 + x_2
2.81/1.42	   POL(true) = 1
2.81/1.42	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.81/1.42	
2.81/1.42	   if(true, X, Y) -> X
2.81/1.42	
2.81/1.42	
2.81/1.42	
2.81/1.42	
2.81/1.42	----------------------------------------
2.81/1.42	
2.81/1.42	(6)
2.81/1.42	Obligation:
2.81/1.42	Context-sensitive rewrite system:
2.81/1.42	R is empty.
2.81/1.42	
2.81/1.42	----------------------------------------
2.81/1.42	
2.81/1.42	(7) RisEmptyProof (EQUIVALENT)
2.81/1.42	The CSR R is empty. Hence, termination is trivially proven.
2.81/1.42	----------------------------------------
2.81/1.42	
2.81/1.42	(8)
2.81/1.42	YES
2.92/1.46	EOF