3.44/1.66	YES
3.44/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.44/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.44/1.67	
3.44/1.67	
3.44/1.67	Termination of the given CSR could be proven:
3.44/1.67	
3.44/1.67	(0) CSR
3.44/1.67	(1) CSRRRRProof [EQUIVALENT, 66 ms]
3.44/1.67	(2) CSR
3.44/1.67	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.44/1.67	(4) CSR
3.44/1.67	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.44/1.67	(6) CSR
3.44/1.67	(7) CSRRRRProof [EQUIVALENT, 2 ms]
3.44/1.67	(8) CSR
3.44/1.67	(9) CSRRRRProof [EQUIVALENT, 0 ms]
3.44/1.67	(10) CSR
3.44/1.67	(11) RisEmptyProof [EQUIVALENT, 0 ms]
3.44/1.67	(12) YES
3.44/1.67	
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(0)
3.44/1.67	Obligation:
3.44/1.67	Context-sensitive rewrite system:
3.44/1.67	The TRS R consists of the following rules:
3.44/1.67	
3.44/1.67	   minus(0, Y) -> 0
3.44/1.67	   minus(s(X), s(Y)) -> minus(X, Y)
3.44/1.67	   geq(X, 0) -> true
3.44/1.67	   geq(0, s(Y)) -> false
3.44/1.67	   geq(s(X), s(Y)) -> geq(X, Y)
3.44/1.67	   div(0, s(Y)) -> 0
3.44/1.67	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.44/1.67	   if(true, X, Y) -> X
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	The replacement map contains the following entries:
3.44/1.67	
3.44/1.67	minus: empty set
3.44/1.67	0: empty set
3.44/1.67	s: {1}
3.44/1.67	geq: empty set
3.44/1.67	true: empty set
3.44/1.67	false: empty set
3.44/1.67	div: {1}
3.44/1.67	if: {1}
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(1) CSRRRRProof (EQUIVALENT)
3.44/1.67	The following CSR is given: Context-sensitive rewrite system:
3.44/1.67	The TRS R consists of the following rules:
3.44/1.67	
3.44/1.67	   minus(0, Y) -> 0
3.44/1.67	   minus(s(X), s(Y)) -> minus(X, Y)
3.44/1.67	   geq(X, 0) -> true
3.44/1.67	   geq(0, s(Y)) -> false
3.44/1.67	   geq(s(X), s(Y)) -> geq(X, Y)
3.44/1.67	   div(0, s(Y)) -> 0
3.44/1.67	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.44/1.67	   if(true, X, Y) -> X
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	The replacement map contains the following entries:
3.44/1.67	
3.44/1.67	minus: empty set
3.44/1.67	0: empty set
3.44/1.67	s: {1}
3.44/1.67	geq: empty set
3.44/1.67	true: empty set
3.44/1.67	false: empty set
3.44/1.67	div: {1}
3.44/1.67	if: {1}
3.44/1.67	Used ordering:
3.44/1.67	Polynomial interpretation [POLO]:
3.44/1.67	
3.44/1.67	   POL(0) = 0
3.44/1.67	   POL(div(x_1, x_2)) = 2*x_1
3.44/1.67	   POL(false) = 0
3.44/1.67	   POL(geq(x_1, x_2)) = x_1
3.44/1.67	   POL(if(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
3.44/1.67	   POL(minus(x_1, x_2)) = 0
3.44/1.67	   POL(s(x_1)) = 1 + 2*x_1
3.44/1.67	   POL(true) = 0
3.44/1.67	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.44/1.67	
3.44/1.67	   geq(s(X), s(Y)) -> geq(X, Y)
3.44/1.67	
3.44/1.67	
3.44/1.67	
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(2)
3.44/1.67	Obligation:
3.44/1.67	Context-sensitive rewrite system:
3.44/1.67	The TRS R consists of the following rules:
3.44/1.67	
3.44/1.67	   minus(0, Y) -> 0
3.44/1.67	   minus(s(X), s(Y)) -> minus(X, Y)
3.44/1.67	   geq(X, 0) -> true
3.44/1.67	   geq(0, s(Y)) -> false
3.44/1.67	   div(0, s(Y)) -> 0
3.44/1.67	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.44/1.67	   if(true, X, Y) -> X
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	The replacement map contains the following entries:
3.44/1.67	
3.44/1.67	minus: empty set
3.44/1.67	0: empty set
3.44/1.67	s: {1}
3.44/1.67	geq: empty set
3.44/1.67	true: empty set
3.44/1.67	false: empty set
3.44/1.67	div: {1}
3.44/1.67	if: {1}
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(3) CSRRRRProof (EQUIVALENT)
3.44/1.67	The following CSR is given: Context-sensitive rewrite system:
3.44/1.67	The TRS R consists of the following rules:
3.44/1.67	
3.44/1.67	   minus(0, Y) -> 0
3.44/1.67	   minus(s(X), s(Y)) -> minus(X, Y)
3.44/1.67	   geq(X, 0) -> true
3.44/1.67	   geq(0, s(Y)) -> false
3.44/1.67	   div(0, s(Y)) -> 0
3.44/1.67	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.44/1.67	   if(true, X, Y) -> X
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	The replacement map contains the following entries:
3.44/1.67	
3.44/1.67	minus: empty set
3.44/1.67	0: empty set
3.44/1.67	s: {1}
3.44/1.67	geq: empty set
3.44/1.67	true: empty set
3.44/1.67	false: empty set
3.44/1.67	div: {1}
3.44/1.67	if: {1}
3.44/1.67	Used ordering:
3.44/1.67	Polynomial interpretation [POLO]:
3.44/1.67	
3.44/1.67	   POL(0) = 0
3.44/1.67	   POL(div(x_1, x_2)) = 2 + 2*x_1 + x_2
3.44/1.67	   POL(false) = 0
3.44/1.67	   POL(geq(x_1, x_2)) = 0
3.44/1.67	   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
3.44/1.67	   POL(minus(x_1, x_2)) = 0
3.44/1.67	   POL(s(x_1)) = 1 + x_1
3.44/1.67	   POL(true) = 0
3.44/1.67	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.44/1.67	
3.44/1.67	   div(0, s(Y)) -> 0
3.44/1.67	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.44/1.67	
3.44/1.67	
3.44/1.67	
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(4)
3.44/1.67	Obligation:
3.44/1.67	Context-sensitive rewrite system:
3.44/1.67	The TRS R consists of the following rules:
3.44/1.67	
3.44/1.67	   minus(0, Y) -> 0
3.44/1.67	   minus(s(X), s(Y)) -> minus(X, Y)
3.44/1.67	   geq(X, 0) -> true
3.44/1.67	   geq(0, s(Y)) -> false
3.44/1.67	   if(true, X, Y) -> X
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	The replacement map contains the following entries:
3.44/1.67	
3.44/1.67	minus: empty set
3.44/1.67	0: empty set
3.44/1.67	s: {1}
3.44/1.67	geq: empty set
3.44/1.67	true: empty set
3.44/1.67	false: empty set
3.44/1.67	if: {1}
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(5) CSRRRRProof (EQUIVALENT)
3.44/1.67	The following CSR is given: Context-sensitive rewrite system:
3.44/1.67	The TRS R consists of the following rules:
3.44/1.67	
3.44/1.67	   minus(0, Y) -> 0
3.44/1.67	   minus(s(X), s(Y)) -> minus(X, Y)
3.44/1.67	   geq(X, 0) -> true
3.44/1.67	   geq(0, s(Y)) -> false
3.44/1.67	   if(true, X, Y) -> X
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	The replacement map contains the following entries:
3.44/1.67	
3.44/1.67	minus: empty set
3.44/1.67	0: empty set
3.44/1.67	s: {1}
3.44/1.67	geq: empty set
3.44/1.67	true: empty set
3.44/1.67	false: empty set
3.44/1.67	if: {1}
3.44/1.67	Used ordering:
3.44/1.67	Polynomial interpretation [POLO]:
3.44/1.67	
3.44/1.67	   POL(0) = 1
3.44/1.67	   POL(false) = 0
3.44/1.67	   POL(geq(x_1, x_2)) = 1 + 2*x_1 + x_2
3.44/1.67	   POL(if(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
3.44/1.67	   POL(minus(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
3.44/1.67	   POL(s(x_1)) = 2 + x_1
3.44/1.67	   POL(true) = 2
3.44/1.67	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.44/1.67	
3.44/1.67	   minus(0, Y) -> 0
3.44/1.67	   minus(s(X), s(Y)) -> minus(X, Y)
3.44/1.67	   geq(0, s(Y)) -> false
3.44/1.67	   if(true, X, Y) -> X
3.44/1.67	
3.44/1.67	
3.44/1.67	
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(6)
3.44/1.67	Obligation:
3.44/1.67	Context-sensitive rewrite system:
3.44/1.67	The TRS R consists of the following rules:
3.44/1.67	
3.44/1.67	   geq(X, 0) -> true
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	The replacement map contains the following entries:
3.44/1.67	
3.44/1.67	0: empty set
3.44/1.67	geq: empty set
3.44/1.67	true: empty set
3.44/1.67	false: empty set
3.44/1.67	if: {1}
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(7) CSRRRRProof (EQUIVALENT)
3.44/1.67	The following CSR is given: Context-sensitive rewrite system:
3.44/1.67	The TRS R consists of the following rules:
3.44/1.67	
3.44/1.67	   geq(X, 0) -> true
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	The replacement map contains the following entries:
3.44/1.67	
3.44/1.67	0: empty set
3.44/1.67	geq: empty set
3.44/1.67	true: empty set
3.44/1.67	false: empty set
3.44/1.67	if: {1}
3.44/1.67	Used ordering:
3.44/1.67	Polynomial interpretation [POLO]:
3.44/1.67	
3.44/1.67	   POL(0) = 1
3.44/1.67	   POL(false) = 0
3.44/1.67	   POL(geq(x_1, x_2)) = 1 + x_2
3.44/1.67	   POL(if(x_1, x_2, x_3)) = x_1 + x_3
3.44/1.67	   POL(true) = 0
3.44/1.67	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.44/1.67	
3.44/1.67	   geq(X, 0) -> true
3.44/1.67	
3.44/1.67	
3.44/1.67	
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(8)
3.44/1.67	Obligation:
3.44/1.67	Context-sensitive rewrite system:
3.44/1.67	The TRS R consists of the following rules:
3.44/1.67	
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	The replacement map contains the following entries:
3.44/1.67	
3.44/1.67	false: empty set
3.44/1.67	if: {1}
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(9) CSRRRRProof (EQUIVALENT)
3.44/1.67	The following CSR is given: Context-sensitive rewrite system:
3.44/1.67	The TRS R consists of the following rules:
3.44/1.67	
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	The replacement map contains the following entries:
3.44/1.67	
3.44/1.67	false: empty set
3.44/1.67	if: {1}
3.44/1.67	Used ordering:
3.44/1.67	Polynomial interpretation [POLO]:
3.44/1.67	
3.44/1.67	   POL(false) = 1
3.44/1.67	   POL(if(x_1, x_2, x_3)) = 2 + x_1 + x_3
3.44/1.67	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.44/1.67	
3.44/1.67	   if(false, X, Y) -> Y
3.44/1.67	
3.44/1.67	
3.44/1.67	
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(10)
3.44/1.67	Obligation:
3.44/1.67	Context-sensitive rewrite system:
3.44/1.67	R is empty.
3.44/1.67	
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(11) RisEmptyProof (EQUIVALENT)
3.44/1.67	The CSR R is empty. Hence, termination is trivially proven.
3.44/1.67	----------------------------------------
3.44/1.67	
3.44/1.67	(12)
3.44/1.67	YES
3.44/1.69	EOF