3.51/1.53	YES
3.51/1.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.51/1.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.51/1.54	
3.51/1.54	
3.51/1.54	Termination of the given CSR could be proven:
3.51/1.54	
3.51/1.54	(0) CSR
3.51/1.54	(1) CSRRRRProof [EQUIVALENT, 58 ms]
3.51/1.54	(2) CSR
3.51/1.54	(3) CSRRRRProof [EQUIVALENT, 7 ms]
3.51/1.54	(4) CSR
3.51/1.54	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.51/1.54	(6) CSR
3.51/1.54	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.51/1.54	(8) CSR
3.51/1.54	(9) CSRRRRProof [EQUIVALENT, 6 ms]
3.51/1.54	(10) CSR
3.51/1.54	(11) RisEmptyProof [EQUIVALENT, 0 ms]
3.51/1.54	(12) YES
3.51/1.54	
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(0)
3.51/1.54	Obligation:
3.51/1.54	Context-sensitive rewrite system:
3.51/1.54	The TRS R consists of the following rules:
3.51/1.54	
3.51/1.54	   incr(nil) -> nil
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(nil) -> nil
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	   nats -> adx(zeros)
3.51/1.54	   zeros -> cons(0, zeros)
3.51/1.54	   head(cons(X, L)) -> X
3.51/1.54	   tail(cons(X, L)) -> L
3.51/1.54	
3.51/1.54	The replacement map contains the following entries:
3.51/1.54	
3.51/1.54	incr: {1}
3.51/1.54	nil: empty set
3.51/1.54	cons: {1}
3.51/1.54	s: {1}
3.51/1.54	adx: {1}
3.51/1.54	nats: empty set
3.51/1.54	zeros: empty set
3.51/1.54	0: empty set
3.51/1.54	head: {1}
3.51/1.54	tail: {1}
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(1) CSRRRRProof (EQUIVALENT)
3.51/1.54	The following CSR is given: Context-sensitive rewrite system:
3.51/1.54	The TRS R consists of the following rules:
3.51/1.54	
3.51/1.54	   incr(nil) -> nil
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(nil) -> nil
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	   nats -> adx(zeros)
3.51/1.54	   zeros -> cons(0, zeros)
3.51/1.54	   head(cons(X, L)) -> X
3.51/1.54	   tail(cons(X, L)) -> L
3.51/1.54	
3.51/1.54	The replacement map contains the following entries:
3.51/1.54	
3.51/1.54	incr: {1}
3.51/1.54	nil: empty set
3.51/1.54	cons: {1}
3.51/1.54	s: {1}
3.51/1.54	adx: {1}
3.51/1.54	nats: empty set
3.51/1.54	zeros: empty set
3.51/1.54	0: empty set
3.51/1.54	head: {1}
3.51/1.54	tail: {1}
3.51/1.54	Used ordering:
3.51/1.54	Polynomial interpretation [POLO]:
3.51/1.54	
3.51/1.54	   POL(0) = 0
3.51/1.54	   POL(adx(x_1)) = x_1
3.51/1.54	   POL(cons(x_1, x_2)) = x_1 + x_2
3.51/1.54	   POL(head(x_1)) = 2*x_1
3.51/1.54	   POL(incr(x_1)) = x_1
3.51/1.54	   POL(nats) = 2
3.51/1.54	   POL(nil) = 0
3.51/1.54	   POL(s(x_1)) = x_1
3.51/1.54	   POL(tail(x_1)) = x_1
3.51/1.54	   POL(zeros) = 0
3.51/1.54	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.51/1.54	
3.51/1.54	   nats -> adx(zeros)
3.51/1.54	
3.51/1.54	
3.51/1.54	
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(2)
3.51/1.54	Obligation:
3.51/1.54	Context-sensitive rewrite system:
3.51/1.54	The TRS R consists of the following rules:
3.51/1.54	
3.51/1.54	   incr(nil) -> nil
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(nil) -> nil
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	   zeros -> cons(0, zeros)
3.51/1.54	   head(cons(X, L)) -> X
3.51/1.54	   tail(cons(X, L)) -> L
3.51/1.54	
3.51/1.54	The replacement map contains the following entries:
3.51/1.54	
3.51/1.54	incr: {1}
3.51/1.54	nil: empty set
3.51/1.54	cons: {1}
3.51/1.54	s: {1}
3.51/1.54	adx: {1}
3.51/1.54	zeros: empty set
3.51/1.54	0: empty set
3.51/1.54	head: {1}
3.51/1.54	tail: {1}
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(3) CSRRRRProof (EQUIVALENT)
3.51/1.54	The following CSR is given: Context-sensitive rewrite system:
3.51/1.54	The TRS R consists of the following rules:
3.51/1.54	
3.51/1.54	   incr(nil) -> nil
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(nil) -> nil
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	   zeros -> cons(0, zeros)
3.51/1.54	   head(cons(X, L)) -> X
3.51/1.54	   tail(cons(X, L)) -> L
3.51/1.54	
3.51/1.54	The replacement map contains the following entries:
3.51/1.54	
3.51/1.54	incr: {1}
3.51/1.54	nil: empty set
3.51/1.54	cons: {1}
3.51/1.54	s: {1}
3.51/1.54	adx: {1}
3.51/1.54	zeros: empty set
3.51/1.54	0: empty set
3.51/1.54	head: {1}
3.51/1.54	tail: {1}
3.51/1.54	Used ordering:
3.51/1.54	Polynomial interpretation [POLO]:
3.51/1.54	
3.51/1.54	   POL(0) = 0
3.51/1.54	   POL(adx(x_1)) = x_1
3.51/1.54	   POL(cons(x_1, x_2)) = x_1 + x_2
3.51/1.54	   POL(head(x_1)) = 1 + x_1
3.51/1.54	   POL(incr(x_1)) = x_1
3.51/1.54	   POL(nil) = 0
3.51/1.54	   POL(s(x_1)) = x_1
3.51/1.54	   POL(tail(x_1)) = x_1
3.51/1.54	   POL(zeros) = 0
3.51/1.54	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.51/1.54	
3.51/1.54	   head(cons(X, L)) -> X
3.51/1.54	
3.51/1.54	
3.51/1.54	
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(4)
3.51/1.54	Obligation:
3.51/1.54	Context-sensitive rewrite system:
3.51/1.54	The TRS R consists of the following rules:
3.51/1.54	
3.51/1.54	   incr(nil) -> nil
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(nil) -> nil
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	   zeros -> cons(0, zeros)
3.51/1.54	   tail(cons(X, L)) -> L
3.51/1.54	
3.51/1.54	The replacement map contains the following entries:
3.51/1.54	
3.51/1.54	incr: {1}
3.51/1.54	nil: empty set
3.51/1.54	cons: {1}
3.51/1.54	s: {1}
3.51/1.54	adx: {1}
3.51/1.54	zeros: empty set
3.51/1.54	0: empty set
3.51/1.54	tail: {1}
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(5) CSRRRRProof (EQUIVALENT)
3.51/1.54	The following CSR is given: Context-sensitive rewrite system:
3.51/1.54	The TRS R consists of the following rules:
3.51/1.54	
3.51/1.54	   incr(nil) -> nil
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(nil) -> nil
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	   zeros -> cons(0, zeros)
3.51/1.54	   tail(cons(X, L)) -> L
3.51/1.54	
3.51/1.54	The replacement map contains the following entries:
3.51/1.54	
3.51/1.54	incr: {1}
3.51/1.54	nil: empty set
3.51/1.54	cons: {1}
3.51/1.54	s: {1}
3.51/1.54	adx: {1}
3.51/1.54	zeros: empty set
3.51/1.54	0: empty set
3.51/1.54	tail: {1}
3.51/1.54	Used ordering:
3.51/1.54	Polynomial interpretation [POLO]:
3.51/1.54	
3.51/1.54	   POL(0) = 0
3.51/1.54	   POL(adx(x_1)) = x_1
3.51/1.54	   POL(cons(x_1, x_2)) = x_1 + x_2
3.51/1.54	   POL(incr(x_1)) = x_1
3.51/1.54	   POL(nil) = 0
3.51/1.54	   POL(s(x_1)) = x_1
3.51/1.54	   POL(tail(x_1)) = 1 + x_1
3.51/1.54	   POL(zeros) = 0
3.51/1.54	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.51/1.54	
3.51/1.54	   tail(cons(X, L)) -> L
3.51/1.54	
3.51/1.54	
3.51/1.54	
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(6)
3.51/1.54	Obligation:
3.51/1.54	Context-sensitive rewrite system:
3.51/1.54	The TRS R consists of the following rules:
3.51/1.54	
3.51/1.54	   incr(nil) -> nil
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(nil) -> nil
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	   zeros -> cons(0, zeros)
3.51/1.54	
3.51/1.54	The replacement map contains the following entries:
3.51/1.54	
3.51/1.54	incr: {1}
3.51/1.54	nil: empty set
3.51/1.54	cons: {1}
3.51/1.54	s: {1}
3.51/1.54	adx: {1}
3.51/1.54	zeros: empty set
3.51/1.54	0: empty set
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(7) CSRRRRProof (EQUIVALENT)
3.51/1.54	The following CSR is given: Context-sensitive rewrite system:
3.51/1.54	The TRS R consists of the following rules:
3.51/1.54	
3.51/1.54	   incr(nil) -> nil
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(nil) -> nil
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	   zeros -> cons(0, zeros)
3.51/1.54	
3.51/1.54	The replacement map contains the following entries:
3.51/1.54	
3.51/1.54	incr: {1}
3.51/1.54	nil: empty set
3.51/1.54	cons: {1}
3.51/1.54	s: {1}
3.51/1.54	adx: {1}
3.51/1.54	zeros: empty set
3.51/1.54	0: empty set
3.51/1.54	Used ordering:
3.51/1.54	Polynomial interpretation [POLO]:
3.51/1.54	
3.51/1.54	   POL(0) = 0
3.51/1.54	   POL(adx(x_1)) = 2*x_1
3.51/1.54	   POL(cons(x_1, x_2)) = x_1
3.51/1.54	   POL(incr(x_1)) = 2*x_1
3.51/1.54	   POL(nil) = 1
3.51/1.54	   POL(s(x_1)) = 2*x_1
3.51/1.54	   POL(zeros) = 1
3.51/1.54	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.51/1.54	
3.51/1.54	   incr(nil) -> nil
3.51/1.54	   adx(nil) -> nil
3.51/1.54	   zeros -> cons(0, zeros)
3.51/1.54	
3.51/1.54	
3.51/1.54	
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(8)
3.51/1.54	Obligation:
3.51/1.54	Context-sensitive rewrite system:
3.51/1.54	The TRS R consists of the following rules:
3.51/1.54	
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	
3.51/1.54	The replacement map contains the following entries:
3.51/1.54	
3.51/1.54	incr: {1}
3.51/1.54	cons: {1}
3.51/1.54	s: {1}
3.51/1.54	adx: {1}
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(9) CSRRRRProof (EQUIVALENT)
3.51/1.54	The following CSR is given: Context-sensitive rewrite system:
3.51/1.54	The TRS R consists of the following rules:
3.51/1.54	
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	
3.51/1.54	The replacement map contains the following entries:
3.51/1.54	
3.51/1.54	incr: {1}
3.51/1.54	cons: {1}
3.51/1.54	s: {1}
3.51/1.54	adx: {1}
3.51/1.54	Used ordering:
3.51/1.54	Polynomial interpretation [POLO]:
3.51/1.54	
3.51/1.54	   POL(adx(x_1)) = 2 + 2*x_1
3.51/1.54	   POL(cons(x_1, x_2)) = 1 + x_1
3.51/1.54	   POL(incr(x_1)) = 1 + 2*x_1
3.51/1.54	   POL(s(x_1)) = x_1
3.51/1.54	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.51/1.54	
3.51/1.54	   incr(cons(X, L)) -> cons(s(X), incr(L))
3.51/1.54	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
3.51/1.54	
3.51/1.54	
3.51/1.54	
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(10)
3.51/1.54	Obligation:
3.51/1.54	Context-sensitive rewrite system:
3.51/1.54	R is empty.
3.51/1.54	
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(11) RisEmptyProof (EQUIVALENT)
3.51/1.54	The CSR R is empty. Hence, termination is trivially proven.
3.51/1.54	----------------------------------------
3.51/1.54	
3.51/1.54	(12)
3.51/1.54	YES
3.69/1.58	EOF