2.86/1.34	YES
2.86/1.35	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.86/1.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.86/1.35	
2.86/1.35	
2.86/1.35	Termination of the given CSR could be proven:
2.86/1.35	
2.86/1.35	(0) CSR
2.86/1.35	(1) CSRRRRProof [EQUIVALENT, 63 ms]
2.86/1.35	(2) CSR
2.86/1.35	(3) RisEmptyProof [EQUIVALENT, 0 ms]
2.86/1.35	(4) YES
2.86/1.35	
2.86/1.35	
2.86/1.35	----------------------------------------
2.86/1.35	
2.86/1.35	(0)
2.86/1.35	Obligation:
2.86/1.35	Context-sensitive rewrite system:
2.86/1.35	The TRS R consists of the following rules:
2.86/1.35	
2.86/1.35	   fst(0, Z) -> nil
2.86/1.35	   fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
2.86/1.35	   from(X) -> cons(X, from(s(X)))
2.86/1.35	   add(0, X) -> X
2.86/1.35	   add(s(X), Y) -> s(add(X, Y))
2.86/1.35	   len(nil) -> 0
2.86/1.35	   len(cons(X, Z)) -> s(len(Z))
2.86/1.35	
2.86/1.35	The replacement map contains the following entries:
2.86/1.35	
2.86/1.35	fst: {1, 2}
2.86/1.35	0: empty set
2.86/1.35	nil: empty set
2.86/1.35	s: empty set
2.86/1.35	cons: {1}
2.86/1.35	from: {1}
2.86/1.35	add: {1, 2}
2.86/1.35	len: {1}
2.86/1.35	
2.86/1.35	----------------------------------------
2.86/1.35	
2.86/1.35	(1) CSRRRRProof (EQUIVALENT)
2.86/1.35	The following CSR is given: Context-sensitive rewrite system:
2.86/1.35	The TRS R consists of the following rules:
2.86/1.35	
2.86/1.35	   fst(0, Z) -> nil
2.86/1.35	   fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
2.86/1.35	   from(X) -> cons(X, from(s(X)))
2.86/1.35	   add(0, X) -> X
2.86/1.35	   add(s(X), Y) -> s(add(X, Y))
2.86/1.35	   len(nil) -> 0
2.86/1.35	   len(cons(X, Z)) -> s(len(Z))
2.86/1.35	
2.86/1.35	The replacement map contains the following entries:
2.86/1.35	
2.86/1.35	fst: {1, 2}
2.86/1.35	0: empty set
2.86/1.35	nil: empty set
2.86/1.35	s: empty set
2.86/1.35	cons: {1}
2.86/1.35	from: {1}
2.86/1.35	add: {1, 2}
2.86/1.35	len: {1}
2.86/1.35	Used ordering:
2.86/1.35	Polynomial interpretation [POLO]:
2.86/1.35	
2.86/1.35	   POL(0) = 2
2.86/1.35	   POL(add(x_1, x_2)) = 1 + x_1 + 2*x_2
2.86/1.35	   POL(cons(x_1, x_2)) = 1 + 2*x_1
2.86/1.35	   POL(from(x_1)) = 2 + 2*x_1
2.86/1.35	   POL(fst(x_1, x_2)) = 2*x_1 + 2*x_2
2.86/1.35	   POL(len(x_1)) = 2 + x_1
2.86/1.35	   POL(nil) = 2
2.86/1.35	   POL(s(x_1)) = 0
2.86/1.35	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.86/1.35	
2.86/1.35	   fst(0, Z) -> nil
2.86/1.35	   fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
2.86/1.35	   from(X) -> cons(X, from(s(X)))
2.86/1.35	   add(0, X) -> X
2.86/1.35	   add(s(X), Y) -> s(add(X, Y))
2.86/1.35	   len(nil) -> 0
2.86/1.35	   len(cons(X, Z)) -> s(len(Z))
2.86/1.35	
2.86/1.35	
2.86/1.35	
2.86/1.35	
2.86/1.35	----------------------------------------
2.86/1.35	
2.86/1.35	(2)
2.86/1.35	Obligation:
2.86/1.35	Context-sensitive rewrite system:
2.86/1.35	R is empty.
2.86/1.35	
2.86/1.35	----------------------------------------
2.86/1.35	
2.86/1.35	(3) RisEmptyProof (EQUIVALENT)
2.86/1.35	The CSR R is empty. Hence, termination is trivially proven.
2.86/1.35	----------------------------------------
2.86/1.35	
2.86/1.35	(4)
2.86/1.35	YES
3.16/1.38	EOF