980.63/291.46	WORST_CASE(Omega(n^1), ?)
980.63/291.47	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
980.63/291.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
980.63/291.47	
980.63/291.47	
980.63/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
980.63/291.47	
980.63/291.47	(0) CpxTRS
980.63/291.47	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
980.63/291.47	(2) CpxTRS
980.63/291.47	(3) SlicingProof [LOWER BOUND(ID), 0 ms]
980.63/291.47	(4) CpxTRS
980.63/291.47	(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
980.63/291.47	(6) typed CpxTrs
980.63/291.47	(7) OrderProof [LOWER BOUND(ID), 0 ms]
980.63/291.47	(8) typed CpxTrs
980.63/291.47	(9) RewriteLemmaProof [LOWER BOUND(ID), 288 ms]
980.63/291.47	(10) BEST
980.63/291.47	    (11) proven lower bound
980.63/291.47	        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
980.63/291.47	        (13) BOUNDS(n^1, INF)
980.63/291.47	    (14) typed CpxTrs
980.63/291.47	        (15) RewriteLemmaProof [LOWER BOUND(ID), 86 ms]
980.63/291.47	        (16) BOUNDS(1, INF)
980.63/291.47	
980.63/291.47	
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(0)
980.63/291.47	Obligation:
980.63/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
980.63/291.47	
980.63/291.47	
980.63/291.47	The TRS R consists of the following rules:
980.63/291.47	
980.63/291.47	   int(0, 0) -> .(0, nil)
980.63/291.47	   int(0, s(y)) -> .(0, int(s(0), s(y)))
980.63/291.47	   int(s(x), 0) -> nil
980.63/291.47	   int(s(x), s(y)) -> int_list(int(x, y))
980.63/291.47	   int_list(nil) -> nil
980.63/291.47	   int_list(.(x, y)) -> .(s(x), int_list(y))
980.63/291.47	
980.63/291.47	S is empty.
980.63/291.47	Rewrite Strategy: FULL
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(1) RenamingProof (BOTH BOUNDS(ID, ID))
980.63/291.47	Renamed function symbols to avoid clashes with predefined symbol.
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(2)
980.63/291.47	Obligation:
980.63/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
980.63/291.47	
980.63/291.47	
980.63/291.47	The TRS R consists of the following rules:
980.63/291.47	
980.63/291.47	   int(0', 0') -> .(0', nil)
980.63/291.47	   int(0', s(y)) -> .(0', int(s(0'), s(y)))
980.63/291.47	   int(s(x), 0') -> nil
980.63/291.47	   int(s(x), s(y)) -> int_list(int(x, y))
980.63/291.47	   int_list(nil) -> nil
980.63/291.47	   int_list(.(x, y)) -> .(s(x), int_list(y))
980.63/291.47	
980.63/291.47	S is empty.
980.63/291.47	Rewrite Strategy: FULL
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(3) SlicingProof (LOWER BOUND(ID))
980.63/291.47	Sliced the following arguments:
980.63/291.47	./0
980.63/291.47	
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(4)
980.63/291.47	Obligation:
980.63/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
980.63/291.47	
980.63/291.47	
980.63/291.47	The TRS R consists of the following rules:
980.63/291.47	
980.63/291.47	   int(0', 0') -> .(nil)
980.63/291.47	   int(0', s(y)) -> .(int(s(0'), s(y)))
980.63/291.47	   int(s(x), 0') -> nil
980.63/291.47	   int(s(x), s(y)) -> int_list(int(x, y))
980.63/291.47	   int_list(nil) -> nil
980.63/291.47	   int_list(.(y)) -> .(int_list(y))
980.63/291.47	
980.63/291.47	S is empty.
980.63/291.47	Rewrite Strategy: FULL
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
980.63/291.47	Infered types.
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(6)
980.63/291.47	Obligation:
980.63/291.47	TRS:
980.63/291.47	Rules:
980.63/291.47	int(0', 0') -> .(nil)
980.63/291.47	int(0', s(y)) -> .(int(s(0'), s(y)))
980.63/291.47	int(s(x), 0') -> nil
980.63/291.47	int(s(x), s(y)) -> int_list(int(x, y))
980.63/291.47	int_list(nil) -> nil
980.63/291.47	int_list(.(y)) -> .(int_list(y))
980.63/291.47	
980.63/291.47	Types:
980.63/291.47	int :: 0':s -> 0':s -> nil:.
980.63/291.47	0' :: 0':s
980.63/291.47	. :: nil:. -> nil:.
980.63/291.47	nil :: nil:.
980.63/291.47	s :: 0':s -> 0':s
980.63/291.47	int_list :: nil:. -> nil:.
980.63/291.47	hole_nil:.1_0 :: nil:.
980.63/291.47	hole_0':s2_0 :: 0':s
980.63/291.47	gen_nil:.3_0 :: Nat -> nil:.
980.63/291.47	gen_0':s4_0 :: Nat -> 0':s
980.63/291.47	
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(7) OrderProof (LOWER BOUND(ID))
980.63/291.47	Heuristically decided to analyse the following defined symbols:
980.63/291.47	int, int_list
980.63/291.47	
980.63/291.47	They will be analysed ascendingly in the following order:
980.63/291.47	int_list < int
980.63/291.47	
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(8)
980.63/291.47	Obligation:
980.63/291.47	TRS:
980.63/291.47	Rules:
980.63/291.47	int(0', 0') -> .(nil)
980.63/291.47	int(0', s(y)) -> .(int(s(0'), s(y)))
980.63/291.47	int(s(x), 0') -> nil
980.63/291.47	int(s(x), s(y)) -> int_list(int(x, y))
980.63/291.47	int_list(nil) -> nil
980.63/291.47	int_list(.(y)) -> .(int_list(y))
980.63/291.47	
980.63/291.47	Types:
980.63/291.47	int :: 0':s -> 0':s -> nil:.
980.63/291.47	0' :: 0':s
980.63/291.47	. :: nil:. -> nil:.
980.63/291.47	nil :: nil:.
980.63/291.47	s :: 0':s -> 0':s
980.63/291.47	int_list :: nil:. -> nil:.
980.63/291.47	hole_nil:.1_0 :: nil:.
980.63/291.47	hole_0':s2_0 :: 0':s
980.63/291.47	gen_nil:.3_0 :: Nat -> nil:.
980.63/291.47	gen_0':s4_0 :: Nat -> 0':s
980.63/291.47	
980.63/291.47	
980.63/291.47	Generator Equations:
980.63/291.47	gen_nil:.3_0(0) <=> nil
980.63/291.47	gen_nil:.3_0(+(x, 1)) <=> .(gen_nil:.3_0(x))
980.63/291.47	gen_0':s4_0(0) <=> 0'
980.63/291.47	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
980.63/291.47	
980.63/291.47	
980.63/291.47	The following defined symbols remain to be analysed:
980.63/291.47	int_list, int
980.63/291.47	
980.63/291.47	They will be analysed ascendingly in the following order:
980.63/291.47	int_list < int
980.63/291.47	
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(9) RewriteLemmaProof (LOWER BOUND(ID))
980.63/291.47	Proved the following rewrite lemma:
980.63/291.47	int_list(gen_nil:.3_0(n6_0)) -> gen_nil:.3_0(n6_0), rt in Omega(1 + n6_0)
980.63/291.47	
980.63/291.47	Induction Base:
980.63/291.47	int_list(gen_nil:.3_0(0)) ->_R^Omega(1)
980.63/291.47	nil
980.63/291.47	
980.63/291.47	Induction Step:
980.63/291.47	int_list(gen_nil:.3_0(+(n6_0, 1))) ->_R^Omega(1)
980.63/291.47	.(int_list(gen_nil:.3_0(n6_0))) ->_IH
980.63/291.47	.(gen_nil:.3_0(c7_0))
980.63/291.47	
980.63/291.47	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(10)
980.63/291.47	Complex Obligation (BEST)
980.63/291.47	
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(11)
980.63/291.47	Obligation:
980.63/291.47	Proved the lower bound n^1 for the following obligation:
980.63/291.47	
980.63/291.47	TRS:
980.63/291.47	Rules:
980.63/291.47	int(0', 0') -> .(nil)
980.63/291.47	int(0', s(y)) -> .(int(s(0'), s(y)))
980.63/291.47	int(s(x), 0') -> nil
980.63/291.47	int(s(x), s(y)) -> int_list(int(x, y))
980.63/291.47	int_list(nil) -> nil
980.63/291.47	int_list(.(y)) -> .(int_list(y))
980.63/291.47	
980.63/291.47	Types:
980.63/291.47	int :: 0':s -> 0':s -> nil:.
980.63/291.47	0' :: 0':s
980.63/291.47	. :: nil:. -> nil:.
980.63/291.47	nil :: nil:.
980.63/291.47	s :: 0':s -> 0':s
980.63/291.47	int_list :: nil:. -> nil:.
980.63/291.47	hole_nil:.1_0 :: nil:.
980.63/291.47	hole_0':s2_0 :: 0':s
980.63/291.47	gen_nil:.3_0 :: Nat -> nil:.
980.63/291.47	gen_0':s4_0 :: Nat -> 0':s
980.63/291.47	
980.63/291.47	
980.63/291.47	Generator Equations:
980.63/291.47	gen_nil:.3_0(0) <=> nil
980.63/291.47	gen_nil:.3_0(+(x, 1)) <=> .(gen_nil:.3_0(x))
980.63/291.47	gen_0':s4_0(0) <=> 0'
980.63/291.47	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
980.63/291.47	
980.63/291.47	
980.63/291.47	The following defined symbols remain to be analysed:
980.63/291.47	int_list, int
980.63/291.47	
980.63/291.47	They will be analysed ascendingly in the following order:
980.63/291.47	int_list < int
980.63/291.47	
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(12) LowerBoundPropagationProof (FINISHED)
980.63/291.47	Propagated lower bound.
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(13)
980.63/291.47	BOUNDS(n^1, INF)
980.63/291.47	
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(14)
980.63/291.47	Obligation:
980.63/291.47	TRS:
980.63/291.47	Rules:
980.63/291.47	int(0', 0') -> .(nil)
980.63/291.47	int(0', s(y)) -> .(int(s(0'), s(y)))
980.63/291.47	int(s(x), 0') -> nil
980.63/291.47	int(s(x), s(y)) -> int_list(int(x, y))
980.63/291.47	int_list(nil) -> nil
980.63/291.47	int_list(.(y)) -> .(int_list(y))
980.63/291.47	
980.63/291.47	Types:
980.63/291.47	int :: 0':s -> 0':s -> nil:.
980.63/291.47	0' :: 0':s
980.63/291.47	. :: nil:. -> nil:.
980.63/291.47	nil :: nil:.
980.63/291.47	s :: 0':s -> 0':s
980.63/291.47	int_list :: nil:. -> nil:.
980.63/291.47	hole_nil:.1_0 :: nil:.
980.63/291.47	hole_0':s2_0 :: 0':s
980.63/291.47	gen_nil:.3_0 :: Nat -> nil:.
980.63/291.47	gen_0':s4_0 :: Nat -> 0':s
980.63/291.47	
980.63/291.47	
980.63/291.47	Lemmas:
980.63/291.47	int_list(gen_nil:.3_0(n6_0)) -> gen_nil:.3_0(n6_0), rt in Omega(1 + n6_0)
980.63/291.47	
980.63/291.47	
980.63/291.47	Generator Equations:
980.63/291.47	gen_nil:.3_0(0) <=> nil
980.63/291.47	gen_nil:.3_0(+(x, 1)) <=> .(gen_nil:.3_0(x))
980.63/291.47	gen_0':s4_0(0) <=> 0'
980.63/291.47	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
980.63/291.47	
980.63/291.47	
980.63/291.47	The following defined symbols remain to be analysed:
980.63/291.47	int
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(15) RewriteLemmaProof (LOWER BOUND(ID))
980.63/291.47	Proved the following rewrite lemma:
980.63/291.47	int(gen_0':s4_0(n184_0), gen_0':s4_0(n184_0)) -> gen_nil:.3_0(1), rt in Omega(1 + n184_0)
980.63/291.47	
980.63/291.47	Induction Base:
980.63/291.47	int(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
980.63/291.47	.(nil)
980.63/291.47	
980.63/291.47	Induction Step:
980.63/291.47	int(gen_0':s4_0(+(n184_0, 1)), gen_0':s4_0(+(n184_0, 1))) ->_R^Omega(1)
980.63/291.47	int_list(int(gen_0':s4_0(n184_0), gen_0':s4_0(n184_0))) ->_IH
980.63/291.47	int_list(gen_nil:.3_0(1)) ->_L^Omega(2)
980.63/291.47	gen_nil:.3_0(1)
980.63/291.47	
980.63/291.47	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
980.63/291.47	----------------------------------------
980.63/291.47	
980.63/291.47	(16)
980.63/291.47	BOUNDS(1, INF)
980.79/291.53	EOF