6.13/2.32	WORST_CASE(NON_POLY, ?)
6.13/2.33	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.13/2.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.13/2.33	
6.13/2.33	
6.13/2.33	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.13/2.33	
6.13/2.33	(0) CpxTRS
6.13/2.33	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
6.13/2.33	(2) CpxTRS
6.13/2.33	(3) SlicingProof [LOWER BOUND(ID), 0 ms]
6.13/2.33	(4) CpxTRS
6.13/2.33	(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
6.13/2.33	(6) typed CpxTrs
6.13/2.33	(7) OrderProof [LOWER BOUND(ID), 0 ms]
6.13/2.33	(8) typed CpxTrs
6.13/2.33	(9) RewriteLemmaProof [LOWER BOUND(ID), 239 ms]
6.13/2.33	(10) BEST
6.13/2.33	    (11) proven lower bound
6.13/2.33	        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
6.13/2.33	        (13) BOUNDS(n^1, INF)
6.13/2.33	    (14) typed CpxTrs
6.13/2.33	        (15) RewriteLemmaProof [FINISHED, 115 ms]
6.13/2.33	        (16) BOUNDS(EXP, INF)
6.13/2.33	
6.13/2.33	
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(0)
6.13/2.33	Obligation:
6.13/2.33	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.13/2.33	
6.13/2.33	
6.13/2.33	The TRS R consists of the following rules:
6.13/2.33	
6.13/2.33	   empty(nil) -> true
6.13/2.33	   empty(cons(x, l)) -> false
6.13/2.33	   head(cons(x, l)) -> x
6.13/2.33	   tail(nil) -> nil
6.13/2.33	   tail(cons(x, l)) -> l
6.13/2.33	   rev(nil) -> nil
6.13/2.33	   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
6.13/2.33	   last(x, l) -> if(empty(l), x, l)
6.13/2.33	   if(true, x, l) -> x
6.13/2.33	   if(false, x, l) -> last(head(l), tail(l))
6.13/2.33	   rev2(x, nil) -> nil
6.13/2.33	   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.13/2.33	
6.13/2.33	S is empty.
6.13/2.33	Rewrite Strategy: INNERMOST
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(1) RenamingProof (BOTH BOUNDS(ID, ID))
6.13/2.33	Renamed function symbols to avoid clashes with predefined symbol.
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(2)
6.13/2.33	Obligation:
6.13/2.33	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.13/2.33	
6.13/2.33	
6.13/2.33	The TRS R consists of the following rules:
6.13/2.33	
6.13/2.33	   empty(nil) -> true
6.13/2.33	   empty(cons(x, l)) -> false
6.13/2.33	   head(cons(x, l)) -> x
6.13/2.33	   tail(nil) -> nil
6.13/2.33	   tail(cons(x, l)) -> l
6.13/2.33	   rev(nil) -> nil
6.13/2.33	   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
6.13/2.33	   last(x, l) -> if(empty(l), x, l)
6.13/2.33	   if(true, x, l) -> x
6.13/2.33	   if(false, x, l) -> last(head(l), tail(l))
6.13/2.33	   rev2(x, nil) -> nil
6.13/2.33	   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.13/2.33	
6.13/2.33	S is empty.
6.13/2.33	Rewrite Strategy: INNERMOST
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(3) SlicingProof (LOWER BOUND(ID))
6.13/2.33	Sliced the following arguments:
6.13/2.33	rev1/0
6.13/2.33	rev1/1
6.13/2.33	
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(4)
6.13/2.33	Obligation:
6.13/2.33	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.13/2.33	
6.13/2.33	
6.13/2.33	The TRS R consists of the following rules:
6.13/2.33	
6.13/2.33	   empty(nil) -> true
6.13/2.33	   empty(cons(x, l)) -> false
6.13/2.33	   head(cons(x, l)) -> x
6.13/2.33	   tail(nil) -> nil
6.13/2.33	   tail(cons(x, l)) -> l
6.13/2.33	   rev(nil) -> nil
6.13/2.33	   rev(cons(x, l)) -> cons(rev1, rev2(x, l))
6.13/2.33	   last(x, l) -> if(empty(l), x, l)
6.13/2.33	   if(true, x, l) -> x
6.13/2.33	   if(false, x, l) -> last(head(l), tail(l))
6.13/2.33	   rev2(x, nil) -> nil
6.13/2.33	   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.13/2.33	
6.13/2.33	S is empty.
6.13/2.33	Rewrite Strategy: INNERMOST
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
6.13/2.33	Infered types.
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(6)
6.13/2.33	Obligation:
6.13/2.33	Innermost TRS:
6.13/2.33	Rules:
6.13/2.33	empty(nil) -> true
6.13/2.33	empty(cons(x, l)) -> false
6.13/2.33	head(cons(x, l)) -> x
6.13/2.33	tail(nil) -> nil
6.13/2.33	tail(cons(x, l)) -> l
6.13/2.33	rev(nil) -> nil
6.13/2.33	rev(cons(x, l)) -> cons(rev1, rev2(x, l))
6.13/2.33	last(x, l) -> if(empty(l), x, l)
6.13/2.33	if(true, x, l) -> x
6.13/2.33	if(false, x, l) -> last(head(l), tail(l))
6.13/2.33	rev2(x, nil) -> nil
6.13/2.33	rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.13/2.33	
6.13/2.33	Types:
6.13/2.33	empty :: nil:cons -> true:false
6.13/2.33	nil :: nil:cons
6.13/2.33	true :: true:false
6.13/2.33	cons :: rev1 -> nil:cons -> nil:cons
6.13/2.33	false :: true:false
6.13/2.33	head :: nil:cons -> rev1
6.13/2.33	tail :: nil:cons -> nil:cons
6.13/2.33	rev :: nil:cons -> nil:cons
6.13/2.33	rev1 :: rev1
6.13/2.33	rev2 :: rev1 -> nil:cons -> nil:cons
6.13/2.33	last :: rev1 -> nil:cons -> rev1
6.13/2.33	if :: true:false -> rev1 -> nil:cons -> rev1
6.13/2.33	hole_true:false1_0 :: true:false
6.13/2.33	hole_nil:cons2_0 :: nil:cons
6.13/2.33	hole_rev13_0 :: rev1
6.13/2.33	gen_nil:cons4_0 :: Nat -> nil:cons
6.13/2.33	
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(7) OrderProof (LOWER BOUND(ID))
6.13/2.33	Heuristically decided to analyse the following defined symbols:
6.13/2.33	rev, rev2, last
6.13/2.33	
6.13/2.33	They will be analysed ascendingly in the following order:
6.13/2.33	rev = rev2
6.13/2.33	
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(8)
6.13/2.33	Obligation:
6.13/2.33	Innermost TRS:
6.13/2.33	Rules:
6.13/2.33	empty(nil) -> true
6.13/2.33	empty(cons(x, l)) -> false
6.13/2.33	head(cons(x, l)) -> x
6.13/2.33	tail(nil) -> nil
6.13/2.33	tail(cons(x, l)) -> l
6.13/2.33	rev(nil) -> nil
6.13/2.33	rev(cons(x, l)) -> cons(rev1, rev2(x, l))
6.13/2.33	last(x, l) -> if(empty(l), x, l)
6.13/2.33	if(true, x, l) -> x
6.13/2.33	if(false, x, l) -> last(head(l), tail(l))
6.13/2.33	rev2(x, nil) -> nil
6.13/2.33	rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.13/2.33	
6.13/2.33	Types:
6.13/2.33	empty :: nil:cons -> true:false
6.13/2.33	nil :: nil:cons
6.13/2.33	true :: true:false
6.13/2.33	cons :: rev1 -> nil:cons -> nil:cons
6.13/2.33	false :: true:false
6.13/2.33	head :: nil:cons -> rev1
6.13/2.33	tail :: nil:cons -> nil:cons
6.13/2.33	rev :: nil:cons -> nil:cons
6.13/2.33	rev1 :: rev1
6.13/2.33	rev2 :: rev1 -> nil:cons -> nil:cons
6.13/2.33	last :: rev1 -> nil:cons -> rev1
6.13/2.33	if :: true:false -> rev1 -> nil:cons -> rev1
6.13/2.33	hole_true:false1_0 :: true:false
6.13/2.33	hole_nil:cons2_0 :: nil:cons
6.13/2.33	hole_rev13_0 :: rev1
6.13/2.33	gen_nil:cons4_0 :: Nat -> nil:cons
6.13/2.33	
6.13/2.33	
6.13/2.33	Generator Equations:
6.13/2.33	gen_nil:cons4_0(0) <=> nil
6.13/2.33	gen_nil:cons4_0(+(x, 1)) <=> cons(rev1, gen_nil:cons4_0(x))
6.13/2.33	
6.13/2.33	
6.13/2.33	The following defined symbols remain to be analysed:
6.13/2.33	last, rev, rev2
6.13/2.33	
6.13/2.33	They will be analysed ascendingly in the following order:
6.13/2.33	rev = rev2
6.13/2.33	
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(9) RewriteLemmaProof (LOWER BOUND(ID))
6.13/2.33	Proved the following rewrite lemma:
6.13/2.33	last(rev1, gen_nil:cons4_0(n6_0)) -> rev1, rt in Omega(1 + n6_0)
6.13/2.33	
6.13/2.33	Induction Base:
6.13/2.33	last(rev1, gen_nil:cons4_0(0)) ->_R^Omega(1)
6.13/2.33	if(empty(gen_nil:cons4_0(0)), rev1, gen_nil:cons4_0(0)) ->_R^Omega(1)
6.13/2.33	if(true, rev1, gen_nil:cons4_0(0)) ->_R^Omega(1)
6.13/2.33	rev1
6.13/2.33	
6.13/2.33	Induction Step:
6.13/2.33	last(rev1, gen_nil:cons4_0(+(n6_0, 1))) ->_R^Omega(1)
6.13/2.33	if(empty(gen_nil:cons4_0(+(n6_0, 1))), rev1, gen_nil:cons4_0(+(n6_0, 1))) ->_R^Omega(1)
6.13/2.33	if(false, rev1, gen_nil:cons4_0(+(1, n6_0))) ->_R^Omega(1)
6.13/2.33	last(head(gen_nil:cons4_0(+(1, n6_0))), tail(gen_nil:cons4_0(+(1, n6_0)))) ->_R^Omega(1)
6.13/2.33	last(rev1, tail(gen_nil:cons4_0(+(1, n6_0)))) ->_R^Omega(1)
6.13/2.33	last(rev1, gen_nil:cons4_0(n6_0)) ->_IH
6.13/2.33	rev1
6.13/2.33	
6.13/2.33	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(10)
6.13/2.33	Complex Obligation (BEST)
6.13/2.33	
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(11)
6.13/2.33	Obligation:
6.13/2.33	Proved the lower bound n^1 for the following obligation:
6.13/2.33	
6.13/2.33	Innermost TRS:
6.13/2.33	Rules:
6.13/2.33	empty(nil) -> true
6.13/2.33	empty(cons(x, l)) -> false
6.13/2.33	head(cons(x, l)) -> x
6.13/2.33	tail(nil) -> nil
6.13/2.33	tail(cons(x, l)) -> l
6.13/2.33	rev(nil) -> nil
6.13/2.33	rev(cons(x, l)) -> cons(rev1, rev2(x, l))
6.13/2.33	last(x, l) -> if(empty(l), x, l)
6.13/2.33	if(true, x, l) -> x
6.13/2.33	if(false, x, l) -> last(head(l), tail(l))
6.13/2.33	rev2(x, nil) -> nil
6.13/2.33	rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.13/2.33	
6.13/2.33	Types:
6.13/2.33	empty :: nil:cons -> true:false
6.13/2.33	nil :: nil:cons
6.13/2.33	true :: true:false
6.13/2.33	cons :: rev1 -> nil:cons -> nil:cons
6.13/2.33	false :: true:false
6.13/2.33	head :: nil:cons -> rev1
6.13/2.33	tail :: nil:cons -> nil:cons
6.13/2.33	rev :: nil:cons -> nil:cons
6.13/2.33	rev1 :: rev1
6.13/2.33	rev2 :: rev1 -> nil:cons -> nil:cons
6.13/2.33	last :: rev1 -> nil:cons -> rev1
6.13/2.33	if :: true:false -> rev1 -> nil:cons -> rev1
6.13/2.33	hole_true:false1_0 :: true:false
6.13/2.33	hole_nil:cons2_0 :: nil:cons
6.13/2.33	hole_rev13_0 :: rev1
6.13/2.33	gen_nil:cons4_0 :: Nat -> nil:cons
6.13/2.33	
6.13/2.33	
6.13/2.33	Generator Equations:
6.13/2.33	gen_nil:cons4_0(0) <=> nil
6.13/2.33	gen_nil:cons4_0(+(x, 1)) <=> cons(rev1, gen_nil:cons4_0(x))
6.13/2.33	
6.13/2.33	
6.13/2.33	The following defined symbols remain to be analysed:
6.13/2.33	last, rev, rev2
6.13/2.33	
6.13/2.33	They will be analysed ascendingly in the following order:
6.13/2.33	rev = rev2
6.13/2.33	
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(12) LowerBoundPropagationProof (FINISHED)
6.13/2.33	Propagated lower bound.
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(13)
6.13/2.33	BOUNDS(n^1, INF)
6.13/2.33	
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(14)
6.13/2.33	Obligation:
6.13/2.33	Innermost TRS:
6.13/2.33	Rules:
6.13/2.33	empty(nil) -> true
6.13/2.33	empty(cons(x, l)) -> false
6.13/2.33	head(cons(x, l)) -> x
6.13/2.33	tail(nil) -> nil
6.13/2.33	tail(cons(x, l)) -> l
6.13/2.33	rev(nil) -> nil
6.13/2.33	rev(cons(x, l)) -> cons(rev1, rev2(x, l))
6.13/2.33	last(x, l) -> if(empty(l), x, l)
6.13/2.33	if(true, x, l) -> x
6.13/2.33	if(false, x, l) -> last(head(l), tail(l))
6.13/2.33	rev2(x, nil) -> nil
6.13/2.33	rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.13/2.33	
6.13/2.33	Types:
6.13/2.33	empty :: nil:cons -> true:false
6.13/2.33	nil :: nil:cons
6.13/2.33	true :: true:false
6.13/2.33	cons :: rev1 -> nil:cons -> nil:cons
6.13/2.33	false :: true:false
6.13/2.33	head :: nil:cons -> rev1
6.13/2.33	tail :: nil:cons -> nil:cons
6.13/2.33	rev :: nil:cons -> nil:cons
6.13/2.33	rev1 :: rev1
6.13/2.33	rev2 :: rev1 -> nil:cons -> nil:cons
6.13/2.33	last :: rev1 -> nil:cons -> rev1
6.13/2.33	if :: true:false -> rev1 -> nil:cons -> rev1
6.13/2.33	hole_true:false1_0 :: true:false
6.13/2.33	hole_nil:cons2_0 :: nil:cons
6.13/2.33	hole_rev13_0 :: rev1
6.13/2.33	gen_nil:cons4_0 :: Nat -> nil:cons
6.13/2.33	
6.13/2.33	
6.13/2.33	Lemmas:
6.13/2.33	last(rev1, gen_nil:cons4_0(n6_0)) -> rev1, rt in Omega(1 + n6_0)
6.13/2.33	
6.13/2.33	
6.13/2.33	Generator Equations:
6.13/2.33	gen_nil:cons4_0(0) <=> nil
6.13/2.33	gen_nil:cons4_0(+(x, 1)) <=> cons(rev1, gen_nil:cons4_0(x))
6.13/2.33	
6.13/2.33	
6.13/2.33	The following defined symbols remain to be analysed:
6.13/2.33	rev2, rev
6.13/2.33	
6.13/2.33	They will be analysed ascendingly in the following order:
6.13/2.33	rev = rev2
6.13/2.33	
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(15) RewriteLemmaProof (FINISHED)
6.13/2.33	Proved the following rewrite lemma:
6.13/2.33	rev2(rev1, gen_nil:cons4_0(n341_0)) -> gen_nil:cons4_0(n341_0), rt in Omega(EXP)
6.13/2.33	
6.13/2.33	Induction Base:
6.13/2.33	rev2(rev1, gen_nil:cons4_0(0)) ->_R^Omega(1)
6.13/2.33	nil
6.13/2.33	
6.13/2.33	Induction Step:
6.13/2.33	rev2(rev1, gen_nil:cons4_0(+(n341_0, 1))) ->_R^Omega(1)
6.13/2.33	rev(cons(rev1, rev2(rev1, gen_nil:cons4_0(n341_0)))) ->_IH
6.13/2.33	rev(cons(rev1, gen_nil:cons4_0(c342_0))) ->_R^Omega(1)
6.13/2.33	cons(rev1, rev2(rev1, gen_nil:cons4_0(n341_0))) ->_IH
6.13/2.33	cons(rev1, gen_nil:cons4_0(c342_0))
6.13/2.33	
6.13/2.33	We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP
6.13/2.33	----------------------------------------
6.13/2.33	
6.13/2.33	(16)
6.13/2.33	BOUNDS(EXP, INF)
6.13/2.37	EOF