1116.46/291.55	WORST_CASE(Omega(n^1), ?)
1129.25/294.78	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1129.25/294.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).
1129.25/294.78	
1129.25/294.78	(0) CpxRelTRS
1129.25/294.78	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 235 ms]
1129.25/294.78	(2) CpxRelTRS
1129.25/294.78	(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1129.25/294.78	(4) CpxRelTRS
1129.25/294.78	(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1129.25/294.78	(6) typed CpxTrs
1129.25/294.78	(7) OrderProof [LOWER BOUND(ID), 0 ms]
1129.25/294.78	(8) typed CpxTrs
1129.25/294.78	(9) RewriteLemmaProof [LOWER BOUND(ID), 290 ms]
1129.25/294.78	(10) typed CpxTrs
1129.25/294.78	(11) RewriteLemmaProof [LOWER BOUND(ID), 48 ms]
1129.25/294.78	(12) BEST
1129.25/294.78	    (13) proven lower bound
1129.25/294.78	        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
1129.25/294.78	        (15) BOUNDS(n^1, INF)
1129.25/294.78	    (16) typed CpxTrs
1129.25/294.78	        (17) RewriteLemmaProof [LOWER BOUND(ID), 48 ms]
1129.25/294.78	        (18) typed CpxTrs
1129.25/294.78	        (19) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
1129.25/294.78	        (20) typed CpxTrs
1129.25/294.78	        (21) RewriteLemmaProof [LOWER BOUND(ID), 2 ms]
1129.25/294.78	        (22) typed CpxTrs
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(0)
1129.25/294.78	Obligation:
1129.25/294.78	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The TRS R consists of the following rules:
1129.25/294.78	
1129.25/294.78	   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	   quicksort(Nil) -> Nil
1129.25/294.78	   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	   partLt(x, Nil) -> Nil
1129.25/294.78	   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	   partGt(x, Nil) -> Nil
1129.25/294.78	   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	   app(Nil, ys) -> ys
1129.25/294.78	   notEmpty(Cons(x, xs)) -> True
1129.25/294.78	   notEmpty(Nil) -> False
1129.25/294.78	   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	   goal(xs) -> quicksort(xs)
1129.25/294.78	
1129.25/294.78	The (relative) TRS S consists of the following rules:
1129.25/294.78	
1129.25/294.78	   <(S(x), S(y)) -> <(x, y)
1129.25/294.78	   <(0, S(y)) -> True
1129.25/294.78	   <(x, 0) -> False
1129.25/294.78	   >(S(x), S(y)) -> >(x, y)
1129.25/294.78	   >(0, y) -> False
1129.25/294.78	   >(S(x), 0) -> True
1129.25/294.78	   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Rewrite Strategy: INNERMOST
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
1129.25/294.78	proved termination of relative rules
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(2)
1129.25/294.78	Obligation:
1129.25/294.78	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The TRS R consists of the following rules:
1129.25/294.78	
1129.25/294.78	   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	   quicksort(Nil) -> Nil
1129.25/294.78	   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	   partLt(x, Nil) -> Nil
1129.25/294.78	   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	   partGt(x, Nil) -> Nil
1129.25/294.78	   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	   app(Nil, ys) -> ys
1129.25/294.78	   notEmpty(Cons(x, xs)) -> True
1129.25/294.78	   notEmpty(Nil) -> False
1129.25/294.78	   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	   goal(xs) -> quicksort(xs)
1129.25/294.78	
1129.25/294.78	The (relative) TRS S consists of the following rules:
1129.25/294.78	
1129.25/294.78	   <(S(x), S(y)) -> <(x, y)
1129.25/294.78	   <(0, S(y)) -> True
1129.25/294.78	   <(x, 0) -> False
1129.25/294.78	   >(S(x), S(y)) -> >(x, y)
1129.25/294.78	   >(0, y) -> False
1129.25/294.78	   >(S(x), 0) -> True
1129.25/294.78	   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Rewrite Strategy: INNERMOST
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(3) RenamingProof (BOTH BOUNDS(ID, ID))
1129.25/294.78	Renamed function symbols to avoid clashes with predefined symbol.
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(4)
1129.25/294.78	Obligation:
1129.25/294.78	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The TRS R consists of the following rules:
1129.25/294.78	
1129.25/294.78	   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	   quicksort(Nil) -> Nil
1129.25/294.78	   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	   partLt(x, Nil) -> Nil
1129.25/294.78	   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	   partGt(x, Nil) -> Nil
1129.25/294.78	   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	   app(Nil, ys) -> ys
1129.25/294.78	   notEmpty(Cons(x, xs)) -> True
1129.25/294.78	   notEmpty(Nil) -> False
1129.25/294.78	   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	   goal(xs) -> quicksort(xs)
1129.25/294.78	
1129.25/294.78	The (relative) TRS S consists of the following rules:
1129.25/294.78	
1129.25/294.78	   <(S(x), S(y)) -> <(x, y)
1129.25/294.78	   <(0', S(y)) -> True
1129.25/294.78	   <(x, 0') -> False
1129.25/294.78	   >(S(x), S(y)) -> >(x, y)
1129.25/294.78	   >(0', y) -> False
1129.25/294.78	   >(S(x), 0') -> True
1129.25/294.78	   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Rewrite Strategy: INNERMOST
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1129.25/294.78	Infered types.
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(6)
1129.25/294.78	Obligation:
1129.25/294.78	Innermost TRS:
1129.25/294.78	Rules:
1129.25/294.78	quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	quicksort(Nil) -> Nil
1129.25/294.78	partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	partLt(x, Nil) -> Nil
1129.25/294.78	partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	partGt(x, Nil) -> Nil
1129.25/294.78	app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	app(Nil, ys) -> ys
1129.25/294.78	notEmpty(Cons(x, xs)) -> True
1129.25/294.78	notEmpty(Nil) -> False
1129.25/294.78	part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	goal(xs) -> quicksort(xs)
1129.25/294.78	<(S(x), S(y)) -> <(x, y)
1129.25/294.78	<(0', S(y)) -> True
1129.25/294.78	<(x, 0') -> False
1129.25/294.78	>(S(x), S(y)) -> >(x, y)
1129.25/294.78	>(0', y) -> False
1129.25/294.78	>(S(x), 0') -> True
1129.25/294.78	partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Types:
1129.25/294.78	quicksort :: Cons:Nil -> Cons:Nil
1129.25/294.78	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	part :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	Nil :: Cons:Nil
1129.25/294.78	partLt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partLt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	< :: S:0' -> S:0' -> True:False
1129.25/294.78	partGt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partGt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	> :: S:0' -> S:0' -> True:False
1129.25/294.78	app :: Cons:Nil -> Cons:Nil -> Cons:Nil
1129.25/294.78	notEmpty :: Cons:Nil -> True:False
1129.25/294.78	True :: True:False
1129.25/294.78	False :: True:False
1129.25/294.78	goal :: Cons:Nil -> Cons:Nil
1129.25/294.78	S :: S:0' -> S:0'
1129.25/294.78	0' :: S:0'
1129.25/294.78	hole_Cons:Nil1_0 :: Cons:Nil
1129.25/294.78	hole_S:0'2_0 :: S:0'
1129.25/294.78	hole_True:False3_0 :: True:False
1129.25/294.78	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1129.25/294.78	gen_S:0'5_0 :: Nat -> S:0'
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(7) OrderProof (LOWER BOUND(ID))
1129.25/294.78	Heuristically decided to analyse the following defined symbols:
1129.25/294.78	quicksort, partLt, <, partGt, >, app
1129.25/294.78	
1129.25/294.78	They will be analysed ascendingly in the following order:
1129.25/294.78	partLt < quicksort
1129.25/294.78	partGt < quicksort
1129.25/294.78	app < quicksort
1129.25/294.78	< < partLt
1129.25/294.78	> < partGt
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(8)
1129.25/294.78	Obligation:
1129.25/294.78	Innermost TRS:
1129.25/294.78	Rules:
1129.25/294.78	quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	quicksort(Nil) -> Nil
1129.25/294.78	partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	partLt(x, Nil) -> Nil
1129.25/294.78	partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	partGt(x, Nil) -> Nil
1129.25/294.78	app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	app(Nil, ys) -> ys
1129.25/294.78	notEmpty(Cons(x, xs)) -> True
1129.25/294.78	notEmpty(Nil) -> False
1129.25/294.78	part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	goal(xs) -> quicksort(xs)
1129.25/294.78	<(S(x), S(y)) -> <(x, y)
1129.25/294.78	<(0', S(y)) -> True
1129.25/294.78	<(x, 0') -> False
1129.25/294.78	>(S(x), S(y)) -> >(x, y)
1129.25/294.78	>(0', y) -> False
1129.25/294.78	>(S(x), 0') -> True
1129.25/294.78	partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Types:
1129.25/294.78	quicksort :: Cons:Nil -> Cons:Nil
1129.25/294.78	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	part :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	Nil :: Cons:Nil
1129.25/294.78	partLt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partLt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	< :: S:0' -> S:0' -> True:False
1129.25/294.78	partGt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partGt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	> :: S:0' -> S:0' -> True:False
1129.25/294.78	app :: Cons:Nil -> Cons:Nil -> Cons:Nil
1129.25/294.78	notEmpty :: Cons:Nil -> True:False
1129.25/294.78	True :: True:False
1129.25/294.78	False :: True:False
1129.25/294.78	goal :: Cons:Nil -> Cons:Nil
1129.25/294.78	S :: S:0' -> S:0'
1129.25/294.78	0' :: S:0'
1129.25/294.78	hole_Cons:Nil1_0 :: Cons:Nil
1129.25/294.78	hole_S:0'2_0 :: S:0'
1129.25/294.78	hole_True:False3_0 :: True:False
1129.25/294.78	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1129.25/294.78	gen_S:0'5_0 :: Nat -> S:0'
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Generator Equations:
1129.25/294.78	gen_Cons:Nil4_0(0) <=> Nil
1129.25/294.78	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1129.25/294.78	gen_S:0'5_0(0) <=> 0'
1129.25/294.78	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The following defined symbols remain to be analysed:
1129.25/294.78	<, quicksort, partLt, partGt, >, app
1129.25/294.78	
1129.25/294.78	They will be analysed ascendingly in the following order:
1129.25/294.78	partLt < quicksort
1129.25/294.78	partGt < quicksort
1129.25/294.78	app < quicksort
1129.25/294.78	< < partLt
1129.25/294.78	> < partGt
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(9) RewriteLemmaProof (LOWER BOUND(ID))
1129.25/294.78	Proved the following rewrite lemma:
1129.25/294.78	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1129.25/294.78	
1129.25/294.78	Induction Base:
1129.25/294.78	<(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) ->_R^Omega(0)
1129.25/294.78	True
1129.25/294.78	
1129.25/294.78	Induction Step:
1129.25/294.78	<(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) ->_R^Omega(0)
1129.25/294.78	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) ->_IH
1129.25/294.78	True
1129.25/294.78	
1129.25/294.78	We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(10)
1129.25/294.78	Obligation:
1129.25/294.78	Innermost TRS:
1129.25/294.78	Rules:
1129.25/294.78	quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	quicksort(Nil) -> Nil
1129.25/294.78	partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	partLt(x, Nil) -> Nil
1129.25/294.78	partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	partGt(x, Nil) -> Nil
1129.25/294.78	app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	app(Nil, ys) -> ys
1129.25/294.78	notEmpty(Cons(x, xs)) -> True
1129.25/294.78	notEmpty(Nil) -> False
1129.25/294.78	part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	goal(xs) -> quicksort(xs)
1129.25/294.78	<(S(x), S(y)) -> <(x, y)
1129.25/294.78	<(0', S(y)) -> True
1129.25/294.78	<(x, 0') -> False
1129.25/294.78	>(S(x), S(y)) -> >(x, y)
1129.25/294.78	>(0', y) -> False
1129.25/294.78	>(S(x), 0') -> True
1129.25/294.78	partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Types:
1129.25/294.78	quicksort :: Cons:Nil -> Cons:Nil
1129.25/294.78	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	part :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	Nil :: Cons:Nil
1129.25/294.78	partLt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partLt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	< :: S:0' -> S:0' -> True:False
1129.25/294.78	partGt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partGt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	> :: S:0' -> S:0' -> True:False
1129.25/294.78	app :: Cons:Nil -> Cons:Nil -> Cons:Nil
1129.25/294.78	notEmpty :: Cons:Nil -> True:False
1129.25/294.78	True :: True:False
1129.25/294.78	False :: True:False
1129.25/294.78	goal :: Cons:Nil -> Cons:Nil
1129.25/294.78	S :: S:0' -> S:0'
1129.25/294.78	0' :: S:0'
1129.25/294.78	hole_Cons:Nil1_0 :: Cons:Nil
1129.25/294.78	hole_S:0'2_0 :: S:0'
1129.25/294.78	hole_True:False3_0 :: True:False
1129.25/294.78	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1129.25/294.78	gen_S:0'5_0 :: Nat -> S:0'
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Lemmas:
1129.25/294.78	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Generator Equations:
1129.25/294.78	gen_Cons:Nil4_0(0) <=> Nil
1129.25/294.78	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1129.25/294.78	gen_S:0'5_0(0) <=> 0'
1129.25/294.78	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The following defined symbols remain to be analysed:
1129.25/294.78	partLt, quicksort, partGt, >, app
1129.25/294.78	
1129.25/294.78	They will be analysed ascendingly in the following order:
1129.25/294.78	partLt < quicksort
1129.25/294.78	partGt < quicksort
1129.25/294.78	app < quicksort
1129.25/294.78	> < partGt
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(11) RewriteLemmaProof (LOWER BOUND(ID))
1129.25/294.78	Proved the following rewrite lemma:
1129.25/294.78	partLt(gen_S:0'5_0(1), gen_Cons:Nil4_0(n299_0)) -> gen_Cons:Nil4_0(n299_0), rt in Omega(1 + n299_0)
1129.25/294.78	
1129.25/294.78	Induction Base:
1129.25/294.78	partLt(gen_S:0'5_0(1), gen_Cons:Nil4_0(0)) ->_R^Omega(1)
1129.25/294.78	Nil
1129.25/294.78	
1129.25/294.78	Induction Step:
1129.25/294.78	partLt(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(n299_0, 1))) ->_R^Omega(1)
1129.25/294.78	partLt[Ite][True][Ite](<(0', gen_S:0'5_0(1)), gen_S:0'5_0(1), Cons(0', gen_Cons:Nil4_0(n299_0))) ->_L^Omega(0)
1129.25/294.78	partLt[Ite][True][Ite](True, gen_S:0'5_0(1), Cons(0', gen_Cons:Nil4_0(n299_0))) ->_R^Omega(0)
1129.25/294.78	Cons(0', partLt(gen_S:0'5_0(1), gen_Cons:Nil4_0(n299_0))) ->_IH
1129.25/294.78	Cons(0', gen_Cons:Nil4_0(c300_0))
1129.25/294.78	
1129.25/294.78	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(12)
1129.25/294.78	Complex Obligation (BEST)
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(13)
1129.25/294.78	Obligation:
1129.25/294.78	Proved the lower bound n^1 for the following obligation:
1129.25/294.78	
1129.25/294.78	Innermost TRS:
1129.25/294.78	Rules:
1129.25/294.78	quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	quicksort(Nil) -> Nil
1129.25/294.78	partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	partLt(x, Nil) -> Nil
1129.25/294.78	partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	partGt(x, Nil) -> Nil
1129.25/294.78	app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	app(Nil, ys) -> ys
1129.25/294.78	notEmpty(Cons(x, xs)) -> True
1129.25/294.78	notEmpty(Nil) -> False
1129.25/294.78	part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	goal(xs) -> quicksort(xs)
1129.25/294.78	<(S(x), S(y)) -> <(x, y)
1129.25/294.78	<(0', S(y)) -> True
1129.25/294.78	<(x, 0') -> False
1129.25/294.78	>(S(x), S(y)) -> >(x, y)
1129.25/294.78	>(0', y) -> False
1129.25/294.78	>(S(x), 0') -> True
1129.25/294.78	partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Types:
1129.25/294.78	quicksort :: Cons:Nil -> Cons:Nil
1129.25/294.78	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	part :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	Nil :: Cons:Nil
1129.25/294.78	partLt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partLt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	< :: S:0' -> S:0' -> True:False
1129.25/294.78	partGt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partGt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	> :: S:0' -> S:0' -> True:False
1129.25/294.78	app :: Cons:Nil -> Cons:Nil -> Cons:Nil
1129.25/294.78	notEmpty :: Cons:Nil -> True:False
1129.25/294.78	True :: True:False
1129.25/294.78	False :: True:False
1129.25/294.78	goal :: Cons:Nil -> Cons:Nil
1129.25/294.78	S :: S:0' -> S:0'
1129.25/294.78	0' :: S:0'
1129.25/294.78	hole_Cons:Nil1_0 :: Cons:Nil
1129.25/294.78	hole_S:0'2_0 :: S:0'
1129.25/294.78	hole_True:False3_0 :: True:False
1129.25/294.78	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1129.25/294.78	gen_S:0'5_0 :: Nat -> S:0'
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Lemmas:
1129.25/294.78	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Generator Equations:
1129.25/294.78	gen_Cons:Nil4_0(0) <=> Nil
1129.25/294.78	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1129.25/294.78	gen_S:0'5_0(0) <=> 0'
1129.25/294.78	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The following defined symbols remain to be analysed:
1129.25/294.78	partLt, quicksort, partGt, >, app
1129.25/294.78	
1129.25/294.78	They will be analysed ascendingly in the following order:
1129.25/294.78	partLt < quicksort
1129.25/294.78	partGt < quicksort
1129.25/294.78	app < quicksort
1129.25/294.78	> < partGt
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(14) LowerBoundPropagationProof (FINISHED)
1129.25/294.78	Propagated lower bound.
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(15)
1129.25/294.78	BOUNDS(n^1, INF)
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(16)
1129.25/294.78	Obligation:
1129.25/294.78	Innermost TRS:
1129.25/294.78	Rules:
1129.25/294.78	quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	quicksort(Nil) -> Nil
1129.25/294.78	partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	partLt(x, Nil) -> Nil
1129.25/294.78	partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	partGt(x, Nil) -> Nil
1129.25/294.78	app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	app(Nil, ys) -> ys
1129.25/294.78	notEmpty(Cons(x, xs)) -> True
1129.25/294.78	notEmpty(Nil) -> False
1129.25/294.78	part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	goal(xs) -> quicksort(xs)
1129.25/294.78	<(S(x), S(y)) -> <(x, y)
1129.25/294.78	<(0', S(y)) -> True
1129.25/294.78	<(x, 0') -> False
1129.25/294.78	>(S(x), S(y)) -> >(x, y)
1129.25/294.78	>(0', y) -> False
1129.25/294.78	>(S(x), 0') -> True
1129.25/294.78	partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Types:
1129.25/294.78	quicksort :: Cons:Nil -> Cons:Nil
1129.25/294.78	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	part :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	Nil :: Cons:Nil
1129.25/294.78	partLt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partLt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	< :: S:0' -> S:0' -> True:False
1129.25/294.78	partGt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partGt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	> :: S:0' -> S:0' -> True:False
1129.25/294.78	app :: Cons:Nil -> Cons:Nil -> Cons:Nil
1129.25/294.78	notEmpty :: Cons:Nil -> True:False
1129.25/294.78	True :: True:False
1129.25/294.78	False :: True:False
1129.25/294.78	goal :: Cons:Nil -> Cons:Nil
1129.25/294.78	S :: S:0' -> S:0'
1129.25/294.78	0' :: S:0'
1129.25/294.78	hole_Cons:Nil1_0 :: Cons:Nil
1129.25/294.78	hole_S:0'2_0 :: S:0'
1129.25/294.78	hole_True:False3_0 :: True:False
1129.25/294.78	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1129.25/294.78	gen_S:0'5_0 :: Nat -> S:0'
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Lemmas:
1129.25/294.78	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1129.25/294.78	partLt(gen_S:0'5_0(1), gen_Cons:Nil4_0(n299_0)) -> gen_Cons:Nil4_0(n299_0), rt in Omega(1 + n299_0)
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Generator Equations:
1129.25/294.78	gen_Cons:Nil4_0(0) <=> Nil
1129.25/294.78	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1129.25/294.78	gen_S:0'5_0(0) <=> 0'
1129.25/294.78	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The following defined symbols remain to be analysed:
1129.25/294.78	>, quicksort, partGt, app
1129.25/294.78	
1129.25/294.78	They will be analysed ascendingly in the following order:
1129.25/294.78	partGt < quicksort
1129.25/294.78	app < quicksort
1129.25/294.78	> < partGt
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(17) RewriteLemmaProof (LOWER BOUND(ID))
1129.25/294.78	Proved the following rewrite lemma:
1129.25/294.78	>(gen_S:0'5_0(n1026_0), gen_S:0'5_0(n1026_0)) -> False, rt in Omega(0)
1129.25/294.78	
1129.25/294.78	Induction Base:
1129.25/294.78	>(gen_S:0'5_0(0), gen_S:0'5_0(0)) ->_R^Omega(0)
1129.25/294.78	False
1129.25/294.78	
1129.25/294.78	Induction Step:
1129.25/294.78	>(gen_S:0'5_0(+(n1026_0, 1)), gen_S:0'5_0(+(n1026_0, 1))) ->_R^Omega(0)
1129.25/294.78	>(gen_S:0'5_0(n1026_0), gen_S:0'5_0(n1026_0)) ->_IH
1129.25/294.78	False
1129.25/294.78	
1129.25/294.78	We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(18)
1129.25/294.78	Obligation:
1129.25/294.78	Innermost TRS:
1129.25/294.78	Rules:
1129.25/294.78	quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	quicksort(Nil) -> Nil
1129.25/294.78	partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	partLt(x, Nil) -> Nil
1129.25/294.78	partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	partGt(x, Nil) -> Nil
1129.25/294.78	app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	app(Nil, ys) -> ys
1129.25/294.78	notEmpty(Cons(x, xs)) -> True
1129.25/294.78	notEmpty(Nil) -> False
1129.25/294.78	part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	goal(xs) -> quicksort(xs)
1129.25/294.78	<(S(x), S(y)) -> <(x, y)
1129.25/294.78	<(0', S(y)) -> True
1129.25/294.78	<(x, 0') -> False
1129.25/294.78	>(S(x), S(y)) -> >(x, y)
1129.25/294.78	>(0', y) -> False
1129.25/294.78	>(S(x), 0') -> True
1129.25/294.78	partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Types:
1129.25/294.78	quicksort :: Cons:Nil -> Cons:Nil
1129.25/294.78	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	part :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	Nil :: Cons:Nil
1129.25/294.78	partLt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partLt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	< :: S:0' -> S:0' -> True:False
1129.25/294.78	partGt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partGt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	> :: S:0' -> S:0' -> True:False
1129.25/294.78	app :: Cons:Nil -> Cons:Nil -> Cons:Nil
1129.25/294.78	notEmpty :: Cons:Nil -> True:False
1129.25/294.78	True :: True:False
1129.25/294.78	False :: True:False
1129.25/294.78	goal :: Cons:Nil -> Cons:Nil
1129.25/294.78	S :: S:0' -> S:0'
1129.25/294.78	0' :: S:0'
1129.25/294.78	hole_Cons:Nil1_0 :: Cons:Nil
1129.25/294.78	hole_S:0'2_0 :: S:0'
1129.25/294.78	hole_True:False3_0 :: True:False
1129.25/294.78	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1129.25/294.78	gen_S:0'5_0 :: Nat -> S:0'
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Lemmas:
1129.25/294.78	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1129.25/294.78	partLt(gen_S:0'5_0(1), gen_Cons:Nil4_0(n299_0)) -> gen_Cons:Nil4_0(n299_0), rt in Omega(1 + n299_0)
1129.25/294.78	>(gen_S:0'5_0(n1026_0), gen_S:0'5_0(n1026_0)) -> False, rt in Omega(0)
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Generator Equations:
1129.25/294.78	gen_Cons:Nil4_0(0) <=> Nil
1129.25/294.78	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1129.25/294.78	gen_S:0'5_0(0) <=> 0'
1129.25/294.78	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The following defined symbols remain to be analysed:
1129.25/294.78	partGt, quicksort, app
1129.25/294.78	
1129.25/294.78	They will be analysed ascendingly in the following order:
1129.25/294.78	partGt < quicksort
1129.25/294.78	app < quicksort
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(19) RewriteLemmaProof (LOWER BOUND(ID))
1129.25/294.78	Proved the following rewrite lemma:
1129.25/294.78	partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1331_0)) -> gen_Cons:Nil4_0(0), rt in Omega(1 + n1331_0)
1129.25/294.78	
1129.25/294.78	Induction Base:
1129.25/294.78	partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(0)) ->_R^Omega(1)
1129.25/294.78	Nil
1129.25/294.78	
1129.25/294.78	Induction Step:
1129.25/294.78	partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(+(n1331_0, 1))) ->_R^Omega(1)
1129.25/294.78	partGt[Ite][True][Ite](>(0', gen_S:0'5_0(0)), gen_S:0'5_0(0), Cons(0', gen_Cons:Nil4_0(n1331_0))) ->_L^Omega(0)
1129.25/294.78	partGt[Ite][True][Ite](False, gen_S:0'5_0(0), Cons(0', gen_Cons:Nil4_0(n1331_0))) ->_R^Omega(0)
1129.25/294.78	partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1331_0)) ->_IH
1129.25/294.78	gen_Cons:Nil4_0(0)
1129.25/294.78	
1129.25/294.78	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(20)
1129.25/294.78	Obligation:
1129.25/294.78	Innermost TRS:
1129.25/294.78	Rules:
1129.25/294.78	quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	quicksort(Nil) -> Nil
1129.25/294.78	partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	partLt(x, Nil) -> Nil
1129.25/294.78	partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	partGt(x, Nil) -> Nil
1129.25/294.78	app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	app(Nil, ys) -> ys
1129.25/294.78	notEmpty(Cons(x, xs)) -> True
1129.25/294.78	notEmpty(Nil) -> False
1129.25/294.78	part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	goal(xs) -> quicksort(xs)
1129.25/294.78	<(S(x), S(y)) -> <(x, y)
1129.25/294.78	<(0', S(y)) -> True
1129.25/294.78	<(x, 0') -> False
1129.25/294.78	>(S(x), S(y)) -> >(x, y)
1129.25/294.78	>(0', y) -> False
1129.25/294.78	>(S(x), 0') -> True
1129.25/294.78	partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Types:
1129.25/294.78	quicksort :: Cons:Nil -> Cons:Nil
1129.25/294.78	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	part :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	Nil :: Cons:Nil
1129.25/294.78	partLt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partLt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	< :: S:0' -> S:0' -> True:False
1129.25/294.78	partGt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partGt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	> :: S:0' -> S:0' -> True:False
1129.25/294.78	app :: Cons:Nil -> Cons:Nil -> Cons:Nil
1129.25/294.78	notEmpty :: Cons:Nil -> True:False
1129.25/294.78	True :: True:False
1129.25/294.78	False :: True:False
1129.25/294.78	goal :: Cons:Nil -> Cons:Nil
1129.25/294.78	S :: S:0' -> S:0'
1129.25/294.78	0' :: S:0'
1129.25/294.78	hole_Cons:Nil1_0 :: Cons:Nil
1129.25/294.78	hole_S:0'2_0 :: S:0'
1129.25/294.78	hole_True:False3_0 :: True:False
1129.25/294.78	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1129.25/294.78	gen_S:0'5_0 :: Nat -> S:0'
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Lemmas:
1129.25/294.78	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1129.25/294.78	partLt(gen_S:0'5_0(1), gen_Cons:Nil4_0(n299_0)) -> gen_Cons:Nil4_0(n299_0), rt in Omega(1 + n299_0)
1129.25/294.78	>(gen_S:0'5_0(n1026_0), gen_S:0'5_0(n1026_0)) -> False, rt in Omega(0)
1129.25/294.78	partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1331_0)) -> gen_Cons:Nil4_0(0), rt in Omega(1 + n1331_0)
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Generator Equations:
1129.25/294.78	gen_Cons:Nil4_0(0) <=> Nil
1129.25/294.78	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1129.25/294.78	gen_S:0'5_0(0) <=> 0'
1129.25/294.78	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The following defined symbols remain to be analysed:
1129.25/294.78	app, quicksort
1129.25/294.78	
1129.25/294.78	They will be analysed ascendingly in the following order:
1129.25/294.78	app < quicksort
1129.25/294.78	
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(21) RewriteLemmaProof (LOWER BOUND(ID))
1129.25/294.78	Proved the following rewrite lemma:
1129.25/294.78	app(gen_Cons:Nil4_0(n2095_0), gen_Cons:Nil4_0(b)) -> gen_Cons:Nil4_0(+(n2095_0, b)), rt in Omega(1 + n2095_0)
1129.25/294.78	
1129.25/294.78	Induction Base:
1129.25/294.78	app(gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(b)) ->_R^Omega(1)
1129.25/294.78	gen_Cons:Nil4_0(b)
1129.25/294.78	
1129.25/294.78	Induction Step:
1129.25/294.78	app(gen_Cons:Nil4_0(+(n2095_0, 1)), gen_Cons:Nil4_0(b)) ->_R^Omega(1)
1129.25/294.78	Cons(0', app(gen_Cons:Nil4_0(n2095_0), gen_Cons:Nil4_0(b))) ->_IH
1129.25/294.78	Cons(0', gen_Cons:Nil4_0(+(b, c2096_0)))
1129.25/294.78	
1129.25/294.78	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1129.25/294.78	----------------------------------------
1129.25/294.78	
1129.25/294.78	(22)
1129.25/294.78	Obligation:
1129.25/294.78	Innermost TRS:
1129.25/294.78	Rules:
1129.25/294.78	quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
1129.25/294.78	quicksort(Cons(x, Nil)) -> Cons(x, Nil)
1129.25/294.78	quicksort(Nil) -> Nil
1129.25/294.78	partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
1129.25/294.78	partLt(x, Nil) -> Nil
1129.25/294.78	partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
1129.25/294.78	partGt(x, Nil) -> Nil
1129.25/294.78	app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
1129.25/294.78	app(Nil, ys) -> ys
1129.25/294.78	notEmpty(Cons(x, xs)) -> True
1129.25/294.78	notEmpty(Nil) -> False
1129.25/294.78	part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
1129.25/294.78	goal(xs) -> quicksort(xs)
1129.25/294.78	<(S(x), S(y)) -> <(x, y)
1129.25/294.78	<(0', S(y)) -> True
1129.25/294.78	<(x, 0') -> False
1129.25/294.78	>(S(x), S(y)) -> >(x, y)
1129.25/294.78	>(0', y) -> False
1129.25/294.78	>(S(x), 0') -> True
1129.25/294.78	partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
1129.25/294.78	partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
1129.25/294.78	partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
1129.25/294.78	partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)
1129.25/294.78	
1129.25/294.78	Types:
1129.25/294.78	quicksort :: Cons:Nil -> Cons:Nil
1129.25/294.78	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	part :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	Nil :: Cons:Nil
1129.25/294.78	partLt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partLt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	< :: S:0' -> S:0' -> True:False
1129.25/294.78	partGt :: S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	partGt[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1129.25/294.78	> :: S:0' -> S:0' -> True:False
1129.25/294.78	app :: Cons:Nil -> Cons:Nil -> Cons:Nil
1129.25/294.78	notEmpty :: Cons:Nil -> True:False
1129.25/294.78	True :: True:False
1129.25/294.78	False :: True:False
1129.25/294.78	goal :: Cons:Nil -> Cons:Nil
1129.25/294.78	S :: S:0' -> S:0'
1129.25/294.78	0' :: S:0'
1129.25/294.78	hole_Cons:Nil1_0 :: Cons:Nil
1129.25/294.78	hole_S:0'2_0 :: S:0'
1129.25/294.78	hole_True:False3_0 :: True:False
1129.25/294.78	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1129.25/294.78	gen_S:0'5_0 :: Nat -> S:0'
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Lemmas:
1129.25/294.78	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1129.25/294.78	partLt(gen_S:0'5_0(1), gen_Cons:Nil4_0(n299_0)) -> gen_Cons:Nil4_0(n299_0), rt in Omega(1 + n299_0)
1129.25/294.78	>(gen_S:0'5_0(n1026_0), gen_S:0'5_0(n1026_0)) -> False, rt in Omega(0)
1129.25/294.78	partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1331_0)) -> gen_Cons:Nil4_0(0), rt in Omega(1 + n1331_0)
1129.25/294.78	app(gen_Cons:Nil4_0(n2095_0), gen_Cons:Nil4_0(b)) -> gen_Cons:Nil4_0(+(n2095_0, b)), rt in Omega(1 + n2095_0)
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	Generator Equations:
1129.25/294.78	gen_Cons:Nil4_0(0) <=> Nil
1129.25/294.78	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1129.25/294.78	gen_S:0'5_0(0) <=> 0'
1129.25/294.78	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1129.25/294.78	
1129.25/294.78	
1129.25/294.78	The following defined symbols remain to be analysed:
1129.25/294.78	quicksort
1129.64/294.87	EOF