1132.82/291.50	WORST_CASE(Omega(n^1), ?)
1132.82/291.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1132.82/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).
1132.82/291.53	
1132.82/291.53	(0) CpxRelTRS
1132.82/291.53	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 169 ms]
1132.82/291.53	(2) CpxRelTRS
1132.82/291.53	(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1132.82/291.53	(4) CpxRelTRS
1132.82/291.53	(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1132.82/291.53	(6) typed CpxTrs
1132.82/291.53	(7) OrderProof [LOWER BOUND(ID), 0 ms]
1132.82/291.53	(8) typed CpxTrs
1132.82/291.53	(9) RewriteLemmaProof [LOWER BOUND(ID), 286 ms]
1132.82/291.53	(10) typed CpxTrs
1132.82/291.53	(11) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
1132.82/291.53	(12) typed CpxTrs
1132.82/291.53	(13) RewriteLemmaProof [LOWER BOUND(ID), 188 ms]
1132.82/291.53	(14) BEST
1132.82/291.53	    (15) proven lower bound
1132.82/291.53	        (16) LowerBoundPropagationProof [FINISHED, 0 ms]
1132.82/291.53	        (17) BOUNDS(n^1, INF)
1132.82/291.53	    (18) typed CpxTrs
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(0)
1132.82/291.53	Obligation:
1132.82/291.53	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	The TRS R consists of the following rules:
1132.82/291.53	
1132.82/291.53	   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
1132.82/291.53	   minsort(Nil) -> Nil
1132.82/291.53	   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
1132.82/291.53	   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
1132.82/291.53	   notEmpty(Cons(x, xs)) -> True
1132.82/291.53	   notEmpty(Nil) -> False
1132.82/291.53	   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))
1132.82/291.53	
1132.82/291.53	The (relative) TRS S consists of the following rules:
1132.82/291.53	
1132.82/291.53	   !EQ(S(x), S(y)) -> !EQ(x, y)
1132.82/291.53	   !EQ(0, S(y)) -> False
1132.82/291.53	   !EQ(S(x), 0) -> False
1132.82/291.53	   !EQ(0, 0) -> True
1132.82/291.53	   <(S(x), S(y)) -> <(x, y)
1132.82/291.53	   <(0, S(y)) -> True
1132.82/291.53	   <(x, 0) -> False
1132.82/291.53	   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
1132.82/291.53	   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
1132.82/291.53	   remove[Ite](True, x', Cons(x, xs)) -> xs
1132.82/291.53	   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')
1132.82/291.53	
1132.82/291.53	Rewrite Strategy: INNERMOST
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
1132.82/291.53	proved termination of relative rules
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(2)
1132.82/291.53	Obligation:
1132.82/291.53	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	The TRS R consists of the following rules:
1132.82/291.53	
1132.82/291.53	   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
1132.82/291.53	   minsort(Nil) -> Nil
1132.82/291.53	   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
1132.82/291.53	   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
1132.82/291.53	   notEmpty(Cons(x, xs)) -> True
1132.82/291.53	   notEmpty(Nil) -> False
1132.82/291.53	   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))
1132.82/291.53	
1132.82/291.53	The (relative) TRS S consists of the following rules:
1132.82/291.53	
1132.82/291.53	   !EQ(S(x), S(y)) -> !EQ(x, y)
1132.82/291.53	   !EQ(0, S(y)) -> False
1132.82/291.53	   !EQ(S(x), 0) -> False
1132.82/291.53	   !EQ(0, 0) -> True
1132.82/291.53	   <(S(x), S(y)) -> <(x, y)
1132.82/291.53	   <(0, S(y)) -> True
1132.82/291.53	   <(x, 0) -> False
1132.82/291.53	   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
1132.82/291.53	   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
1132.82/291.53	   remove[Ite](True, x', Cons(x, xs)) -> xs
1132.82/291.53	   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')
1132.82/291.53	
1132.82/291.53	Rewrite Strategy: INNERMOST
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(3) RenamingProof (BOTH BOUNDS(ID, ID))
1132.82/291.53	Renamed function symbols to avoid clashes with predefined symbol.
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(4)
1132.82/291.53	Obligation:
1132.82/291.53	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	The TRS R consists of the following rules:
1132.82/291.53	
1132.82/291.53	   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
1132.82/291.53	   minsort(Nil) -> Nil
1132.82/291.53	   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
1132.82/291.53	   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
1132.82/291.53	   notEmpty(Cons(x, xs)) -> True
1132.82/291.53	   notEmpty(Nil) -> False
1132.82/291.53	   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))
1132.82/291.53	
1132.82/291.53	The (relative) TRS S consists of the following rules:
1132.82/291.53	
1132.82/291.53	   !EQ(S(x), S(y)) -> !EQ(x, y)
1132.82/291.53	   !EQ(0', S(y)) -> False
1132.82/291.53	   !EQ(S(x), 0') -> False
1132.82/291.53	   !EQ(0', 0') -> True
1132.82/291.53	   <(S(x), S(y)) -> <(x, y)
1132.82/291.53	   <(0', S(y)) -> True
1132.82/291.53	   <(x, 0') -> False
1132.82/291.53	   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
1132.82/291.53	   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
1132.82/291.53	   remove[Ite](True, x', Cons(x, xs)) -> xs
1132.82/291.53	   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')
1132.82/291.53	
1132.82/291.53	Rewrite Strategy: INNERMOST
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1132.82/291.53	Infered types.
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(6)
1132.82/291.53	Obligation:
1132.82/291.53	Innermost TRS:
1132.82/291.53	Rules:
1132.82/291.53	minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
1132.82/291.53	minsort(Nil) -> Nil
1132.82/291.53	appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
1132.82/291.53	appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
1132.82/291.53	notEmpty(Cons(x, xs)) -> True
1132.82/291.53	notEmpty(Nil) -> False
1132.82/291.53	remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))
1132.82/291.53	!EQ(S(x), S(y)) -> !EQ(x, y)
1132.82/291.53	!EQ(0', S(y)) -> False
1132.82/291.53	!EQ(S(x), 0') -> False
1132.82/291.53	!EQ(0', 0') -> True
1132.82/291.53	<(S(x), S(y)) -> <(x, y)
1132.82/291.53	<(0', S(y)) -> True
1132.82/291.53	<(x, 0') -> False
1132.82/291.53	remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
1132.82/291.53	appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
1132.82/291.53	remove[Ite](True, x', Cons(x, xs)) -> xs
1132.82/291.53	appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')
1132.82/291.53	
1132.82/291.53	Types:
1132.82/291.53	minsort :: Cons:Nil -> Cons:Nil
1132.82/291.53	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	appmin :: S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	Nil :: Cons:Nil
1132.82/291.53	appmin[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	< :: S:0' -> S:0' -> True:False
1132.82/291.53	remove :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	notEmpty :: Cons:Nil -> True:False
1132.82/291.53	True :: True:False
1132.82/291.53	False :: True:False
1132.82/291.53	remove[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	!EQ :: S:0' -> S:0' -> True:False
1132.82/291.53	S :: S:0' -> S:0'
1132.82/291.53	0' :: S:0'
1132.82/291.53	hole_Cons:Nil1_0 :: Cons:Nil
1132.82/291.53	hole_S:0'2_0 :: S:0'
1132.82/291.53	hole_True:False3_0 :: True:False
1132.82/291.53	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1132.82/291.53	gen_S:0'5_0 :: Nat -> S:0'
1132.82/291.53	
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(7) OrderProof (LOWER BOUND(ID))
1132.82/291.53	Heuristically decided to analyse the following defined symbols:
1132.82/291.53	minsort, appmin, <, remove, !EQ
1132.82/291.53	
1132.82/291.53	They will be analysed ascendingly in the following order:
1132.82/291.53	minsort = appmin
1132.82/291.53	< < appmin
1132.82/291.53	remove < appmin
1132.82/291.53	!EQ < remove
1132.82/291.53	
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(8)
1132.82/291.53	Obligation:
1132.82/291.53	Innermost TRS:
1132.82/291.53	Rules:
1132.82/291.53	minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
1132.82/291.53	minsort(Nil) -> Nil
1132.82/291.53	appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
1132.82/291.53	appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
1132.82/291.53	notEmpty(Cons(x, xs)) -> True
1132.82/291.53	notEmpty(Nil) -> False
1132.82/291.53	remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))
1132.82/291.53	!EQ(S(x), S(y)) -> !EQ(x, y)
1132.82/291.53	!EQ(0', S(y)) -> False
1132.82/291.53	!EQ(S(x), 0') -> False
1132.82/291.53	!EQ(0', 0') -> True
1132.82/291.53	<(S(x), S(y)) -> <(x, y)
1132.82/291.53	<(0', S(y)) -> True
1132.82/291.53	<(x, 0') -> False
1132.82/291.53	remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
1132.82/291.53	appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
1132.82/291.53	remove[Ite](True, x', Cons(x, xs)) -> xs
1132.82/291.53	appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')
1132.82/291.53	
1132.82/291.53	Types:
1132.82/291.53	minsort :: Cons:Nil -> Cons:Nil
1132.82/291.53	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	appmin :: S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	Nil :: Cons:Nil
1132.82/291.53	appmin[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	< :: S:0' -> S:0' -> True:False
1132.82/291.53	remove :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	notEmpty :: Cons:Nil -> True:False
1132.82/291.53	True :: True:False
1132.82/291.53	False :: True:False
1132.82/291.53	remove[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	!EQ :: S:0' -> S:0' -> True:False
1132.82/291.53	S :: S:0' -> S:0'
1132.82/291.53	0' :: S:0'
1132.82/291.53	hole_Cons:Nil1_0 :: Cons:Nil
1132.82/291.53	hole_S:0'2_0 :: S:0'
1132.82/291.53	hole_True:False3_0 :: True:False
1132.82/291.53	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1132.82/291.53	gen_S:0'5_0 :: Nat -> S:0'
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	Generator Equations:
1132.82/291.53	gen_Cons:Nil4_0(0) <=> Nil
1132.82/291.53	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1132.82/291.53	gen_S:0'5_0(0) <=> 0'
1132.82/291.53	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	The following defined symbols remain to be analysed:
1132.82/291.53	<, minsort, appmin, remove, !EQ
1132.82/291.53	
1132.82/291.53	They will be analysed ascendingly in the following order:
1132.82/291.53	minsort = appmin
1132.82/291.53	< < appmin
1132.82/291.53	remove < appmin
1132.82/291.53	!EQ < remove
1132.82/291.53	
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(9) RewriteLemmaProof (LOWER BOUND(ID))
1132.82/291.53	Proved the following rewrite lemma:
1132.82/291.53	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1132.82/291.53	
1132.82/291.53	Induction Base:
1132.82/291.53	<(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) ->_R^Omega(0)
1132.82/291.53	True
1132.82/291.53	
1132.82/291.53	Induction Step:
1132.82/291.53	<(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) ->_R^Omega(0)
1132.82/291.53	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) ->_IH
1132.82/291.53	True
1132.82/291.53	
1132.82/291.53	We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(10)
1132.82/291.53	Obligation:
1132.82/291.53	Innermost TRS:
1132.82/291.53	Rules:
1132.82/291.53	minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
1132.82/291.53	minsort(Nil) -> Nil
1132.82/291.53	appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
1132.82/291.53	appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
1132.82/291.53	notEmpty(Cons(x, xs)) -> True
1132.82/291.53	notEmpty(Nil) -> False
1132.82/291.53	remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))
1132.82/291.53	!EQ(S(x), S(y)) -> !EQ(x, y)
1132.82/291.53	!EQ(0', S(y)) -> False
1132.82/291.53	!EQ(S(x), 0') -> False
1132.82/291.53	!EQ(0', 0') -> True
1132.82/291.53	<(S(x), S(y)) -> <(x, y)
1132.82/291.53	<(0', S(y)) -> True
1132.82/291.53	<(x, 0') -> False
1132.82/291.53	remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
1132.82/291.53	appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
1132.82/291.53	remove[Ite](True, x', Cons(x, xs)) -> xs
1132.82/291.53	appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')
1132.82/291.53	
1132.82/291.53	Types:
1132.82/291.53	minsort :: Cons:Nil -> Cons:Nil
1132.82/291.53	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	appmin :: S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	Nil :: Cons:Nil
1132.82/291.53	appmin[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	< :: S:0' -> S:0' -> True:False
1132.82/291.53	remove :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	notEmpty :: Cons:Nil -> True:False
1132.82/291.53	True :: True:False
1132.82/291.53	False :: True:False
1132.82/291.53	remove[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	!EQ :: S:0' -> S:0' -> True:False
1132.82/291.53	S :: S:0' -> S:0'
1132.82/291.53	0' :: S:0'
1132.82/291.53	hole_Cons:Nil1_0 :: Cons:Nil
1132.82/291.53	hole_S:0'2_0 :: S:0'
1132.82/291.53	hole_True:False3_0 :: True:False
1132.82/291.53	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1132.82/291.53	gen_S:0'5_0 :: Nat -> S:0'
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	Lemmas:
1132.82/291.53	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	Generator Equations:
1132.82/291.53	gen_Cons:Nil4_0(0) <=> Nil
1132.82/291.53	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1132.82/291.53	gen_S:0'5_0(0) <=> 0'
1132.82/291.53	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	The following defined symbols remain to be analysed:
1132.82/291.53	!EQ, minsort, appmin, remove
1132.82/291.53	
1132.82/291.53	They will be analysed ascendingly in the following order:
1132.82/291.53	minsort = appmin
1132.82/291.53	remove < appmin
1132.82/291.53	!EQ < remove
1132.82/291.53	
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(11) RewriteLemmaProof (LOWER BOUND(ID))
1132.82/291.53	Proved the following rewrite lemma:
1132.82/291.53	!EQ(gen_S:0'5_0(n269_0), gen_S:0'5_0(+(1, n269_0))) -> False, rt in Omega(0)
1132.82/291.53	
1132.82/291.53	Induction Base:
1132.82/291.53	!EQ(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) ->_R^Omega(0)
1132.82/291.53	False
1132.82/291.53	
1132.82/291.53	Induction Step:
1132.82/291.53	!EQ(gen_S:0'5_0(+(n269_0, 1)), gen_S:0'5_0(+(1, +(n269_0, 1)))) ->_R^Omega(0)
1132.82/291.53	!EQ(gen_S:0'5_0(n269_0), gen_S:0'5_0(+(1, n269_0))) ->_IH
1132.82/291.53	False
1132.82/291.53	
1132.82/291.53	We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(12)
1132.82/291.53	Obligation:
1132.82/291.53	Innermost TRS:
1132.82/291.53	Rules:
1132.82/291.53	minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
1132.82/291.53	minsort(Nil) -> Nil
1132.82/291.53	appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
1132.82/291.53	appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
1132.82/291.53	notEmpty(Cons(x, xs)) -> True
1132.82/291.53	notEmpty(Nil) -> False
1132.82/291.53	remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))
1132.82/291.53	!EQ(S(x), S(y)) -> !EQ(x, y)
1132.82/291.53	!EQ(0', S(y)) -> False
1132.82/291.53	!EQ(S(x), 0') -> False
1132.82/291.53	!EQ(0', 0') -> True
1132.82/291.53	<(S(x), S(y)) -> <(x, y)
1132.82/291.53	<(0', S(y)) -> True
1132.82/291.53	<(x, 0') -> False
1132.82/291.53	remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
1132.82/291.53	appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
1132.82/291.53	remove[Ite](True, x', Cons(x, xs)) -> xs
1132.82/291.53	appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')
1132.82/291.53	
1132.82/291.53	Types:
1132.82/291.53	minsort :: Cons:Nil -> Cons:Nil
1132.82/291.53	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	appmin :: S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	Nil :: Cons:Nil
1132.82/291.53	appmin[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	< :: S:0' -> S:0' -> True:False
1132.82/291.53	remove :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	notEmpty :: Cons:Nil -> True:False
1132.82/291.53	True :: True:False
1132.82/291.53	False :: True:False
1132.82/291.53	remove[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	!EQ :: S:0' -> S:0' -> True:False
1132.82/291.53	S :: S:0' -> S:0'
1132.82/291.53	0' :: S:0'
1132.82/291.53	hole_Cons:Nil1_0 :: Cons:Nil
1132.82/291.53	hole_S:0'2_0 :: S:0'
1132.82/291.53	hole_True:False3_0 :: True:False
1132.82/291.53	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1132.82/291.53	gen_S:0'5_0 :: Nat -> S:0'
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	Lemmas:
1132.82/291.53	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1132.82/291.53	!EQ(gen_S:0'5_0(n269_0), gen_S:0'5_0(+(1, n269_0))) -> False, rt in Omega(0)
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	Generator Equations:
1132.82/291.53	gen_Cons:Nil4_0(0) <=> Nil
1132.82/291.53	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1132.82/291.53	gen_S:0'5_0(0) <=> 0'
1132.82/291.53	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	The following defined symbols remain to be analysed:
1132.82/291.53	remove, minsort, appmin
1132.82/291.53	
1132.82/291.53	They will be analysed ascendingly in the following order:
1132.82/291.53	minsort = appmin
1132.82/291.53	remove < appmin
1132.82/291.53	
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(13) RewriteLemmaProof (LOWER BOUND(ID))
1132.82/291.53	Proved the following rewrite lemma:
1132.82/291.53	remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(1, n572_0))) -> *6_0, rt in Omega(n572_0)
1132.82/291.53	
1132.82/291.53	Induction Base:
1132.82/291.53	remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(1, 0)))
1132.82/291.53	
1132.82/291.53	Induction Step:
1132.82/291.53	remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(1, +(n572_0, 1)))) ->_R^Omega(1)
1132.82/291.53	remove[Ite](!EQ(gen_S:0'5_0(1), 0'), gen_S:0'5_0(1), Cons(0', gen_Cons:Nil4_0(+(1, n572_0)))) ->_R^Omega(0)
1132.82/291.53	remove[Ite](False, gen_S:0'5_0(1), Cons(0', gen_Cons:Nil4_0(+(1, n572_0)))) ->_R^Omega(0)
1132.82/291.53	Cons(0', remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(1, n572_0)))) ->_IH
1132.82/291.53	Cons(0', *6_0)
1132.82/291.53	
1132.82/291.53	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(14)
1132.82/291.53	Complex Obligation (BEST)
1132.82/291.53	
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(15)
1132.82/291.53	Obligation:
1132.82/291.53	Proved the lower bound n^1 for the following obligation:
1132.82/291.53	
1132.82/291.53	Innermost TRS:
1132.82/291.53	Rules:
1132.82/291.53	minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
1132.82/291.53	minsort(Nil) -> Nil
1132.82/291.53	appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
1132.82/291.53	appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
1132.82/291.53	notEmpty(Cons(x, xs)) -> True
1132.82/291.53	notEmpty(Nil) -> False
1132.82/291.53	remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))
1132.82/291.53	!EQ(S(x), S(y)) -> !EQ(x, y)
1132.82/291.53	!EQ(0', S(y)) -> False
1132.82/291.53	!EQ(S(x), 0') -> False
1132.82/291.53	!EQ(0', 0') -> True
1132.82/291.53	<(S(x), S(y)) -> <(x, y)
1132.82/291.53	<(0', S(y)) -> True
1132.82/291.53	<(x, 0') -> False
1132.82/291.53	remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
1132.82/291.53	appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
1132.82/291.53	remove[Ite](True, x', Cons(x, xs)) -> xs
1132.82/291.53	appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')
1132.82/291.53	
1132.82/291.53	Types:
1132.82/291.53	minsort :: Cons:Nil -> Cons:Nil
1132.82/291.53	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	appmin :: S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	Nil :: Cons:Nil
1132.82/291.53	appmin[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	< :: S:0' -> S:0' -> True:False
1132.82/291.53	remove :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	notEmpty :: Cons:Nil -> True:False
1132.82/291.53	True :: True:False
1132.82/291.53	False :: True:False
1132.82/291.53	remove[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	!EQ :: S:0' -> S:0' -> True:False
1132.82/291.53	S :: S:0' -> S:0'
1132.82/291.53	0' :: S:0'
1132.82/291.53	hole_Cons:Nil1_0 :: Cons:Nil
1132.82/291.53	hole_S:0'2_0 :: S:0'
1132.82/291.53	hole_True:False3_0 :: True:False
1132.82/291.53	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1132.82/291.53	gen_S:0'5_0 :: Nat -> S:0'
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	Lemmas:
1132.82/291.53	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1132.82/291.53	!EQ(gen_S:0'5_0(n269_0), gen_S:0'5_0(+(1, n269_0))) -> False, rt in Omega(0)
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	Generator Equations:
1132.82/291.53	gen_Cons:Nil4_0(0) <=> Nil
1132.82/291.53	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1132.82/291.53	gen_S:0'5_0(0) <=> 0'
1132.82/291.53	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	The following defined symbols remain to be analysed:
1132.82/291.53	remove, minsort, appmin
1132.82/291.53	
1132.82/291.53	They will be analysed ascendingly in the following order:
1132.82/291.53	minsort = appmin
1132.82/291.53	remove < appmin
1132.82/291.53	
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(16) LowerBoundPropagationProof (FINISHED)
1132.82/291.53	Propagated lower bound.
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(17)
1132.82/291.53	BOUNDS(n^1, INF)
1132.82/291.53	
1132.82/291.53	----------------------------------------
1132.82/291.53	
1132.82/291.53	(18)
1132.82/291.53	Obligation:
1132.82/291.53	Innermost TRS:
1132.82/291.53	Rules:
1132.82/291.53	minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
1132.82/291.53	minsort(Nil) -> Nil
1132.82/291.53	appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
1132.82/291.53	appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
1132.82/291.53	notEmpty(Cons(x, xs)) -> True
1132.82/291.53	notEmpty(Nil) -> False
1132.82/291.53	remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))
1132.82/291.53	!EQ(S(x), S(y)) -> !EQ(x, y)
1132.82/291.53	!EQ(0', S(y)) -> False
1132.82/291.53	!EQ(S(x), 0') -> False
1132.82/291.53	!EQ(0', 0') -> True
1132.82/291.53	<(S(x), S(y)) -> <(x, y)
1132.82/291.53	<(0', S(y)) -> True
1132.82/291.53	<(x, 0') -> False
1132.82/291.53	remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
1132.82/291.53	appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
1132.82/291.53	remove[Ite](True, x', Cons(x, xs)) -> xs
1132.82/291.53	appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')
1132.82/291.53	
1132.82/291.53	Types:
1132.82/291.53	minsort :: Cons:Nil -> Cons:Nil
1132.82/291.53	Cons :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	appmin :: S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	Nil :: Cons:Nil
1132.82/291.53	appmin[Ite][True][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil -> Cons:Nil
1132.82/291.53	< :: S:0' -> S:0' -> True:False
1132.82/291.53	remove :: S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	notEmpty :: Cons:Nil -> True:False
1132.82/291.53	True :: True:False
1132.82/291.53	False :: True:False
1132.82/291.53	remove[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
1132.82/291.53	!EQ :: S:0' -> S:0' -> True:False
1132.82/291.53	S :: S:0' -> S:0'
1132.82/291.53	0' :: S:0'
1132.82/291.53	hole_Cons:Nil1_0 :: Cons:Nil
1132.82/291.53	hole_S:0'2_0 :: S:0'
1132.82/291.53	hole_True:False3_0 :: True:False
1132.82/291.53	gen_Cons:Nil4_0 :: Nat -> Cons:Nil
1132.82/291.53	gen_S:0'5_0 :: Nat -> S:0'
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	Lemmas:
1132.82/291.53	<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
1132.82/291.53	!EQ(gen_S:0'5_0(n269_0), gen_S:0'5_0(+(1, n269_0))) -> False, rt in Omega(0)
1132.82/291.53	remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(1, n572_0))) -> *6_0, rt in Omega(n572_0)
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	Generator Equations:
1132.82/291.53	gen_Cons:Nil4_0(0) <=> Nil
1132.82/291.53	gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
1132.82/291.53	gen_S:0'5_0(0) <=> 0'
1132.82/291.53	gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))
1132.82/291.53	
1132.82/291.53	
1132.82/291.53	The following defined symbols remain to be analysed:
1132.82/291.53	appmin, minsort
1132.82/291.53	
1132.82/291.53	They will be analysed ascendingly in the following order:
1132.82/291.53	minsort = appmin
1133.23/291.59	EOF