309.26/291.60	WORST_CASE(Omega(n^2), ?)
309.26/291.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
309.26/291.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
309.26/291.61	
309.26/291.61	
309.26/291.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
309.26/291.61	
309.26/291.61	(0) CpxTRS
309.26/291.61	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
309.26/291.61	(2) CpxTRS
309.26/291.61	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
309.26/291.61	(4) typed CpxTrs
309.26/291.61	(5) OrderProof [LOWER BOUND(ID), 0 ms]
309.26/291.61	(6) typed CpxTrs
309.26/291.61	(7) RewriteLemmaProof [LOWER BOUND(ID), 314 ms]
309.26/291.61	(8) BEST
309.26/291.61	    (9) proven lower bound
309.26/291.61	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
309.26/291.61	        (11) BOUNDS(n^1, INF)
309.26/291.61	    (12) typed CpxTrs
309.26/291.61	        (13) RewriteLemmaProof [LOWER BOUND(ID), 11 ms]
309.26/291.61	        (14) typed CpxTrs
309.26/291.61	        (15) RewriteLemmaProof [LOWER BOUND(ID), 29 ms]
309.26/291.61	        (16) typed CpxTrs
309.26/291.61	        (17) RewriteLemmaProof [LOWER BOUND(ID), 62 ms]
309.26/291.61	        (18) typed CpxTrs
309.26/291.61	        (19) RewriteLemmaProof [LOWER BOUND(ID), 37 ms]
309.26/291.61	        (20) proven lower bound
309.26/291.61	        (21) LowerBoundPropagationProof [FINISHED, 0 ms]
309.26/291.61	        (22) BOUNDS(n^2, INF)
309.26/291.61	
309.26/291.61	
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(0)
309.26/291.61	Obligation:
309.26/291.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
309.26/291.61	
309.26/291.61	
309.26/291.61	The TRS R consists of the following rules:
309.26/291.61	
309.26/291.61	   max(nil) -> 0
309.26/291.61	   max(cons(x, nil)) -> x
309.26/291.61	   max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs)
309.26/291.61	   if1(true, x, y, xs) -> max(cons(x, xs))
309.26/291.61	   if1(false, x, y, xs) -> max(cons(y, xs))
309.26/291.61	   del(x, nil) -> nil
309.26/291.61	   del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
309.26/291.61	   if2(true, x, y, xs) -> xs
309.26/291.61	   if2(false, x, y, xs) -> cons(y, del(x, xs))
309.26/291.61	   eq(0, 0) -> true
309.26/291.61	   eq(0, s(y)) -> false
309.26/291.61	   eq(s(x), 0) -> false
309.26/291.61	   eq(s(x), s(y)) -> eq(x, y)
309.26/291.61	   sort(nil) -> nil
309.26/291.61	   sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(h(del(max(cons(x, xs)), cons(x, xs)))))
309.26/291.61	   ge(0, 0) -> true
309.26/291.61	   ge(s(x), 0) -> true
309.26/291.61	   ge(0, s(x)) -> false
309.26/291.61	   ge(s(x), s(y)) -> ge(x, y)
309.26/291.61	   h(nil) -> nil
309.26/291.61	   h(cons(x, xs)) -> cons(x, h(xs))
309.26/291.61	
309.26/291.61	S is empty.
309.26/291.61	Rewrite Strategy: FULL
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(1) RenamingProof (BOTH BOUNDS(ID, ID))
309.26/291.61	Renamed function symbols to avoid clashes with predefined symbol.
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(2)
309.26/291.61	Obligation:
309.26/291.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
309.26/291.61	
309.26/291.61	
309.26/291.61	The TRS R consists of the following rules:
309.26/291.61	
309.26/291.61	   max(nil) -> 0'
309.26/291.61	   max(cons(x, nil)) -> x
309.26/291.61	   max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs)
309.26/291.61	   if1(true, x, y, xs) -> max(cons(x, xs))
309.26/291.61	   if1(false, x, y, xs) -> max(cons(y, xs))
309.26/291.61	   del(x, nil) -> nil
309.26/291.61	   del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
309.26/291.61	   if2(true, x, y, xs) -> xs
309.26/291.61	   if2(false, x, y, xs) -> cons(y, del(x, xs))
309.26/291.61	   eq(0', 0') -> true
309.26/291.61	   eq(0', s(y)) -> false
309.26/291.61	   eq(s(x), 0') -> false
309.26/291.61	   eq(s(x), s(y)) -> eq(x, y)
309.26/291.61	   sort(nil) -> nil
309.26/291.61	   sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(h(del(max(cons(x, xs)), cons(x, xs)))))
309.26/291.61	   ge(0', 0') -> true
309.26/291.61	   ge(s(x), 0') -> true
309.26/291.61	   ge(0', s(x)) -> false
309.26/291.61	   ge(s(x), s(y)) -> ge(x, y)
309.26/291.61	   h(nil) -> nil
309.26/291.61	   h(cons(x, xs)) -> cons(x, h(xs))
309.26/291.61	
309.26/291.61	S is empty.
309.26/291.61	Rewrite Strategy: FULL
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
309.26/291.61	Infered types.
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(4)
309.26/291.61	Obligation:
309.26/291.61	TRS:
309.26/291.61	Rules:
309.26/291.61	max(nil) -> 0'
309.26/291.61	max(cons(x, nil)) -> x
309.26/291.61	max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs)
309.26/291.61	if1(true, x, y, xs) -> max(cons(x, xs))
309.26/291.61	if1(false, x, y, xs) -> max(cons(y, xs))
309.26/291.61	del(x, nil) -> nil
309.26/291.61	del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
309.26/291.61	if2(true, x, y, xs) -> xs
309.26/291.61	if2(false, x, y, xs) -> cons(y, del(x, xs))
309.26/291.61	eq(0', 0') -> true
309.26/291.61	eq(0', s(y)) -> false
309.26/291.61	eq(s(x), 0') -> false
309.26/291.61	eq(s(x), s(y)) -> eq(x, y)
309.26/291.61	sort(nil) -> nil
309.26/291.61	sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(h(del(max(cons(x, xs)), cons(x, xs)))))
309.26/291.61	ge(0', 0') -> true
309.26/291.61	ge(s(x), 0') -> true
309.26/291.61	ge(0', s(x)) -> false
309.26/291.61	ge(s(x), s(y)) -> ge(x, y)
309.26/291.61	h(nil) -> nil
309.26/291.61	h(cons(x, xs)) -> cons(x, h(xs))
309.26/291.61	
309.26/291.61	Types:
309.26/291.61	max :: nil:cons -> 0':s
309.26/291.61	nil :: nil:cons
309.26/291.61	0' :: 0':s
309.26/291.61	cons :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
309.26/291.61	ge :: 0':s -> 0':s -> true:false
309.26/291.61	true :: true:false
309.26/291.61	false :: true:false
309.26/291.61	del :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
309.26/291.61	eq :: 0':s -> 0':s -> true:false
309.26/291.61	s :: 0':s -> 0':s
309.26/291.61	sort :: nil:cons -> nil:cons
309.26/291.61	h :: nil:cons -> nil:cons
309.26/291.61	hole_0':s1_0 :: 0':s
309.26/291.61	hole_nil:cons2_0 :: nil:cons
309.26/291.61	hole_true:false3_0 :: true:false
309.26/291.61	gen_0':s4_0 :: Nat -> 0':s
309.26/291.61	gen_nil:cons5_0 :: Nat -> nil:cons
309.26/291.61	
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(5) OrderProof (LOWER BOUND(ID))
309.26/291.61	Heuristically decided to analyse the following defined symbols:
309.26/291.61	max, ge, del, eq, sort, h
309.26/291.61	
309.26/291.61	They will be analysed ascendingly in the following order:
309.26/291.61	ge < max
309.26/291.61	max < sort
309.26/291.61	eq < del
309.26/291.61	del < sort
309.26/291.61	h < sort
309.26/291.61	
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(6)
309.26/291.61	Obligation:
309.26/291.61	TRS:
309.26/291.61	Rules:
309.26/291.61	max(nil) -> 0'
309.26/291.61	max(cons(x, nil)) -> x
309.26/291.61	max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs)
309.26/291.61	if1(true, x, y, xs) -> max(cons(x, xs))
309.26/291.61	if1(false, x, y, xs) -> max(cons(y, xs))
309.26/291.61	del(x, nil) -> nil
309.26/291.61	del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
309.26/291.61	if2(true, x, y, xs) -> xs
309.26/291.61	if2(false, x, y, xs) -> cons(y, del(x, xs))
309.26/291.61	eq(0', 0') -> true
309.26/291.61	eq(0', s(y)) -> false
309.26/291.61	eq(s(x), 0') -> false
309.26/291.61	eq(s(x), s(y)) -> eq(x, y)
309.26/291.61	sort(nil) -> nil
309.26/291.61	sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(h(del(max(cons(x, xs)), cons(x, xs)))))
309.26/291.61	ge(0', 0') -> true
309.26/291.61	ge(s(x), 0') -> true
309.26/291.61	ge(0', s(x)) -> false
309.26/291.61	ge(s(x), s(y)) -> ge(x, y)
309.26/291.61	h(nil) -> nil
309.26/291.61	h(cons(x, xs)) -> cons(x, h(xs))
309.26/291.61	
309.26/291.61	Types:
309.26/291.61	max :: nil:cons -> 0':s
309.26/291.61	nil :: nil:cons
309.26/291.61	0' :: 0':s
309.26/291.61	cons :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
309.26/291.61	ge :: 0':s -> 0':s -> true:false
309.26/291.61	true :: true:false
309.26/291.61	false :: true:false
309.26/291.61	del :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
309.26/291.61	eq :: 0':s -> 0':s -> true:false
309.26/291.61	s :: 0':s -> 0':s
309.26/291.61	sort :: nil:cons -> nil:cons
309.26/291.61	h :: nil:cons -> nil:cons
309.26/291.61	hole_0':s1_0 :: 0':s
309.26/291.61	hole_nil:cons2_0 :: nil:cons
309.26/291.61	hole_true:false3_0 :: true:false
309.26/291.61	gen_0':s4_0 :: Nat -> 0':s
309.26/291.61	gen_nil:cons5_0 :: Nat -> nil:cons
309.26/291.61	
309.26/291.61	
309.26/291.61	Generator Equations:
309.26/291.61	gen_0':s4_0(0) <=> 0'
309.26/291.61	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
309.26/291.61	gen_nil:cons5_0(0) <=> nil
309.26/291.61	gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))
309.26/291.61	
309.26/291.61	
309.26/291.61	The following defined symbols remain to be analysed:
309.26/291.61	ge, max, del, eq, sort, h
309.26/291.61	
309.26/291.61	They will be analysed ascendingly in the following order:
309.26/291.61	ge < max
309.26/291.61	max < sort
309.26/291.61	eq < del
309.26/291.61	del < sort
309.26/291.61	h < sort
309.26/291.61	
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(7) RewriteLemmaProof (LOWER BOUND(ID))
309.26/291.61	Proved the following rewrite lemma:
309.26/291.61	ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
309.26/291.61	
309.26/291.61	Induction Base:
309.26/291.61	ge(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
309.26/291.61	true
309.26/291.61	
309.26/291.61	Induction Step:
309.26/291.61	ge(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1)
309.26/291.61	ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH
309.26/291.61	true
309.26/291.61	
309.26/291.61	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(8)
309.26/291.61	Complex Obligation (BEST)
309.26/291.61	
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(9)
309.26/291.61	Obligation:
309.26/291.61	Proved the lower bound n^1 for the following obligation:
309.26/291.61	
309.26/291.61	TRS:
309.26/291.61	Rules:
309.26/291.61	max(nil) -> 0'
309.26/291.61	max(cons(x, nil)) -> x
309.26/291.61	max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs)
309.26/291.61	if1(true, x, y, xs) -> max(cons(x, xs))
309.26/291.61	if1(false, x, y, xs) -> max(cons(y, xs))
309.26/291.61	del(x, nil) -> nil
309.26/291.61	del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
309.26/291.61	if2(true, x, y, xs) -> xs
309.26/291.61	if2(false, x, y, xs) -> cons(y, del(x, xs))
309.26/291.61	eq(0', 0') -> true
309.26/291.61	eq(0', s(y)) -> false
309.26/291.61	eq(s(x), 0') -> false
309.26/291.61	eq(s(x), s(y)) -> eq(x, y)
309.26/291.61	sort(nil) -> nil
309.26/291.61	sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(h(del(max(cons(x, xs)), cons(x, xs)))))
309.26/291.61	ge(0', 0') -> true
309.26/291.61	ge(s(x), 0') -> true
309.26/291.61	ge(0', s(x)) -> false
309.26/291.61	ge(s(x), s(y)) -> ge(x, y)
309.26/291.61	h(nil) -> nil
309.26/291.61	h(cons(x, xs)) -> cons(x, h(xs))
309.26/291.61	
309.26/291.61	Types:
309.26/291.61	max :: nil:cons -> 0':s
309.26/291.61	nil :: nil:cons
309.26/291.61	0' :: 0':s
309.26/291.61	cons :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
309.26/291.61	ge :: 0':s -> 0':s -> true:false
309.26/291.61	true :: true:false
309.26/291.61	false :: true:false
309.26/291.61	del :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
309.26/291.61	eq :: 0':s -> 0':s -> true:false
309.26/291.61	s :: 0':s -> 0':s
309.26/291.61	sort :: nil:cons -> nil:cons
309.26/291.61	h :: nil:cons -> nil:cons
309.26/291.61	hole_0':s1_0 :: 0':s
309.26/291.61	hole_nil:cons2_0 :: nil:cons
309.26/291.61	hole_true:false3_0 :: true:false
309.26/291.61	gen_0':s4_0 :: Nat -> 0':s
309.26/291.61	gen_nil:cons5_0 :: Nat -> nil:cons
309.26/291.61	
309.26/291.61	
309.26/291.61	Generator Equations:
309.26/291.61	gen_0':s4_0(0) <=> 0'
309.26/291.61	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
309.26/291.61	gen_nil:cons5_0(0) <=> nil
309.26/291.61	gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))
309.26/291.61	
309.26/291.61	
309.26/291.61	The following defined symbols remain to be analysed:
309.26/291.61	ge, max, del, eq, sort, h
309.26/291.61	
309.26/291.61	They will be analysed ascendingly in the following order:
309.26/291.61	ge < max
309.26/291.61	max < sort
309.26/291.61	eq < del
309.26/291.61	del < sort
309.26/291.61	h < sort
309.26/291.61	
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(10) LowerBoundPropagationProof (FINISHED)
309.26/291.61	Propagated lower bound.
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(11)
309.26/291.61	BOUNDS(n^1, INF)
309.26/291.61	
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(12)
309.26/291.61	Obligation:
309.26/291.61	TRS:
309.26/291.61	Rules:
309.26/291.61	max(nil) -> 0'
309.26/291.61	max(cons(x, nil)) -> x
309.26/291.61	max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs)
309.26/291.61	if1(true, x, y, xs) -> max(cons(x, xs))
309.26/291.61	if1(false, x, y, xs) -> max(cons(y, xs))
309.26/291.61	del(x, nil) -> nil
309.26/291.61	del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
309.26/291.61	if2(true, x, y, xs) -> xs
309.26/291.61	if2(false, x, y, xs) -> cons(y, del(x, xs))
309.26/291.61	eq(0', 0') -> true
309.26/291.61	eq(0', s(y)) -> false
309.26/291.61	eq(s(x), 0') -> false
309.26/291.61	eq(s(x), s(y)) -> eq(x, y)
309.26/291.61	sort(nil) -> nil
309.26/291.61	sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(h(del(max(cons(x, xs)), cons(x, xs)))))
309.26/291.61	ge(0', 0') -> true
309.26/291.61	ge(s(x), 0') -> true
309.26/291.61	ge(0', s(x)) -> false
309.26/291.61	ge(s(x), s(y)) -> ge(x, y)
309.26/291.61	h(nil) -> nil
309.26/291.61	h(cons(x, xs)) -> cons(x, h(xs))
309.26/291.61	
309.26/291.61	Types:
309.26/291.61	max :: nil:cons -> 0':s
309.26/291.61	nil :: nil:cons
309.26/291.61	0' :: 0':s
309.26/291.61	cons :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
309.26/291.61	ge :: 0':s -> 0':s -> true:false
309.26/291.61	true :: true:false
309.26/291.61	false :: true:false
309.26/291.61	del :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
309.26/291.61	eq :: 0':s -> 0':s -> true:false
309.26/291.61	s :: 0':s -> 0':s
309.26/291.61	sort :: nil:cons -> nil:cons
309.26/291.61	h :: nil:cons -> nil:cons
309.26/291.61	hole_0':s1_0 :: 0':s
309.26/291.61	hole_nil:cons2_0 :: nil:cons
309.26/291.61	hole_true:false3_0 :: true:false
309.26/291.61	gen_0':s4_0 :: Nat -> 0':s
309.26/291.61	gen_nil:cons5_0 :: Nat -> nil:cons
309.26/291.61	
309.26/291.61	
309.26/291.61	Lemmas:
309.26/291.61	ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
309.26/291.61	
309.26/291.61	
309.26/291.61	Generator Equations:
309.26/291.61	gen_0':s4_0(0) <=> 0'
309.26/291.61	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
309.26/291.61	gen_nil:cons5_0(0) <=> nil
309.26/291.61	gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))
309.26/291.61	
309.26/291.61	
309.26/291.61	The following defined symbols remain to be analysed:
309.26/291.61	max, del, eq, sort, h
309.26/291.61	
309.26/291.61	They will be analysed ascendingly in the following order:
309.26/291.61	max < sort
309.26/291.61	eq < del
309.26/291.61	del < sort
309.26/291.61	h < sort
309.26/291.61	
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(13) RewriteLemmaProof (LOWER BOUND(ID))
309.26/291.61	Proved the following rewrite lemma:
309.26/291.61	max(gen_nil:cons5_0(+(1, n536_0))) -> gen_0':s4_0(0), rt in Omega(1 + n536_0)
309.26/291.61	
309.26/291.61	Induction Base:
309.26/291.61	max(gen_nil:cons5_0(+(1, 0))) ->_R^Omega(1)
309.26/291.61	0'
309.26/291.61	
309.26/291.61	Induction Step:
309.26/291.61	max(gen_nil:cons5_0(+(1, +(n536_0, 1)))) ->_R^Omega(1)
309.26/291.61	if1(ge(0', 0'), 0', 0', gen_nil:cons5_0(n536_0)) ->_L^Omega(1)
309.26/291.61	if1(true, 0', 0', gen_nil:cons5_0(n536_0)) ->_R^Omega(1)
309.26/291.61	max(cons(0', gen_nil:cons5_0(n536_0))) ->_IH
309.26/291.61	gen_0':s4_0(0)
309.26/291.61	
309.26/291.61	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(14)
309.26/291.61	Obligation:
309.26/291.61	TRS:
309.26/291.61	Rules:
309.26/291.61	max(nil) -> 0'
309.26/291.61	max(cons(x, nil)) -> x
309.26/291.61	max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs)
309.26/291.61	if1(true, x, y, xs) -> max(cons(x, xs))
309.26/291.61	if1(false, x, y, xs) -> max(cons(y, xs))
309.26/291.61	del(x, nil) -> nil
309.26/291.61	del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
309.26/291.61	if2(true, x, y, xs) -> xs
309.26/291.61	if2(false, x, y, xs) -> cons(y, del(x, xs))
309.26/291.61	eq(0', 0') -> true
309.26/291.61	eq(0', s(y)) -> false
309.26/291.61	eq(s(x), 0') -> false
309.26/291.61	eq(s(x), s(y)) -> eq(x, y)
309.26/291.61	sort(nil) -> nil
309.26/291.61	sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(h(del(max(cons(x, xs)), cons(x, xs)))))
309.26/291.61	ge(0', 0') -> true
309.26/291.61	ge(s(x), 0') -> true
309.26/291.61	ge(0', s(x)) -> false
309.26/291.61	ge(s(x), s(y)) -> ge(x, y)
309.26/291.61	h(nil) -> nil
309.26/291.61	h(cons(x, xs)) -> cons(x, h(xs))
309.26/291.61	
309.26/291.61	Types:
309.26/291.61	max :: nil:cons -> 0':s
309.26/291.61	nil :: nil:cons
309.26/291.61	0' :: 0':s
309.26/291.61	cons :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
309.26/291.61	ge :: 0':s -> 0':s -> true:false
309.26/291.61	true :: true:false
309.26/291.61	false :: true:false
309.26/291.61	del :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
309.26/291.61	eq :: 0':s -> 0':s -> true:false
309.26/291.61	s :: 0':s -> 0':s
309.26/291.61	sort :: nil:cons -> nil:cons
309.26/291.61	h :: nil:cons -> nil:cons
309.26/291.61	hole_0':s1_0 :: 0':s
309.26/291.61	hole_nil:cons2_0 :: nil:cons
309.26/291.61	hole_true:false3_0 :: true:false
309.26/291.61	gen_0':s4_0 :: Nat -> 0':s
309.26/291.61	gen_nil:cons5_0 :: Nat -> nil:cons
309.26/291.61	
309.26/291.61	
309.26/291.61	Lemmas:
309.26/291.61	ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
309.26/291.61	max(gen_nil:cons5_0(+(1, n536_0))) -> gen_0':s4_0(0), rt in Omega(1 + n536_0)
309.26/291.61	
309.26/291.61	
309.26/291.61	Generator Equations:
309.26/291.61	gen_0':s4_0(0) <=> 0'
309.26/291.61	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
309.26/291.61	gen_nil:cons5_0(0) <=> nil
309.26/291.61	gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))
309.26/291.61	
309.26/291.61	
309.26/291.61	The following defined symbols remain to be analysed:
309.26/291.61	eq, del, sort, h
309.26/291.61	
309.26/291.61	They will be analysed ascendingly in the following order:
309.26/291.61	eq < del
309.26/291.61	del < sort
309.26/291.61	h < sort
309.26/291.61	
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(15) RewriteLemmaProof (LOWER BOUND(ID))
309.26/291.61	Proved the following rewrite lemma:
309.26/291.61	eq(gen_0':s4_0(n985_0), gen_0':s4_0(n985_0)) -> true, rt in Omega(1 + n985_0)
309.26/291.61	
309.26/291.61	Induction Base:
309.26/291.61	eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
309.26/291.61	true
309.26/291.61	
309.26/291.61	Induction Step:
309.26/291.61	eq(gen_0':s4_0(+(n985_0, 1)), gen_0':s4_0(+(n985_0, 1))) ->_R^Omega(1)
309.26/291.61	eq(gen_0':s4_0(n985_0), gen_0':s4_0(n985_0)) ->_IH
309.26/291.61	true
309.26/291.61	
309.26/291.61	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(16)
309.26/291.61	Obligation:
309.26/291.61	TRS:
309.26/291.61	Rules:
309.26/291.61	max(nil) -> 0'
309.26/291.61	max(cons(x, nil)) -> x
309.26/291.61	max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs)
309.26/291.61	if1(true, x, y, xs) -> max(cons(x, xs))
309.26/291.61	if1(false, x, y, xs) -> max(cons(y, xs))
309.26/291.61	del(x, nil) -> nil
309.26/291.61	del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
309.26/291.61	if2(true, x, y, xs) -> xs
309.26/291.61	if2(false, x, y, xs) -> cons(y, del(x, xs))
309.26/291.61	eq(0', 0') -> true
309.26/291.61	eq(0', s(y)) -> false
309.26/291.61	eq(s(x), 0') -> false
309.26/291.61	eq(s(x), s(y)) -> eq(x, y)
309.26/291.61	sort(nil) -> nil
309.26/291.61	sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(h(del(max(cons(x, xs)), cons(x, xs)))))
309.26/291.61	ge(0', 0') -> true
309.26/291.61	ge(s(x), 0') -> true
309.26/291.61	ge(0', s(x)) -> false
309.26/291.61	ge(s(x), s(y)) -> ge(x, y)
309.26/291.61	h(nil) -> nil
309.26/291.61	h(cons(x, xs)) -> cons(x, h(xs))
309.26/291.61	
309.26/291.61	Types:
309.26/291.61	max :: nil:cons -> 0':s
309.26/291.61	nil :: nil:cons
309.26/291.61	0' :: 0':s
309.26/291.61	cons :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
309.26/291.61	ge :: 0':s -> 0':s -> true:false
309.26/291.61	true :: true:false
309.26/291.61	false :: true:false
309.26/291.61	del :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
309.26/291.61	eq :: 0':s -> 0':s -> true:false
309.26/291.61	s :: 0':s -> 0':s
309.26/291.61	sort :: nil:cons -> nil:cons
309.26/291.61	h :: nil:cons -> nil:cons
309.26/291.61	hole_0':s1_0 :: 0':s
309.26/291.61	hole_nil:cons2_0 :: nil:cons
309.26/291.61	hole_true:false3_0 :: true:false
309.26/291.61	gen_0':s4_0 :: Nat -> 0':s
309.26/291.61	gen_nil:cons5_0 :: Nat -> nil:cons
309.26/291.61	
309.26/291.61	
309.26/291.61	Lemmas:
309.26/291.61	ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
309.26/291.61	max(gen_nil:cons5_0(+(1, n536_0))) -> gen_0':s4_0(0), rt in Omega(1 + n536_0)
309.26/291.61	eq(gen_0':s4_0(n985_0), gen_0':s4_0(n985_0)) -> true, rt in Omega(1 + n985_0)
309.26/291.61	
309.26/291.61	
309.26/291.61	Generator Equations:
309.26/291.61	gen_0':s4_0(0) <=> 0'
309.26/291.61	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
309.26/291.61	gen_nil:cons5_0(0) <=> nil
309.26/291.61	gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))
309.26/291.61	
309.26/291.61	
309.26/291.61	The following defined symbols remain to be analysed:
309.26/291.61	del, sort, h
309.26/291.61	
309.26/291.61	They will be analysed ascendingly in the following order:
309.26/291.61	del < sort
309.26/291.61	h < sort
309.26/291.61	
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(17) RewriteLemmaProof (LOWER BOUND(ID))
309.26/291.61	Proved the following rewrite lemma:
309.26/291.61	h(gen_nil:cons5_0(n1688_0)) -> gen_nil:cons5_0(n1688_0), rt in Omega(1 + n1688_0)
309.26/291.61	
309.26/291.61	Induction Base:
309.26/291.61	h(gen_nil:cons5_0(0)) ->_R^Omega(1)
309.26/291.61	nil
309.26/291.61	
309.26/291.61	Induction Step:
309.26/291.61	h(gen_nil:cons5_0(+(n1688_0, 1))) ->_R^Omega(1)
309.26/291.61	cons(0', h(gen_nil:cons5_0(n1688_0))) ->_IH
309.26/291.61	cons(0', gen_nil:cons5_0(c1689_0))
309.26/291.61	
309.26/291.61	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(18)
309.26/291.61	Obligation:
309.26/291.61	TRS:
309.26/291.61	Rules:
309.26/291.61	max(nil) -> 0'
309.26/291.61	max(cons(x, nil)) -> x
309.26/291.61	max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs)
309.26/291.61	if1(true, x, y, xs) -> max(cons(x, xs))
309.26/291.61	if1(false, x, y, xs) -> max(cons(y, xs))
309.26/291.61	del(x, nil) -> nil
309.26/291.61	del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
309.26/291.61	if2(true, x, y, xs) -> xs
309.26/291.61	if2(false, x, y, xs) -> cons(y, del(x, xs))
309.26/291.61	eq(0', 0') -> true
309.26/291.61	eq(0', s(y)) -> false
309.26/291.61	eq(s(x), 0') -> false
309.26/291.61	eq(s(x), s(y)) -> eq(x, y)
309.26/291.61	sort(nil) -> nil
309.26/291.61	sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(h(del(max(cons(x, xs)), cons(x, xs)))))
309.26/291.61	ge(0', 0') -> true
309.26/291.61	ge(s(x), 0') -> true
309.26/291.61	ge(0', s(x)) -> false
309.26/291.61	ge(s(x), s(y)) -> ge(x, y)
309.26/291.61	h(nil) -> nil
309.26/291.61	h(cons(x, xs)) -> cons(x, h(xs))
309.26/291.61	
309.26/291.61	Types:
309.26/291.61	max :: nil:cons -> 0':s
309.26/291.61	nil :: nil:cons
309.26/291.61	0' :: 0':s
309.26/291.61	cons :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
309.26/291.61	ge :: 0':s -> 0':s -> true:false
309.26/291.61	true :: true:false
309.26/291.61	false :: true:false
309.26/291.61	del :: 0':s -> nil:cons -> nil:cons
309.26/291.61	if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
309.26/291.61	eq :: 0':s -> 0':s -> true:false
309.26/291.61	s :: 0':s -> 0':s
309.26/291.61	sort :: nil:cons -> nil:cons
309.26/291.61	h :: nil:cons -> nil:cons
309.26/291.61	hole_0':s1_0 :: 0':s
309.26/291.61	hole_nil:cons2_0 :: nil:cons
309.26/291.61	hole_true:false3_0 :: true:false
309.26/291.61	gen_0':s4_0 :: Nat -> 0':s
309.26/291.61	gen_nil:cons5_0 :: Nat -> nil:cons
309.26/291.61	
309.26/291.61	
309.26/291.61	Lemmas:
309.26/291.61	ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
309.26/291.61	max(gen_nil:cons5_0(+(1, n536_0))) -> gen_0':s4_0(0), rt in Omega(1 + n536_0)
309.26/291.61	eq(gen_0':s4_0(n985_0), gen_0':s4_0(n985_0)) -> true, rt in Omega(1 + n985_0)
309.26/291.61	h(gen_nil:cons5_0(n1688_0)) -> gen_nil:cons5_0(n1688_0), rt in Omega(1 + n1688_0)
309.26/291.61	
309.26/291.61	
309.26/291.61	Generator Equations:
309.26/291.61	gen_0':s4_0(0) <=> 0'
309.26/291.61	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
309.26/291.61	gen_nil:cons5_0(0) <=> nil
309.26/291.61	gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))
309.26/291.61	
309.26/291.61	
309.26/291.61	The following defined symbols remain to be analysed:
309.26/291.61	sort
309.26/291.61	----------------------------------------
309.26/291.61	
309.26/291.61	(19) RewriteLemmaProof (LOWER BOUND(ID))
309.26/291.61	Proved the following rewrite lemma:
309.26/291.61	sort(gen_nil:cons5_0(n1990_0)) -> gen_nil:cons5_0(n1990_0), rt in Omega(1 + n1990_0 + n1990_0^2)
309.26/291.61	
309.26/291.61	Induction Base:
309.26/291.61	sort(gen_nil:cons5_0(0)) ->_R^Omega(1)
309.26/291.61	nil
309.26/291.61	
309.26/291.61	Induction Step:
309.26/291.61	sort(gen_nil:cons5_0(+(n1990_0, 1))) ->_R^Omega(1)
309.26/291.61	cons(max(cons(0', gen_nil:cons5_0(n1990_0))), sort(h(del(max(cons(0', gen_nil:cons5_0(n1990_0))), cons(0', gen_nil:cons5_0(n1990_0)))))) ->_L^Omega(1 + n1990_0)
309.26/291.61	cons(gen_0':s4_0(0), sort(h(del(max(cons(0', gen_nil:cons5_0(n1990_0))), cons(0', gen_nil:cons5_0(n1990_0)))))) ->_L^Omega(1 + n1990_0)
309.26/291.61	cons(gen_0':s4_0(0), sort(h(del(gen_0':s4_0(0), cons(0', gen_nil:cons5_0(n1990_0)))))) ->_R^Omega(1)
309.26/291.61	cons(gen_0':s4_0(0), sort(h(if2(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_nil:cons5_0(n1990_0))))) ->_L^Omega(1)
309.26/291.61	cons(gen_0':s4_0(0), sort(h(if2(true, gen_0':s4_0(0), 0', gen_nil:cons5_0(n1990_0))))) ->_R^Omega(1)
309.26/291.62	cons(gen_0':s4_0(0), sort(h(gen_nil:cons5_0(n1990_0)))) ->_L^Omega(1 + n1990_0)
309.26/291.62	cons(gen_0':s4_0(0), sort(gen_nil:cons5_0(n1990_0))) ->_IH
309.26/291.62	cons(gen_0':s4_0(0), gen_nil:cons5_0(c1991_0))
309.26/291.62	
309.26/291.62	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
309.26/291.62	----------------------------------------
309.26/291.62	
309.26/291.62	(20)
309.26/291.62	Obligation:
309.26/291.62	Proved the lower bound n^2 for the following obligation:
309.26/291.62	
309.26/291.62	TRS:
309.26/291.62	Rules:
309.26/291.62	max(nil) -> 0'
309.26/291.62	max(cons(x, nil)) -> x
309.26/291.62	max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs)
309.26/291.62	if1(true, x, y, xs) -> max(cons(x, xs))
309.26/291.62	if1(false, x, y, xs) -> max(cons(y, xs))
309.26/291.62	del(x, nil) -> nil
309.26/291.62	del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
309.26/291.62	if2(true, x, y, xs) -> xs
309.26/291.62	if2(false, x, y, xs) -> cons(y, del(x, xs))
309.26/291.62	eq(0', 0') -> true
309.26/291.62	eq(0', s(y)) -> false
309.26/291.62	eq(s(x), 0') -> false
309.26/291.62	eq(s(x), s(y)) -> eq(x, y)
309.26/291.62	sort(nil) -> nil
309.26/291.62	sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(h(del(max(cons(x, xs)), cons(x, xs)))))
309.26/291.62	ge(0', 0') -> true
309.26/291.62	ge(s(x), 0') -> true
309.26/291.62	ge(0', s(x)) -> false
309.26/291.62	ge(s(x), s(y)) -> ge(x, y)
309.26/291.62	h(nil) -> nil
309.26/291.62	h(cons(x, xs)) -> cons(x, h(xs))
309.26/291.62	
309.26/291.62	Types:
309.26/291.62	max :: nil:cons -> 0':s
309.26/291.62	nil :: nil:cons
309.26/291.62	0' :: 0':s
309.26/291.62	cons :: 0':s -> nil:cons -> nil:cons
309.26/291.62	if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
309.26/291.62	ge :: 0':s -> 0':s -> true:false
309.26/291.62	true :: true:false
309.26/291.62	false :: true:false
309.26/291.62	del :: 0':s -> nil:cons -> nil:cons
309.26/291.62	if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
309.26/291.62	eq :: 0':s -> 0':s -> true:false
309.26/291.62	s :: 0':s -> 0':s
309.26/291.62	sort :: nil:cons -> nil:cons
309.26/291.62	h :: nil:cons -> nil:cons
309.26/291.62	hole_0':s1_0 :: 0':s
309.26/291.62	hole_nil:cons2_0 :: nil:cons
309.26/291.62	hole_true:false3_0 :: true:false
309.26/291.62	gen_0':s4_0 :: Nat -> 0':s
309.26/291.62	gen_nil:cons5_0 :: Nat -> nil:cons
309.26/291.62	
309.26/291.62	
309.26/291.62	Lemmas:
309.26/291.62	ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
309.26/291.62	max(gen_nil:cons5_0(+(1, n536_0))) -> gen_0':s4_0(0), rt in Omega(1 + n536_0)
309.26/291.62	eq(gen_0':s4_0(n985_0), gen_0':s4_0(n985_0)) -> true, rt in Omega(1 + n985_0)
309.26/291.62	h(gen_nil:cons5_0(n1688_0)) -> gen_nil:cons5_0(n1688_0), rt in Omega(1 + n1688_0)
309.26/291.62	
309.26/291.62	
309.26/291.62	Generator Equations:
309.26/291.62	gen_0':s4_0(0) <=> 0'
309.26/291.62	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
309.26/291.62	gen_nil:cons5_0(0) <=> nil
309.26/291.62	gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))
309.26/291.62	
309.26/291.62	
309.26/291.62	The following defined symbols remain to be analysed:
309.26/291.62	sort
309.26/291.62	----------------------------------------
309.26/291.62	
309.26/291.62	(21) LowerBoundPropagationProof (FINISHED)
309.26/291.62	Propagated lower bound.
309.26/291.62	----------------------------------------
309.26/291.62	
309.26/291.62	(22)
309.26/291.62	BOUNDS(n^2, INF)
309.26/291.65	EOF