1010.25/291.52	WORST_CASE(Omega(n^2), ?)
1010.25/291.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1010.25/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1010.25/291.53	
1010.25/291.53	(0) CpxTRS
1010.25/291.53	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1010.25/291.53	(2) CpxTRS
1010.25/291.53	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1010.25/291.53	(4) typed CpxTrs
1010.25/291.53	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1010.25/291.53	(6) typed CpxTrs
1010.25/291.53	(7) RewriteLemmaProof [LOWER BOUND(ID), 273 ms]
1010.25/291.53	(8) BEST
1010.25/291.53	    (9) proven lower bound
1010.25/291.53	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1010.25/291.53	        (11) BOUNDS(n^1, INF)
1010.25/291.53	    (12) typed CpxTrs
1010.25/291.53	        (13) RewriteLemmaProof [LOWER BOUND(ID), 67 ms]
1010.25/291.53	        (14) typed CpxTrs
1010.25/291.53	        (15) RewriteLemmaProof [LOWER BOUND(ID), 55 ms]
1010.25/291.53	        (16) BEST
1010.25/291.53	            (17) proven lower bound
1010.25/291.53	                (18) LowerBoundPropagationProof [FINISHED, 0 ms]
1010.25/291.53	                (19) BOUNDS(n^2, INF)
1010.25/291.53	            (20) typed CpxTrs
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(0)
1010.25/291.53	Obligation:
1010.25/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	The TRS R consists of the following rules:
1010.25/291.53	
1010.25/291.53	   lt(0, s(x)) -> true
1010.25/291.53	   lt(x, 0) -> false
1010.25/291.53	   lt(s(x), s(y)) -> lt(x, y)
1010.25/291.53	   times(0, y) -> 0
1010.25/291.53	   times(s(x), y) -> plus(y, times(x, y))
1010.25/291.53	   plus(0, y) -> y
1010.25/291.53	   plus(s(x), y) -> s(plus(x, y))
1010.25/291.53	   fac(x) -> loop(x, s(0), s(0))
1010.25/291.53	   loop(x, c, y) -> if(lt(x, c), x, c, y)
1010.25/291.53	   if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
1010.25/291.53	   if(true, x, c, y) -> y
1010.25/291.53	
1010.25/291.53	S is empty.
1010.25/291.53	Rewrite Strategy: INNERMOST
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1010.25/291.53	Renamed function symbols to avoid clashes with predefined symbol.
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(2)
1010.25/291.53	Obligation:
1010.25/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	The TRS R consists of the following rules:
1010.25/291.53	
1010.25/291.53	   lt(0', s(x)) -> true
1010.25/291.53	   lt(x, 0') -> false
1010.25/291.53	   lt(s(x), s(y)) -> lt(x, y)
1010.25/291.53	   times(0', y) -> 0'
1010.25/291.53	   times(s(x), y) -> plus(y, times(x, y))
1010.25/291.53	   plus(0', y) -> y
1010.25/291.53	   plus(s(x), y) -> s(plus(x, y))
1010.25/291.53	   fac(x) -> loop(x, s(0'), s(0'))
1010.25/291.53	   loop(x, c, y) -> if(lt(x, c), x, c, y)
1010.25/291.53	   if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
1010.25/291.53	   if(true, x, c, y) -> y
1010.25/291.53	
1010.25/291.53	S is empty.
1010.25/291.53	Rewrite Strategy: INNERMOST
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1010.25/291.53	Infered types.
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(4)
1010.25/291.53	Obligation:
1010.25/291.53	Innermost TRS:
1010.25/291.53	Rules:
1010.25/291.53	lt(0', s(x)) -> true
1010.25/291.53	lt(x, 0') -> false
1010.25/291.53	lt(s(x), s(y)) -> lt(x, y)
1010.25/291.53	times(0', y) -> 0'
1010.25/291.53	times(s(x), y) -> plus(y, times(x, y))
1010.25/291.53	plus(0', y) -> y
1010.25/291.53	plus(s(x), y) -> s(plus(x, y))
1010.25/291.53	fac(x) -> loop(x, s(0'), s(0'))
1010.25/291.53	loop(x, c, y) -> if(lt(x, c), x, c, y)
1010.25/291.53	if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
1010.25/291.53	if(true, x, c, y) -> y
1010.25/291.53	
1010.25/291.53	Types:
1010.25/291.53	lt :: 0':s -> 0':s -> true:false
1010.25/291.53	0' :: 0':s
1010.25/291.53	s :: 0':s -> 0':s
1010.25/291.53	true :: true:false
1010.25/291.53	false :: true:false
1010.25/291.53	times :: 0':s -> 0':s -> 0':s
1010.25/291.53	plus :: 0':s -> 0':s -> 0':s
1010.25/291.53	fac :: 0':s -> 0':s
1010.25/291.53	loop :: 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	hole_true:false1_0 :: true:false
1010.25/291.53	hole_0':s2_0 :: 0':s
1010.25/291.53	gen_0':s3_0 :: Nat -> 0':s
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(5) OrderProof (LOWER BOUND(ID))
1010.25/291.53	Heuristically decided to analyse the following defined symbols:
1010.25/291.53	lt, times, plus, loop
1010.25/291.53	
1010.25/291.53	They will be analysed ascendingly in the following order:
1010.25/291.53	lt < loop
1010.25/291.53	plus < times
1010.25/291.53	times < loop
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(6)
1010.25/291.53	Obligation:
1010.25/291.53	Innermost TRS:
1010.25/291.53	Rules:
1010.25/291.53	lt(0', s(x)) -> true
1010.25/291.53	lt(x, 0') -> false
1010.25/291.53	lt(s(x), s(y)) -> lt(x, y)
1010.25/291.53	times(0', y) -> 0'
1010.25/291.53	times(s(x), y) -> plus(y, times(x, y))
1010.25/291.53	plus(0', y) -> y
1010.25/291.53	plus(s(x), y) -> s(plus(x, y))
1010.25/291.53	fac(x) -> loop(x, s(0'), s(0'))
1010.25/291.53	loop(x, c, y) -> if(lt(x, c), x, c, y)
1010.25/291.53	if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
1010.25/291.53	if(true, x, c, y) -> y
1010.25/291.53	
1010.25/291.53	Types:
1010.25/291.53	lt :: 0':s -> 0':s -> true:false
1010.25/291.53	0' :: 0':s
1010.25/291.53	s :: 0':s -> 0':s
1010.25/291.53	true :: true:false
1010.25/291.53	false :: true:false
1010.25/291.53	times :: 0':s -> 0':s -> 0':s
1010.25/291.53	plus :: 0':s -> 0':s -> 0':s
1010.25/291.53	fac :: 0':s -> 0':s
1010.25/291.53	loop :: 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	hole_true:false1_0 :: true:false
1010.25/291.53	hole_0':s2_0 :: 0':s
1010.25/291.53	gen_0':s3_0 :: Nat -> 0':s
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	Generator Equations:
1010.25/291.53	gen_0':s3_0(0) <=> 0'
1010.25/291.53	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	The following defined symbols remain to be analysed:
1010.25/291.53	lt, times, plus, loop
1010.25/291.53	
1010.25/291.53	They will be analysed ascendingly in the following order:
1010.25/291.53	lt < loop
1010.25/291.53	plus < times
1010.25/291.53	times < loop
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(7) RewriteLemmaProof (LOWER BOUND(ID))
1010.25/291.53	Proved the following rewrite lemma:
1010.25/291.53	lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
1010.25/291.53	
1010.25/291.53	Induction Base:
1010.25/291.53	lt(gen_0':s3_0(0), gen_0':s3_0(+(1, 0))) ->_R^Omega(1)
1010.25/291.53	true
1010.25/291.53	
1010.25/291.53	Induction Step:
1010.25/291.53	lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
1010.25/291.53	lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) ->_IH
1010.25/291.53	true
1010.25/291.53	
1010.25/291.53	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(8)
1010.25/291.53	Complex Obligation (BEST)
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(9)
1010.25/291.53	Obligation:
1010.25/291.53	Proved the lower bound n^1 for the following obligation:
1010.25/291.53	
1010.25/291.53	Innermost TRS:
1010.25/291.53	Rules:
1010.25/291.53	lt(0', s(x)) -> true
1010.25/291.53	lt(x, 0') -> false
1010.25/291.53	lt(s(x), s(y)) -> lt(x, y)
1010.25/291.53	times(0', y) -> 0'
1010.25/291.53	times(s(x), y) -> plus(y, times(x, y))
1010.25/291.53	plus(0', y) -> y
1010.25/291.53	plus(s(x), y) -> s(plus(x, y))
1010.25/291.53	fac(x) -> loop(x, s(0'), s(0'))
1010.25/291.53	loop(x, c, y) -> if(lt(x, c), x, c, y)
1010.25/291.53	if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
1010.25/291.53	if(true, x, c, y) -> y
1010.25/291.53	
1010.25/291.53	Types:
1010.25/291.53	lt :: 0':s -> 0':s -> true:false
1010.25/291.53	0' :: 0':s
1010.25/291.53	s :: 0':s -> 0':s
1010.25/291.53	true :: true:false
1010.25/291.53	false :: true:false
1010.25/291.53	times :: 0':s -> 0':s -> 0':s
1010.25/291.53	plus :: 0':s -> 0':s -> 0':s
1010.25/291.53	fac :: 0':s -> 0':s
1010.25/291.53	loop :: 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	hole_true:false1_0 :: true:false
1010.25/291.53	hole_0':s2_0 :: 0':s
1010.25/291.53	gen_0':s3_0 :: Nat -> 0':s
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	Generator Equations:
1010.25/291.53	gen_0':s3_0(0) <=> 0'
1010.25/291.53	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	The following defined symbols remain to be analysed:
1010.25/291.53	lt, times, plus, loop
1010.25/291.53	
1010.25/291.53	They will be analysed ascendingly in the following order:
1010.25/291.53	lt < loop
1010.25/291.53	plus < times
1010.25/291.53	times < loop
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(10) LowerBoundPropagationProof (FINISHED)
1010.25/291.53	Propagated lower bound.
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(11)
1010.25/291.53	BOUNDS(n^1, INF)
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(12)
1010.25/291.53	Obligation:
1010.25/291.53	Innermost TRS:
1010.25/291.53	Rules:
1010.25/291.53	lt(0', s(x)) -> true
1010.25/291.53	lt(x, 0') -> false
1010.25/291.53	lt(s(x), s(y)) -> lt(x, y)
1010.25/291.53	times(0', y) -> 0'
1010.25/291.53	times(s(x), y) -> plus(y, times(x, y))
1010.25/291.53	plus(0', y) -> y
1010.25/291.53	plus(s(x), y) -> s(plus(x, y))
1010.25/291.53	fac(x) -> loop(x, s(0'), s(0'))
1010.25/291.53	loop(x, c, y) -> if(lt(x, c), x, c, y)
1010.25/291.53	if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
1010.25/291.53	if(true, x, c, y) -> y
1010.25/291.53	
1010.25/291.53	Types:
1010.25/291.53	lt :: 0':s -> 0':s -> true:false
1010.25/291.53	0' :: 0':s
1010.25/291.53	s :: 0':s -> 0':s
1010.25/291.53	true :: true:false
1010.25/291.53	false :: true:false
1010.25/291.53	times :: 0':s -> 0':s -> 0':s
1010.25/291.53	plus :: 0':s -> 0':s -> 0':s
1010.25/291.53	fac :: 0':s -> 0':s
1010.25/291.53	loop :: 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	hole_true:false1_0 :: true:false
1010.25/291.53	hole_0':s2_0 :: 0':s
1010.25/291.53	gen_0':s3_0 :: Nat -> 0':s
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	Lemmas:
1010.25/291.53	lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	Generator Equations:
1010.25/291.53	gen_0':s3_0(0) <=> 0'
1010.25/291.53	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	The following defined symbols remain to be analysed:
1010.25/291.53	plus, times, loop
1010.25/291.53	
1010.25/291.53	They will be analysed ascendingly in the following order:
1010.25/291.53	plus < times
1010.25/291.53	times < loop
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(13) RewriteLemmaProof (LOWER BOUND(ID))
1010.25/291.53	Proved the following rewrite lemma:
1010.25/291.53	plus(gen_0':s3_0(n305_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n305_0, b)), rt in Omega(1 + n305_0)
1010.25/291.53	
1010.25/291.53	Induction Base:
1010.25/291.53	plus(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
1010.25/291.53	gen_0':s3_0(b)
1010.25/291.53	
1010.25/291.53	Induction Step:
1010.25/291.53	plus(gen_0':s3_0(+(n305_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
1010.25/291.53	s(plus(gen_0':s3_0(n305_0), gen_0':s3_0(b))) ->_IH
1010.25/291.53	s(gen_0':s3_0(+(b, c306_0)))
1010.25/291.53	
1010.25/291.53	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(14)
1010.25/291.53	Obligation:
1010.25/291.53	Innermost TRS:
1010.25/291.53	Rules:
1010.25/291.53	lt(0', s(x)) -> true
1010.25/291.53	lt(x, 0') -> false
1010.25/291.53	lt(s(x), s(y)) -> lt(x, y)
1010.25/291.53	times(0', y) -> 0'
1010.25/291.53	times(s(x), y) -> plus(y, times(x, y))
1010.25/291.53	plus(0', y) -> y
1010.25/291.53	plus(s(x), y) -> s(plus(x, y))
1010.25/291.53	fac(x) -> loop(x, s(0'), s(0'))
1010.25/291.53	loop(x, c, y) -> if(lt(x, c), x, c, y)
1010.25/291.53	if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
1010.25/291.53	if(true, x, c, y) -> y
1010.25/291.53	
1010.25/291.53	Types:
1010.25/291.53	lt :: 0':s -> 0':s -> true:false
1010.25/291.53	0' :: 0':s
1010.25/291.53	s :: 0':s -> 0':s
1010.25/291.53	true :: true:false
1010.25/291.53	false :: true:false
1010.25/291.53	times :: 0':s -> 0':s -> 0':s
1010.25/291.53	plus :: 0':s -> 0':s -> 0':s
1010.25/291.53	fac :: 0':s -> 0':s
1010.25/291.53	loop :: 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	hole_true:false1_0 :: true:false
1010.25/291.53	hole_0':s2_0 :: 0':s
1010.25/291.53	gen_0':s3_0 :: Nat -> 0':s
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	Lemmas:
1010.25/291.53	lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
1010.25/291.53	plus(gen_0':s3_0(n305_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n305_0, b)), rt in Omega(1 + n305_0)
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	Generator Equations:
1010.25/291.53	gen_0':s3_0(0) <=> 0'
1010.25/291.53	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	The following defined symbols remain to be analysed:
1010.25/291.53	times, loop
1010.25/291.53	
1010.25/291.53	They will be analysed ascendingly in the following order:
1010.25/291.53	times < loop
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(15) RewriteLemmaProof (LOWER BOUND(ID))
1010.25/291.53	Proved the following rewrite lemma:
1010.25/291.53	times(gen_0':s3_0(n912_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n912_0, b)), rt in Omega(1 + b*n912_0 + n912_0)
1010.25/291.53	
1010.25/291.53	Induction Base:
1010.25/291.53	times(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
1010.25/291.53	0'
1010.25/291.53	
1010.25/291.53	Induction Step:
1010.25/291.53	times(gen_0':s3_0(+(n912_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
1010.25/291.53	plus(gen_0':s3_0(b), times(gen_0':s3_0(n912_0), gen_0':s3_0(b))) ->_IH
1010.25/291.53	plus(gen_0':s3_0(b), gen_0':s3_0(*(c913_0, b))) ->_L^Omega(1 + b)
1010.25/291.53	gen_0':s3_0(+(b, *(n912_0, b)))
1010.25/291.53	
1010.25/291.53	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(16)
1010.25/291.53	Complex Obligation (BEST)
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(17)
1010.25/291.53	Obligation:
1010.25/291.53	Proved the lower bound n^2 for the following obligation:
1010.25/291.53	
1010.25/291.53	Innermost TRS:
1010.25/291.53	Rules:
1010.25/291.53	lt(0', s(x)) -> true
1010.25/291.53	lt(x, 0') -> false
1010.25/291.53	lt(s(x), s(y)) -> lt(x, y)
1010.25/291.53	times(0', y) -> 0'
1010.25/291.53	times(s(x), y) -> plus(y, times(x, y))
1010.25/291.53	plus(0', y) -> y
1010.25/291.53	plus(s(x), y) -> s(plus(x, y))
1010.25/291.53	fac(x) -> loop(x, s(0'), s(0'))
1010.25/291.53	loop(x, c, y) -> if(lt(x, c), x, c, y)
1010.25/291.53	if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
1010.25/291.53	if(true, x, c, y) -> y
1010.25/291.53	
1010.25/291.53	Types:
1010.25/291.53	lt :: 0':s -> 0':s -> true:false
1010.25/291.53	0' :: 0':s
1010.25/291.53	s :: 0':s -> 0':s
1010.25/291.53	true :: true:false
1010.25/291.53	false :: true:false
1010.25/291.53	times :: 0':s -> 0':s -> 0':s
1010.25/291.53	plus :: 0':s -> 0':s -> 0':s
1010.25/291.53	fac :: 0':s -> 0':s
1010.25/291.53	loop :: 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	hole_true:false1_0 :: true:false
1010.25/291.53	hole_0':s2_0 :: 0':s
1010.25/291.53	gen_0':s3_0 :: Nat -> 0':s
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	Lemmas:
1010.25/291.53	lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
1010.25/291.53	plus(gen_0':s3_0(n305_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n305_0, b)), rt in Omega(1 + n305_0)
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	Generator Equations:
1010.25/291.53	gen_0':s3_0(0) <=> 0'
1010.25/291.53	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	The following defined symbols remain to be analysed:
1010.25/291.53	times, loop
1010.25/291.53	
1010.25/291.53	They will be analysed ascendingly in the following order:
1010.25/291.53	times < loop
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(18) LowerBoundPropagationProof (FINISHED)
1010.25/291.53	Propagated lower bound.
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(19)
1010.25/291.53	BOUNDS(n^2, INF)
1010.25/291.53	
1010.25/291.53	----------------------------------------
1010.25/291.53	
1010.25/291.53	(20)
1010.25/291.53	Obligation:
1010.25/291.53	Innermost TRS:
1010.25/291.53	Rules:
1010.25/291.53	lt(0', s(x)) -> true
1010.25/291.53	lt(x, 0') -> false
1010.25/291.53	lt(s(x), s(y)) -> lt(x, y)
1010.25/291.53	times(0', y) -> 0'
1010.25/291.53	times(s(x), y) -> plus(y, times(x, y))
1010.25/291.53	plus(0', y) -> y
1010.25/291.53	plus(s(x), y) -> s(plus(x, y))
1010.25/291.53	fac(x) -> loop(x, s(0'), s(0'))
1010.25/291.53	loop(x, c, y) -> if(lt(x, c), x, c, y)
1010.25/291.53	if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
1010.25/291.53	if(true, x, c, y) -> y
1010.25/291.53	
1010.25/291.53	Types:
1010.25/291.53	lt :: 0':s -> 0':s -> true:false
1010.25/291.53	0' :: 0':s
1010.25/291.53	s :: 0':s -> 0':s
1010.25/291.53	true :: true:false
1010.25/291.53	false :: true:false
1010.25/291.53	times :: 0':s -> 0':s -> 0':s
1010.25/291.53	plus :: 0':s -> 0':s -> 0':s
1010.25/291.53	fac :: 0':s -> 0':s
1010.25/291.53	loop :: 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1010.25/291.53	hole_true:false1_0 :: true:false
1010.25/291.53	hole_0':s2_0 :: 0':s
1010.25/291.53	gen_0':s3_0 :: Nat -> 0':s
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	Lemmas:
1010.25/291.53	lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
1010.25/291.53	plus(gen_0':s3_0(n305_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n305_0, b)), rt in Omega(1 + n305_0)
1010.25/291.53	times(gen_0':s3_0(n912_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n912_0, b)), rt in Omega(1 + b*n912_0 + n912_0)
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	Generator Equations:
1010.25/291.53	gen_0':s3_0(0) <=> 0'
1010.25/291.53	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1010.25/291.53	
1010.25/291.53	
1010.25/291.53	The following defined symbols remain to be analysed:
1010.25/291.53	loop
1010.46/291.59	EOF