942.54/291.50	WORST_CASE(Omega(n^2), ?)
942.62/291.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
942.62/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
942.62/291.52	
942.62/291.52	
942.62/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
942.62/291.52	
942.62/291.52	(0) CpxTRS
942.62/291.52	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
942.62/291.52	(2) CpxTRS
942.62/291.52	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
942.62/291.52	(4) typed CpxTrs
942.62/291.52	(5) OrderProof [LOWER BOUND(ID), 0 ms]
942.62/291.52	(6) typed CpxTrs
942.62/291.52	(7) RewriteLemmaProof [LOWER BOUND(ID), 304 ms]
942.62/291.52	(8) BEST
942.62/291.52	    (9) proven lower bound
942.62/291.52	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
942.62/291.52	        (11) BOUNDS(n^1, INF)
942.62/291.52	    (12) typed CpxTrs
942.62/291.52	        (13) RewriteLemmaProof [LOWER BOUND(ID), 116 ms]
942.62/291.52	        (14) BEST
942.62/291.52	            (15) proven lower bound
942.62/291.52	                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
942.62/291.52	                (17) BOUNDS(n^2, INF)
942.62/291.52	            (18) typed CpxTrs
942.62/291.52	                (19) RewriteLemmaProof [LOWER BOUND(ID), 592 ms]
942.62/291.52	                (20) typed CpxTrs
942.62/291.52	                (21) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
942.62/291.52	                (22) typed CpxTrs
942.62/291.52	
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(0)
942.62/291.52	Obligation:
942.62/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
942.62/291.52	
942.62/291.52	
942.62/291.52	The TRS R consists of the following rules:
942.62/291.52	
942.62/291.52	   plus(0, x) -> x
942.62/291.52	   plus(s(x), y) -> s(plus(x, y))
942.62/291.52	   times(0, y) -> 0
942.62/291.52	   times(s(x), y) -> plus(y, times(x, y))
942.62/291.52	   exp(x, 0) -> s(0)
942.62/291.52	   exp(x, s(y)) -> times(x, exp(x, y))
942.62/291.52	   ge(x, 0) -> true
942.62/291.52	   ge(0, s(x)) -> false
942.62/291.52	   ge(s(x), s(y)) -> ge(x, y)
942.62/291.52	   tower(x, y) -> towerIter(0, x, y, s(0))
942.62/291.52	   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
942.62/291.52	   help(true, c, x, y, z) -> z
942.62/291.52	   help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
942.62/291.52	
942.62/291.52	S is empty.
942.62/291.52	Rewrite Strategy: INNERMOST
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(1) RenamingProof (BOTH BOUNDS(ID, ID))
942.62/291.52	Renamed function symbols to avoid clashes with predefined symbol.
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(2)
942.62/291.52	Obligation:
942.62/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
942.62/291.52	
942.62/291.52	
942.62/291.52	The TRS R consists of the following rules:
942.62/291.52	
942.62/291.52	   plus(0', x) -> x
942.62/291.52	   plus(s(x), y) -> s(plus(x, y))
942.62/291.52	   times(0', y) -> 0'
942.62/291.52	   times(s(x), y) -> plus(y, times(x, y))
942.62/291.52	   exp(x, 0') -> s(0')
942.62/291.52	   exp(x, s(y)) -> times(x, exp(x, y))
942.62/291.52	   ge(x, 0') -> true
942.62/291.52	   ge(0', s(x)) -> false
942.62/291.52	   ge(s(x), s(y)) -> ge(x, y)
942.62/291.52	   tower(x, y) -> towerIter(0', x, y, s(0'))
942.62/291.52	   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
942.62/291.52	   help(true, c, x, y, z) -> z
942.62/291.52	   help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
942.62/291.52	
942.62/291.52	S is empty.
942.62/291.52	Rewrite Strategy: INNERMOST
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
942.62/291.52	Infered types.
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(4)
942.62/291.52	Obligation:
942.62/291.52	Innermost TRS:
942.62/291.52	Rules:
942.62/291.52	plus(0', x) -> x
942.62/291.52	plus(s(x), y) -> s(plus(x, y))
942.62/291.52	times(0', y) -> 0'
942.62/291.52	times(s(x), y) -> plus(y, times(x, y))
942.62/291.52	exp(x, 0') -> s(0')
942.62/291.52	exp(x, s(y)) -> times(x, exp(x, y))
942.62/291.52	ge(x, 0') -> true
942.62/291.52	ge(0', s(x)) -> false
942.62/291.52	ge(s(x), s(y)) -> ge(x, y)
942.62/291.52	tower(x, y) -> towerIter(0', x, y, s(0'))
942.62/291.52	towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
942.62/291.52	help(true, c, x, y, z) -> z
942.62/291.52	help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
942.62/291.52	
942.62/291.52	Types:
942.62/291.52	plus :: 0':s -> 0':s -> 0':s
942.62/291.52	0' :: 0':s
942.62/291.52	s :: 0':s -> 0':s
942.62/291.52	times :: 0':s -> 0':s -> 0':s
942.62/291.52	exp :: 0':s -> 0':s -> 0':s
942.62/291.52	ge :: 0':s -> 0':s -> true:false
942.62/291.52	true :: true:false
942.62/291.52	false :: true:false
942.62/291.52	tower :: 0':s -> 0':s -> 0':s
942.62/291.52	towerIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	help :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	hole_0':s1_0 :: 0':s
942.62/291.52	hole_true:false2_0 :: true:false
942.62/291.52	gen_0':s3_0 :: Nat -> 0':s
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(5) OrderProof (LOWER BOUND(ID))
942.62/291.52	Heuristically decided to analyse the following defined symbols:
942.62/291.52	plus, times, exp, ge, towerIter
942.62/291.52	
942.62/291.52	They will be analysed ascendingly in the following order:
942.62/291.52	plus < times
942.62/291.52	times < exp
942.62/291.52	exp < towerIter
942.62/291.52	ge < towerIter
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(6)
942.62/291.52	Obligation:
942.62/291.52	Innermost TRS:
942.62/291.52	Rules:
942.62/291.52	plus(0', x) -> x
942.62/291.52	plus(s(x), y) -> s(plus(x, y))
942.62/291.52	times(0', y) -> 0'
942.62/291.52	times(s(x), y) -> plus(y, times(x, y))
942.62/291.52	exp(x, 0') -> s(0')
942.62/291.52	exp(x, s(y)) -> times(x, exp(x, y))
942.62/291.52	ge(x, 0') -> true
942.62/291.52	ge(0', s(x)) -> false
942.62/291.52	ge(s(x), s(y)) -> ge(x, y)
942.62/291.52	tower(x, y) -> towerIter(0', x, y, s(0'))
942.62/291.52	towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
942.62/291.52	help(true, c, x, y, z) -> z
942.62/291.52	help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
942.62/291.52	
942.62/291.52	Types:
942.62/291.52	plus :: 0':s -> 0':s -> 0':s
942.62/291.52	0' :: 0':s
942.62/291.52	s :: 0':s -> 0':s
942.62/291.52	times :: 0':s -> 0':s -> 0':s
942.62/291.52	exp :: 0':s -> 0':s -> 0':s
942.62/291.52	ge :: 0':s -> 0':s -> true:false
942.62/291.52	true :: true:false
942.62/291.52	false :: true:false
942.62/291.52	tower :: 0':s -> 0':s -> 0':s
942.62/291.52	towerIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	help :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	hole_0':s1_0 :: 0':s
942.62/291.52	hole_true:false2_0 :: true:false
942.62/291.52	gen_0':s3_0 :: Nat -> 0':s
942.62/291.52	
942.62/291.52	
942.62/291.52	Generator Equations:
942.62/291.52	gen_0':s3_0(0) <=> 0'
942.62/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
942.62/291.52	
942.62/291.52	
942.62/291.52	The following defined symbols remain to be analysed:
942.62/291.52	plus, times, exp, ge, towerIter
942.62/291.52	
942.62/291.52	They will be analysed ascendingly in the following order:
942.62/291.52	plus < times
942.62/291.52	times < exp
942.62/291.52	exp < towerIter
942.62/291.52	ge < towerIter
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(7) RewriteLemmaProof (LOWER BOUND(ID))
942.62/291.52	Proved the following rewrite lemma:
942.62/291.52	plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
942.62/291.52	
942.62/291.52	Induction Base:
942.62/291.52	plus(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
942.62/291.52	gen_0':s3_0(b)
942.62/291.52	
942.62/291.52	Induction Step:
942.62/291.52	plus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
942.62/291.52	s(plus(gen_0':s3_0(n5_0), gen_0':s3_0(b))) ->_IH
942.62/291.52	s(gen_0':s3_0(+(b, c6_0)))
942.62/291.52	
942.62/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(8)
942.62/291.52	Complex Obligation (BEST)
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(9)
942.62/291.52	Obligation:
942.62/291.52	Proved the lower bound n^1 for the following obligation:
942.62/291.52	
942.62/291.52	Innermost TRS:
942.62/291.52	Rules:
942.62/291.52	plus(0', x) -> x
942.62/291.52	plus(s(x), y) -> s(plus(x, y))
942.62/291.52	times(0', y) -> 0'
942.62/291.52	times(s(x), y) -> plus(y, times(x, y))
942.62/291.52	exp(x, 0') -> s(0')
942.62/291.52	exp(x, s(y)) -> times(x, exp(x, y))
942.62/291.52	ge(x, 0') -> true
942.62/291.52	ge(0', s(x)) -> false
942.62/291.52	ge(s(x), s(y)) -> ge(x, y)
942.62/291.52	tower(x, y) -> towerIter(0', x, y, s(0'))
942.62/291.52	towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
942.62/291.52	help(true, c, x, y, z) -> z
942.62/291.52	help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
942.62/291.52	
942.62/291.52	Types:
942.62/291.52	plus :: 0':s -> 0':s -> 0':s
942.62/291.52	0' :: 0':s
942.62/291.52	s :: 0':s -> 0':s
942.62/291.52	times :: 0':s -> 0':s -> 0':s
942.62/291.52	exp :: 0':s -> 0':s -> 0':s
942.62/291.52	ge :: 0':s -> 0':s -> true:false
942.62/291.52	true :: true:false
942.62/291.52	false :: true:false
942.62/291.52	tower :: 0':s -> 0':s -> 0':s
942.62/291.52	towerIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	help :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	hole_0':s1_0 :: 0':s
942.62/291.52	hole_true:false2_0 :: true:false
942.62/291.52	gen_0':s3_0 :: Nat -> 0':s
942.62/291.52	
942.62/291.52	
942.62/291.52	Generator Equations:
942.62/291.52	gen_0':s3_0(0) <=> 0'
942.62/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
942.62/291.52	
942.62/291.52	
942.62/291.52	The following defined symbols remain to be analysed:
942.62/291.52	plus, times, exp, ge, towerIter
942.62/291.52	
942.62/291.52	They will be analysed ascendingly in the following order:
942.62/291.52	plus < times
942.62/291.52	times < exp
942.62/291.52	exp < towerIter
942.62/291.52	ge < towerIter
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(10) LowerBoundPropagationProof (FINISHED)
942.62/291.52	Propagated lower bound.
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(11)
942.62/291.52	BOUNDS(n^1, INF)
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(12)
942.62/291.52	Obligation:
942.62/291.52	Innermost TRS:
942.62/291.52	Rules:
942.62/291.52	plus(0', x) -> x
942.62/291.52	plus(s(x), y) -> s(plus(x, y))
942.62/291.52	times(0', y) -> 0'
942.62/291.52	times(s(x), y) -> plus(y, times(x, y))
942.62/291.52	exp(x, 0') -> s(0')
942.62/291.52	exp(x, s(y)) -> times(x, exp(x, y))
942.62/291.52	ge(x, 0') -> true
942.62/291.52	ge(0', s(x)) -> false
942.62/291.52	ge(s(x), s(y)) -> ge(x, y)
942.62/291.52	tower(x, y) -> towerIter(0', x, y, s(0'))
942.62/291.52	towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
942.62/291.52	help(true, c, x, y, z) -> z
942.62/291.52	help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
942.62/291.52	
942.62/291.52	Types:
942.62/291.52	plus :: 0':s -> 0':s -> 0':s
942.62/291.52	0' :: 0':s
942.62/291.52	s :: 0':s -> 0':s
942.62/291.52	times :: 0':s -> 0':s -> 0':s
942.62/291.52	exp :: 0':s -> 0':s -> 0':s
942.62/291.52	ge :: 0':s -> 0':s -> true:false
942.62/291.52	true :: true:false
942.62/291.52	false :: true:false
942.62/291.52	tower :: 0':s -> 0':s -> 0':s
942.62/291.52	towerIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	help :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	hole_0':s1_0 :: 0':s
942.62/291.52	hole_true:false2_0 :: true:false
942.62/291.52	gen_0':s3_0 :: Nat -> 0':s
942.62/291.52	
942.62/291.52	
942.62/291.52	Lemmas:
942.62/291.52	plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
942.62/291.52	
942.62/291.52	
942.62/291.52	Generator Equations:
942.62/291.52	gen_0':s3_0(0) <=> 0'
942.62/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
942.62/291.52	
942.62/291.52	
942.62/291.52	The following defined symbols remain to be analysed:
942.62/291.52	times, exp, ge, towerIter
942.62/291.52	
942.62/291.52	They will be analysed ascendingly in the following order:
942.62/291.52	times < exp
942.62/291.52	exp < towerIter
942.62/291.52	ge < towerIter
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(13) RewriteLemmaProof (LOWER BOUND(ID))
942.62/291.52	Proved the following rewrite lemma:
942.62/291.52	times(gen_0':s3_0(n646_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n646_0, b)), rt in Omega(1 + b*n646_0 + n646_0)
942.62/291.52	
942.62/291.52	Induction Base:
942.62/291.52	times(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
942.62/291.52	0'
942.62/291.52	
942.62/291.52	Induction Step:
942.62/291.52	times(gen_0':s3_0(+(n646_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
942.62/291.52	plus(gen_0':s3_0(b), times(gen_0':s3_0(n646_0), gen_0':s3_0(b))) ->_IH
942.62/291.52	plus(gen_0':s3_0(b), gen_0':s3_0(*(c647_0, b))) ->_L^Omega(1 + b)
942.62/291.52	gen_0':s3_0(+(b, *(n646_0, b)))
942.62/291.52	
942.62/291.52	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(14)
942.62/291.52	Complex Obligation (BEST)
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(15)
942.62/291.52	Obligation:
942.62/291.52	Proved the lower bound n^2 for the following obligation:
942.62/291.52	
942.62/291.52	Innermost TRS:
942.62/291.52	Rules:
942.62/291.52	plus(0', x) -> x
942.62/291.52	plus(s(x), y) -> s(plus(x, y))
942.62/291.52	times(0', y) -> 0'
942.62/291.52	times(s(x), y) -> plus(y, times(x, y))
942.62/291.52	exp(x, 0') -> s(0')
942.62/291.52	exp(x, s(y)) -> times(x, exp(x, y))
942.62/291.52	ge(x, 0') -> true
942.62/291.52	ge(0', s(x)) -> false
942.62/291.52	ge(s(x), s(y)) -> ge(x, y)
942.62/291.52	tower(x, y) -> towerIter(0', x, y, s(0'))
942.62/291.52	towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
942.62/291.52	help(true, c, x, y, z) -> z
942.62/291.52	help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
942.62/291.52	
942.62/291.52	Types:
942.62/291.52	plus :: 0':s -> 0':s -> 0':s
942.62/291.52	0' :: 0':s
942.62/291.52	s :: 0':s -> 0':s
942.62/291.52	times :: 0':s -> 0':s -> 0':s
942.62/291.52	exp :: 0':s -> 0':s -> 0':s
942.62/291.52	ge :: 0':s -> 0':s -> true:false
942.62/291.52	true :: true:false
942.62/291.52	false :: true:false
942.62/291.52	tower :: 0':s -> 0':s -> 0':s
942.62/291.52	towerIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	help :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	hole_0':s1_0 :: 0':s
942.62/291.52	hole_true:false2_0 :: true:false
942.62/291.52	gen_0':s3_0 :: Nat -> 0':s
942.62/291.52	
942.62/291.52	
942.62/291.52	Lemmas:
942.62/291.52	plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
942.62/291.52	
942.62/291.52	
942.62/291.52	Generator Equations:
942.62/291.52	gen_0':s3_0(0) <=> 0'
942.62/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
942.62/291.52	
942.62/291.52	
942.62/291.52	The following defined symbols remain to be analysed:
942.62/291.52	times, exp, ge, towerIter
942.62/291.52	
942.62/291.52	They will be analysed ascendingly in the following order:
942.62/291.52	times < exp
942.62/291.52	exp < towerIter
942.62/291.52	ge < towerIter
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(16) LowerBoundPropagationProof (FINISHED)
942.62/291.52	Propagated lower bound.
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(17)
942.62/291.52	BOUNDS(n^2, INF)
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(18)
942.62/291.52	Obligation:
942.62/291.52	Innermost TRS:
942.62/291.52	Rules:
942.62/291.52	plus(0', x) -> x
942.62/291.52	plus(s(x), y) -> s(plus(x, y))
942.62/291.52	times(0', y) -> 0'
942.62/291.52	times(s(x), y) -> plus(y, times(x, y))
942.62/291.52	exp(x, 0') -> s(0')
942.62/291.52	exp(x, s(y)) -> times(x, exp(x, y))
942.62/291.52	ge(x, 0') -> true
942.62/291.52	ge(0', s(x)) -> false
942.62/291.52	ge(s(x), s(y)) -> ge(x, y)
942.62/291.52	tower(x, y) -> towerIter(0', x, y, s(0'))
942.62/291.52	towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
942.62/291.52	help(true, c, x, y, z) -> z
942.62/291.52	help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
942.62/291.52	
942.62/291.52	Types:
942.62/291.52	plus :: 0':s -> 0':s -> 0':s
942.62/291.52	0' :: 0':s
942.62/291.52	s :: 0':s -> 0':s
942.62/291.52	times :: 0':s -> 0':s -> 0':s
942.62/291.52	exp :: 0':s -> 0':s -> 0':s
942.62/291.52	ge :: 0':s -> 0':s -> true:false
942.62/291.52	true :: true:false
942.62/291.52	false :: true:false
942.62/291.52	tower :: 0':s -> 0':s -> 0':s
942.62/291.52	towerIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	help :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	hole_0':s1_0 :: 0':s
942.62/291.52	hole_true:false2_0 :: true:false
942.62/291.52	gen_0':s3_0 :: Nat -> 0':s
942.62/291.52	
942.62/291.52	
942.62/291.52	Lemmas:
942.62/291.52	plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
942.62/291.52	times(gen_0':s3_0(n646_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n646_0, b)), rt in Omega(1 + b*n646_0 + n646_0)
942.62/291.52	
942.62/291.52	
942.62/291.52	Generator Equations:
942.62/291.52	gen_0':s3_0(0) <=> 0'
942.62/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
942.62/291.52	
942.62/291.52	
942.62/291.52	The following defined symbols remain to be analysed:
942.62/291.52	exp, ge, towerIter
942.62/291.52	
942.62/291.52	They will be analysed ascendingly in the following order:
942.62/291.52	exp < towerIter
942.62/291.52	ge < towerIter
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(19) RewriteLemmaProof (LOWER BOUND(ID))
942.62/291.52	Proved the following rewrite lemma:
942.62/291.52	exp(gen_0':s3_0(a), gen_0':s3_0(+(1, n1458_0))) -> *4_0, rt in Omega(n1458_0)
942.62/291.52	
942.62/291.52	Induction Base:
942.62/291.52	exp(gen_0':s3_0(a), gen_0':s3_0(+(1, 0)))
942.62/291.52	
942.62/291.52	Induction Step:
942.62/291.52	exp(gen_0':s3_0(a), gen_0':s3_0(+(1, +(n1458_0, 1)))) ->_R^Omega(1)
942.62/291.52	times(gen_0':s3_0(a), exp(gen_0':s3_0(a), gen_0':s3_0(+(1, n1458_0)))) ->_IH
942.62/291.52	times(gen_0':s3_0(a), *4_0)
942.62/291.52	
942.62/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(20)
942.62/291.52	Obligation:
942.62/291.52	Innermost TRS:
942.62/291.52	Rules:
942.62/291.52	plus(0', x) -> x
942.62/291.52	plus(s(x), y) -> s(plus(x, y))
942.62/291.52	times(0', y) -> 0'
942.62/291.52	times(s(x), y) -> plus(y, times(x, y))
942.62/291.52	exp(x, 0') -> s(0')
942.62/291.52	exp(x, s(y)) -> times(x, exp(x, y))
942.62/291.52	ge(x, 0') -> true
942.62/291.52	ge(0', s(x)) -> false
942.62/291.52	ge(s(x), s(y)) -> ge(x, y)
942.62/291.52	tower(x, y) -> towerIter(0', x, y, s(0'))
942.62/291.52	towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
942.62/291.52	help(true, c, x, y, z) -> z
942.62/291.52	help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
942.62/291.52	
942.62/291.52	Types:
942.62/291.52	plus :: 0':s -> 0':s -> 0':s
942.62/291.52	0' :: 0':s
942.62/291.52	s :: 0':s -> 0':s
942.62/291.52	times :: 0':s -> 0':s -> 0':s
942.62/291.52	exp :: 0':s -> 0':s -> 0':s
942.62/291.52	ge :: 0':s -> 0':s -> true:false
942.62/291.52	true :: true:false
942.62/291.52	false :: true:false
942.62/291.52	tower :: 0':s -> 0':s -> 0':s
942.62/291.52	towerIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	help :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	hole_0':s1_0 :: 0':s
942.62/291.52	hole_true:false2_0 :: true:false
942.62/291.52	gen_0':s3_0 :: Nat -> 0':s
942.62/291.52	
942.62/291.52	
942.62/291.52	Lemmas:
942.62/291.52	plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
942.62/291.52	times(gen_0':s3_0(n646_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n646_0, b)), rt in Omega(1 + b*n646_0 + n646_0)
942.62/291.52	exp(gen_0':s3_0(a), gen_0':s3_0(+(1, n1458_0))) -> *4_0, rt in Omega(n1458_0)
942.62/291.52	
942.62/291.52	
942.62/291.52	Generator Equations:
942.62/291.52	gen_0':s3_0(0) <=> 0'
942.62/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
942.62/291.52	
942.62/291.52	
942.62/291.52	The following defined symbols remain to be analysed:
942.62/291.52	ge, towerIter
942.62/291.52	
942.62/291.52	They will be analysed ascendingly in the following order:
942.62/291.52	ge < towerIter
942.62/291.52	
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(21) RewriteLemmaProof (LOWER BOUND(ID))
942.62/291.52	Proved the following rewrite lemma:
942.62/291.52	ge(gen_0':s3_0(n5514_0), gen_0':s3_0(n5514_0)) -> true, rt in Omega(1 + n5514_0)
942.62/291.52	
942.62/291.52	Induction Base:
942.62/291.52	ge(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
942.62/291.52	true
942.62/291.52	
942.62/291.52	Induction Step:
942.62/291.52	ge(gen_0':s3_0(+(n5514_0, 1)), gen_0':s3_0(+(n5514_0, 1))) ->_R^Omega(1)
942.62/291.52	ge(gen_0':s3_0(n5514_0), gen_0':s3_0(n5514_0)) ->_IH
942.62/291.52	true
942.62/291.52	
942.62/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
942.62/291.52	----------------------------------------
942.62/291.52	
942.62/291.52	(22)
942.62/291.52	Obligation:
942.62/291.52	Innermost TRS:
942.62/291.52	Rules:
942.62/291.52	plus(0', x) -> x
942.62/291.52	plus(s(x), y) -> s(plus(x, y))
942.62/291.52	times(0', y) -> 0'
942.62/291.52	times(s(x), y) -> plus(y, times(x, y))
942.62/291.52	exp(x, 0') -> s(0')
942.62/291.52	exp(x, s(y)) -> times(x, exp(x, y))
942.62/291.52	ge(x, 0') -> true
942.62/291.52	ge(0', s(x)) -> false
942.62/291.52	ge(s(x), s(y)) -> ge(x, y)
942.62/291.52	tower(x, y) -> towerIter(0', x, y, s(0'))
942.62/291.52	towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
942.62/291.52	help(true, c, x, y, z) -> z
942.62/291.52	help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
942.62/291.52	
942.62/291.52	Types:
942.62/291.52	plus :: 0':s -> 0':s -> 0':s
942.62/291.52	0' :: 0':s
942.62/291.52	s :: 0':s -> 0':s
942.62/291.52	times :: 0':s -> 0':s -> 0':s
942.62/291.52	exp :: 0':s -> 0':s -> 0':s
942.62/291.52	ge :: 0':s -> 0':s -> true:false
942.62/291.52	true :: true:false
942.62/291.52	false :: true:false
942.62/291.52	tower :: 0':s -> 0':s -> 0':s
942.62/291.52	towerIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	help :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
942.62/291.52	hole_0':s1_0 :: 0':s
942.62/291.52	hole_true:false2_0 :: true:false
942.62/291.52	gen_0':s3_0 :: Nat -> 0':s
942.62/291.52	
942.62/291.52	
942.62/291.52	Lemmas:
942.62/291.52	plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
942.62/291.52	times(gen_0':s3_0(n646_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n646_0, b)), rt in Omega(1 + b*n646_0 + n646_0)
942.62/291.52	exp(gen_0':s3_0(a), gen_0':s3_0(+(1, n1458_0))) -> *4_0, rt in Omega(n1458_0)
942.62/291.52	ge(gen_0':s3_0(n5514_0), gen_0':s3_0(n5514_0)) -> true, rt in Omega(1 + n5514_0)
942.62/291.52	
942.62/291.52	
942.62/291.52	Generator Equations:
942.62/291.52	gen_0':s3_0(0) <=> 0'
942.62/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
942.62/291.52	
942.62/291.52	
942.62/291.52	The following defined symbols remain to be analysed:
942.62/291.52	towerIter
942.69/291.58	EOF