311.84/291.53	WORST_CASE(Omega(n^2), ?)
311.84/291.54	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
311.84/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
311.84/291.54	
311.84/291.54	
311.84/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
311.84/291.54	
311.84/291.54	(0) CpxTRS
311.84/291.54	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
311.84/291.54	(2) CpxTRS
311.84/291.54	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
311.84/291.54	(4) typed CpxTrs
311.84/291.54	(5) OrderProof [LOWER BOUND(ID), 0 ms]
311.84/291.54	(6) typed CpxTrs
311.84/291.54	(7) RewriteLemmaProof [LOWER BOUND(ID), 295 ms]
311.84/291.54	(8) BEST
311.84/291.54	    (9) proven lower bound
311.84/291.54	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
311.84/291.54	        (11) BOUNDS(n^1, INF)
311.84/291.54	    (12) typed CpxTrs
311.84/291.54	        (13) RewriteLemmaProof [LOWER BOUND(ID), 73 ms]
311.84/291.54	        (14) BEST
311.84/291.54	            (15) proven lower bound
311.84/291.54	                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
311.84/291.54	                (17) BOUNDS(n^2, INF)
311.84/291.54	            (18) typed CpxTrs
311.84/291.54	                (19) RewriteLemmaProof [LOWER BOUND(ID), 65 ms]
311.84/291.54	                (20) typed CpxTrs
311.84/291.54	                (21) RewriteLemmaProof [LOWER BOUND(ID), 27 ms]
311.84/291.54	                (22) typed CpxTrs
311.84/291.54	
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(0)
311.84/291.54	Obligation:
311.84/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
311.84/291.54	
311.84/291.54	
311.84/291.54	The TRS R consists of the following rules:
311.84/291.54	
311.84/291.54	   +(0, y) -> y
311.84/291.54	   +(s(x), y) -> s(+(x, y))
311.84/291.54	   *(x, 0) -> 0
311.84/291.54	   *(x, s(y)) -> +(x, *(x, y))
311.84/291.54	   twice(0) -> 0
311.84/291.54	   twice(s(x)) -> s(s(twice(x)))
311.84/291.54	   -(x, 0) -> x
311.84/291.54	   -(s(x), s(y)) -> -(x, y)
311.84/291.54	   f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0)))))
311.84/291.54	
311.84/291.54	S is empty.
311.84/291.54	Rewrite Strategy: FULL
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(1) RenamingProof (BOTH BOUNDS(ID, ID))
311.84/291.54	Renamed function symbols to avoid clashes with predefined symbol.
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(2)
311.84/291.54	Obligation:
311.84/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
311.84/291.54	
311.84/291.54	
311.84/291.54	The TRS R consists of the following rules:
311.84/291.54	
311.84/291.54	   +'(0', y) -> y
311.84/291.54	   +'(s(x), y) -> s(+'(x, y))
311.84/291.54	   *'(x, 0') -> 0'
311.84/291.54	   *'(x, s(y)) -> +'(x, *'(x, y))
311.84/291.54	   twice(0') -> 0'
311.84/291.54	   twice(s(x)) -> s(s(twice(x)))
311.84/291.54	   -(x, 0') -> x
311.84/291.54	   -(s(x), s(y)) -> -(x, y)
311.84/291.54	   f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))
311.84/291.54	
311.84/291.54	S is empty.
311.84/291.54	Rewrite Strategy: FULL
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
311.84/291.54	Infered types.
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(4)
311.84/291.54	Obligation:
311.84/291.54	TRS:
311.84/291.54	Rules:
311.84/291.54	+'(0', y) -> y
311.84/291.54	+'(s(x), y) -> s(+'(x, y))
311.84/291.54	*'(x, 0') -> 0'
311.84/291.54	*'(x, s(y)) -> +'(x, *'(x, y))
311.84/291.54	twice(0') -> 0'
311.84/291.54	twice(s(x)) -> s(s(twice(x)))
311.84/291.54	-(x, 0') -> x
311.84/291.54	-(s(x), s(y)) -> -(x, y)
311.84/291.54	f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))
311.84/291.54	
311.84/291.54	Types:
311.84/291.54	+' :: 0':s -> 0':s -> 0':s
311.84/291.54	0' :: 0':s
311.84/291.54	s :: 0':s -> 0':s
311.84/291.54	*' :: 0':s -> 0':s -> 0':s
311.84/291.54	twice :: 0':s -> 0':s
311.84/291.54	- :: 0':s -> 0':s -> 0':s
311.84/291.54	f :: 0':s -> f
311.84/291.54	hole_0':s1_0 :: 0':s
311.84/291.54	hole_f2_0 :: f
311.84/291.54	gen_0':s3_0 :: Nat -> 0':s
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(5) OrderProof (LOWER BOUND(ID))
311.84/291.54	Heuristically decided to analyse the following defined symbols:
311.84/291.54	+', *', twice, -, f
311.84/291.54	
311.84/291.54	They will be analysed ascendingly in the following order:
311.84/291.54	+' < *'
311.84/291.54	+' < f
311.84/291.54	*' < f
311.84/291.54	- < f
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(6)
311.84/291.54	Obligation:
311.84/291.54	TRS:
311.84/291.54	Rules:
311.84/291.54	+'(0', y) -> y
311.84/291.54	+'(s(x), y) -> s(+'(x, y))
311.84/291.54	*'(x, 0') -> 0'
311.84/291.54	*'(x, s(y)) -> +'(x, *'(x, y))
311.84/291.54	twice(0') -> 0'
311.84/291.54	twice(s(x)) -> s(s(twice(x)))
311.84/291.54	-(x, 0') -> x
311.84/291.54	-(s(x), s(y)) -> -(x, y)
311.84/291.54	f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))
311.84/291.54	
311.84/291.54	Types:
311.84/291.54	+' :: 0':s -> 0':s -> 0':s
311.84/291.54	0' :: 0':s
311.84/291.54	s :: 0':s -> 0':s
311.84/291.54	*' :: 0':s -> 0':s -> 0':s
311.84/291.54	twice :: 0':s -> 0':s
311.84/291.54	- :: 0':s -> 0':s -> 0':s
311.84/291.54	f :: 0':s -> f
311.84/291.54	hole_0':s1_0 :: 0':s
311.84/291.54	hole_f2_0 :: f
311.84/291.54	gen_0':s3_0 :: Nat -> 0':s
311.84/291.54	
311.84/291.54	
311.84/291.54	Generator Equations:
311.84/291.54	gen_0':s3_0(0) <=> 0'
311.84/291.54	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.84/291.54	
311.84/291.54	
311.84/291.54	The following defined symbols remain to be analysed:
311.84/291.54	+', *', twice, -, f
311.84/291.54	
311.84/291.54	They will be analysed ascendingly in the following order:
311.84/291.54	+' < *'
311.84/291.54	+' < f
311.84/291.54	*' < f
311.84/291.54	- < f
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(7) RewriteLemmaProof (LOWER BOUND(ID))
311.84/291.54	Proved the following rewrite lemma:
311.84/291.54	+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
311.84/291.54	
311.84/291.54	Induction Base:
311.84/291.54	+'(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
311.84/291.54	gen_0':s3_0(b)
311.84/291.54	
311.84/291.54	Induction Step:
311.84/291.54	+'(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
311.84/291.54	s(+'(gen_0':s3_0(n5_0), gen_0':s3_0(b))) ->_IH
311.84/291.54	s(gen_0':s3_0(+(b, c6_0)))
311.84/291.54	
311.84/291.54	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(8)
311.84/291.54	Complex Obligation (BEST)
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(9)
311.84/291.54	Obligation:
311.84/291.54	Proved the lower bound n^1 for the following obligation:
311.84/291.54	
311.84/291.54	TRS:
311.84/291.54	Rules:
311.84/291.54	+'(0', y) -> y
311.84/291.54	+'(s(x), y) -> s(+'(x, y))
311.84/291.54	*'(x, 0') -> 0'
311.84/291.54	*'(x, s(y)) -> +'(x, *'(x, y))
311.84/291.54	twice(0') -> 0'
311.84/291.54	twice(s(x)) -> s(s(twice(x)))
311.84/291.54	-(x, 0') -> x
311.84/291.54	-(s(x), s(y)) -> -(x, y)
311.84/291.54	f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))
311.84/291.54	
311.84/291.54	Types:
311.84/291.54	+' :: 0':s -> 0':s -> 0':s
311.84/291.54	0' :: 0':s
311.84/291.54	s :: 0':s -> 0':s
311.84/291.54	*' :: 0':s -> 0':s -> 0':s
311.84/291.54	twice :: 0':s -> 0':s
311.84/291.54	- :: 0':s -> 0':s -> 0':s
311.84/291.54	f :: 0':s -> f
311.84/291.54	hole_0':s1_0 :: 0':s
311.84/291.54	hole_f2_0 :: f
311.84/291.54	gen_0':s3_0 :: Nat -> 0':s
311.84/291.54	
311.84/291.54	
311.84/291.54	Generator Equations:
311.84/291.54	gen_0':s3_0(0) <=> 0'
311.84/291.54	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.84/291.54	
311.84/291.54	
311.84/291.54	The following defined symbols remain to be analysed:
311.84/291.54	+', *', twice, -, f
311.84/291.54	
311.84/291.54	They will be analysed ascendingly in the following order:
311.84/291.54	+' < *'
311.84/291.54	+' < f
311.84/291.54	*' < f
311.84/291.54	- < f
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(10) LowerBoundPropagationProof (FINISHED)
311.84/291.54	Propagated lower bound.
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(11)
311.84/291.54	BOUNDS(n^1, INF)
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(12)
311.84/291.54	Obligation:
311.84/291.54	TRS:
311.84/291.54	Rules:
311.84/291.54	+'(0', y) -> y
311.84/291.54	+'(s(x), y) -> s(+'(x, y))
311.84/291.54	*'(x, 0') -> 0'
311.84/291.54	*'(x, s(y)) -> +'(x, *'(x, y))
311.84/291.54	twice(0') -> 0'
311.84/291.54	twice(s(x)) -> s(s(twice(x)))
311.84/291.54	-(x, 0') -> x
311.84/291.54	-(s(x), s(y)) -> -(x, y)
311.84/291.54	f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))
311.84/291.54	
311.84/291.54	Types:
311.84/291.54	+' :: 0':s -> 0':s -> 0':s
311.84/291.54	0' :: 0':s
311.84/291.54	s :: 0':s -> 0':s
311.84/291.54	*' :: 0':s -> 0':s -> 0':s
311.84/291.54	twice :: 0':s -> 0':s
311.84/291.54	- :: 0':s -> 0':s -> 0':s
311.84/291.54	f :: 0':s -> f
311.84/291.54	hole_0':s1_0 :: 0':s
311.84/291.54	hole_f2_0 :: f
311.84/291.54	gen_0':s3_0 :: Nat -> 0':s
311.84/291.54	
311.84/291.54	
311.84/291.54	Lemmas:
311.84/291.54	+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
311.84/291.54	
311.84/291.54	
311.84/291.54	Generator Equations:
311.84/291.54	gen_0':s3_0(0) <=> 0'
311.84/291.54	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.84/291.54	
311.84/291.54	
311.84/291.54	The following defined symbols remain to be analysed:
311.84/291.54	*', twice, -, f
311.84/291.54	
311.84/291.54	They will be analysed ascendingly in the following order:
311.84/291.54	*' < f
311.84/291.54	- < f
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(13) RewriteLemmaProof (LOWER BOUND(ID))
311.84/291.54	Proved the following rewrite lemma:
311.84/291.54	*'(gen_0':s3_0(a), gen_0':s3_0(n494_0)) -> gen_0':s3_0(*(n494_0, a)), rt in Omega(1 + a*n494_0 + n494_0)
311.84/291.54	
311.84/291.54	Induction Base:
311.84/291.54	*'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1)
311.84/291.54	0'
311.84/291.54	
311.84/291.54	Induction Step:
311.84/291.54	*'(gen_0':s3_0(a), gen_0':s3_0(+(n494_0, 1))) ->_R^Omega(1)
311.84/291.54	+'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n494_0))) ->_IH
311.84/291.54	+'(gen_0':s3_0(a), gen_0':s3_0(*(c495_0, a))) ->_L^Omega(1 + a)
311.84/291.54	gen_0':s3_0(+(a, *(n494_0, a)))
311.84/291.54	
311.84/291.54	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(14)
311.84/291.54	Complex Obligation (BEST)
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(15)
311.84/291.54	Obligation:
311.84/291.54	Proved the lower bound n^2 for the following obligation:
311.84/291.54	
311.84/291.54	TRS:
311.84/291.54	Rules:
311.84/291.54	+'(0', y) -> y
311.84/291.54	+'(s(x), y) -> s(+'(x, y))
311.84/291.54	*'(x, 0') -> 0'
311.84/291.54	*'(x, s(y)) -> +'(x, *'(x, y))
311.84/291.54	twice(0') -> 0'
311.84/291.54	twice(s(x)) -> s(s(twice(x)))
311.84/291.54	-(x, 0') -> x
311.84/291.54	-(s(x), s(y)) -> -(x, y)
311.84/291.54	f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))
311.84/291.54	
311.84/291.54	Types:
311.84/291.54	+' :: 0':s -> 0':s -> 0':s
311.84/291.54	0' :: 0':s
311.84/291.54	s :: 0':s -> 0':s
311.84/291.54	*' :: 0':s -> 0':s -> 0':s
311.84/291.54	twice :: 0':s -> 0':s
311.84/291.54	- :: 0':s -> 0':s -> 0':s
311.84/291.54	f :: 0':s -> f
311.84/291.54	hole_0':s1_0 :: 0':s
311.84/291.54	hole_f2_0 :: f
311.84/291.54	gen_0':s3_0 :: Nat -> 0':s
311.84/291.54	
311.84/291.54	
311.84/291.54	Lemmas:
311.84/291.54	+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
311.84/291.54	
311.84/291.54	
311.84/291.54	Generator Equations:
311.84/291.54	gen_0':s3_0(0) <=> 0'
311.84/291.54	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.84/291.54	
311.84/291.54	
311.84/291.54	The following defined symbols remain to be analysed:
311.84/291.54	*', twice, -, f
311.84/291.54	
311.84/291.54	They will be analysed ascendingly in the following order:
311.84/291.54	*' < f
311.84/291.54	- < f
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(16) LowerBoundPropagationProof (FINISHED)
311.84/291.54	Propagated lower bound.
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(17)
311.84/291.54	BOUNDS(n^2, INF)
311.84/291.54	
311.84/291.54	----------------------------------------
311.84/291.54	
311.84/291.54	(18)
311.84/291.54	Obligation:
311.84/291.54	TRS:
311.84/291.54	Rules:
311.84/291.54	+'(0', y) -> y
311.84/291.55	+'(s(x), y) -> s(+'(x, y))
311.84/291.55	*'(x, 0') -> 0'
311.84/291.55	*'(x, s(y)) -> +'(x, *'(x, y))
311.84/291.55	twice(0') -> 0'
311.84/291.55	twice(s(x)) -> s(s(twice(x)))
311.84/291.55	-(x, 0') -> x
311.84/291.55	-(s(x), s(y)) -> -(x, y)
311.84/291.55	f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))
311.84/291.55	
311.84/291.55	Types:
311.84/291.55	+' :: 0':s -> 0':s -> 0':s
311.84/291.55	0' :: 0':s
311.84/291.55	s :: 0':s -> 0':s
311.84/291.55	*' :: 0':s -> 0':s -> 0':s
311.84/291.55	twice :: 0':s -> 0':s
311.84/291.55	- :: 0':s -> 0':s -> 0':s
311.84/291.55	f :: 0':s -> f
311.84/291.55	hole_0':s1_0 :: 0':s
311.84/291.55	hole_f2_0 :: f
311.84/291.55	gen_0':s3_0 :: Nat -> 0':s
311.84/291.55	
311.84/291.55	
311.84/291.55	Lemmas:
311.84/291.55	+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
311.84/291.55	*'(gen_0':s3_0(a), gen_0':s3_0(n494_0)) -> gen_0':s3_0(*(n494_0, a)), rt in Omega(1 + a*n494_0 + n494_0)
311.84/291.55	
311.84/291.55	
311.84/291.55	Generator Equations:
311.84/291.55	gen_0':s3_0(0) <=> 0'
311.84/291.55	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.84/291.55	
311.84/291.55	
311.84/291.55	The following defined symbols remain to be analysed:
311.84/291.55	twice, -, f
311.84/291.55	
311.84/291.55	They will be analysed ascendingly in the following order:
311.84/291.55	- < f
311.84/291.55	
311.84/291.55	----------------------------------------
311.84/291.55	
311.84/291.55	(19) RewriteLemmaProof (LOWER BOUND(ID))
311.84/291.55	Proved the following rewrite lemma:
311.84/291.55	twice(gen_0':s3_0(n1110_0)) -> gen_0':s3_0(*(2, n1110_0)), rt in Omega(1 + n1110_0)
311.84/291.55	
311.84/291.55	Induction Base:
311.84/291.55	twice(gen_0':s3_0(0)) ->_R^Omega(1)
311.84/291.55	0'
311.84/291.55	
311.84/291.55	Induction Step:
311.84/291.55	twice(gen_0':s3_0(+(n1110_0, 1))) ->_R^Omega(1)
311.84/291.55	s(s(twice(gen_0':s3_0(n1110_0)))) ->_IH
311.84/291.55	s(s(gen_0':s3_0(*(2, c1111_0))))
311.84/291.55	
311.84/291.55	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
311.84/291.55	----------------------------------------
311.84/291.55	
311.84/291.55	(20)
311.84/291.55	Obligation:
311.84/291.55	TRS:
311.84/291.55	Rules:
311.84/291.55	+'(0', y) -> y
311.84/291.55	+'(s(x), y) -> s(+'(x, y))
311.84/291.55	*'(x, 0') -> 0'
311.84/291.55	*'(x, s(y)) -> +'(x, *'(x, y))
311.84/291.55	twice(0') -> 0'
311.84/291.55	twice(s(x)) -> s(s(twice(x)))
311.84/291.55	-(x, 0') -> x
311.84/291.55	-(s(x), s(y)) -> -(x, y)
311.84/291.55	f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))
311.84/291.55	
311.84/291.55	Types:
311.84/291.55	+' :: 0':s -> 0':s -> 0':s
311.84/291.55	0' :: 0':s
311.84/291.55	s :: 0':s -> 0':s
311.84/291.55	*' :: 0':s -> 0':s -> 0':s
311.84/291.55	twice :: 0':s -> 0':s
311.84/291.55	- :: 0':s -> 0':s -> 0':s
311.84/291.55	f :: 0':s -> f
311.84/291.55	hole_0':s1_0 :: 0':s
311.84/291.55	hole_f2_0 :: f
311.84/291.55	gen_0':s3_0 :: Nat -> 0':s
311.84/291.55	
311.84/291.55	
311.84/291.55	Lemmas:
311.84/291.55	+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
311.84/291.55	*'(gen_0':s3_0(a), gen_0':s3_0(n494_0)) -> gen_0':s3_0(*(n494_0, a)), rt in Omega(1 + a*n494_0 + n494_0)
311.84/291.55	twice(gen_0':s3_0(n1110_0)) -> gen_0':s3_0(*(2, n1110_0)), rt in Omega(1 + n1110_0)
311.84/291.55	
311.84/291.55	
311.84/291.55	Generator Equations:
311.84/291.55	gen_0':s3_0(0) <=> 0'
311.84/291.55	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.84/291.55	
311.84/291.55	
311.84/291.55	The following defined symbols remain to be analysed:
311.84/291.55	-, f
311.84/291.55	
311.84/291.55	They will be analysed ascendingly in the following order:
311.84/291.55	- < f
311.84/291.55	
311.84/291.55	----------------------------------------
311.84/291.55	
311.84/291.55	(21) RewriteLemmaProof (LOWER BOUND(ID))
311.84/291.55	Proved the following rewrite lemma:
311.84/291.55	-(gen_0':s3_0(n1358_0), gen_0':s3_0(n1358_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1358_0)
311.84/291.55	
311.84/291.55	Induction Base:
311.84/291.55	-(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
311.84/291.55	gen_0':s3_0(0)
311.84/291.55	
311.84/291.55	Induction Step:
311.84/291.55	-(gen_0':s3_0(+(n1358_0, 1)), gen_0':s3_0(+(n1358_0, 1))) ->_R^Omega(1)
311.84/291.55	-(gen_0':s3_0(n1358_0), gen_0':s3_0(n1358_0)) ->_IH
311.84/291.55	gen_0':s3_0(0)
311.84/291.55	
311.84/291.55	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
311.84/291.55	----------------------------------------
311.84/291.55	
311.84/291.55	(22)
311.84/291.55	Obligation:
311.84/291.55	TRS:
311.84/291.55	Rules:
311.84/291.55	+'(0', y) -> y
311.84/291.55	+'(s(x), y) -> s(+'(x, y))
311.84/291.55	*'(x, 0') -> 0'
311.84/291.55	*'(x, s(y)) -> +'(x, *'(x, y))
311.84/291.55	twice(0') -> 0'
311.84/291.55	twice(s(x)) -> s(s(twice(x)))
311.84/291.55	-(x, 0') -> x
311.84/291.55	-(s(x), s(y)) -> -(x, y)
311.84/291.55	f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))
311.84/291.55	
311.84/291.55	Types:
311.84/291.55	+' :: 0':s -> 0':s -> 0':s
311.84/291.55	0' :: 0':s
311.84/291.55	s :: 0':s -> 0':s
311.84/291.55	*' :: 0':s -> 0':s -> 0':s
311.84/291.55	twice :: 0':s -> 0':s
311.84/291.55	- :: 0':s -> 0':s -> 0':s
311.84/291.55	f :: 0':s -> f
311.84/291.55	hole_0':s1_0 :: 0':s
311.84/291.55	hole_f2_0 :: f
311.84/291.55	gen_0':s3_0 :: Nat -> 0':s
311.84/291.55	
311.84/291.55	
311.84/291.55	Lemmas:
311.84/291.55	+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
311.84/291.55	*'(gen_0':s3_0(a), gen_0':s3_0(n494_0)) -> gen_0':s3_0(*(n494_0, a)), rt in Omega(1 + a*n494_0 + n494_0)
311.84/291.55	twice(gen_0':s3_0(n1110_0)) -> gen_0':s3_0(*(2, n1110_0)), rt in Omega(1 + n1110_0)
311.84/291.55	-(gen_0':s3_0(n1358_0), gen_0':s3_0(n1358_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1358_0)
311.84/291.55	
311.84/291.55	
311.84/291.55	Generator Equations:
311.84/291.55	gen_0':s3_0(0) <=> 0'
311.84/291.55	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.84/291.55	
311.84/291.55	
311.84/291.55	The following defined symbols remain to be analysed:
311.84/291.55	f
311.93/291.59	EOF