7.43/2.63	WORST_CASE(NON_POLY, ?)
7.43/2.64	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
7.43/2.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.43/2.64	
7.43/2.64	
7.43/2.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
7.43/2.64	
7.43/2.64	(0) CpxTRS
7.43/2.64	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
7.43/2.64	(2) CpxTRS
7.43/2.64	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
7.43/2.64	(4) typed CpxTrs
7.43/2.64	(5) OrderProof [LOWER BOUND(ID), 0 ms]
7.43/2.64	(6) typed CpxTrs
7.43/2.64	(7) RewriteLemmaProof [FINISHED, 376 ms]
7.43/2.64	(8) BOUNDS(EXP, INF)
7.43/2.64	
7.43/2.64	
7.43/2.64	----------------------------------------
7.43/2.64	
7.43/2.64	(0)
7.43/2.64	Obligation:
7.43/2.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
7.43/2.64	
7.43/2.64	
7.43/2.64	The TRS R consists of the following rules:
7.43/2.64	
7.43/2.64	   f(s(x)) -> s(f(f(p(s(x)))))
7.43/2.64	   f(0) -> 0
7.43/2.64	   p(s(x)) -> x
7.43/2.64	
7.43/2.64	S is empty.
7.43/2.64	Rewrite Strategy: INNERMOST
7.43/2.64	----------------------------------------
7.43/2.64	
7.43/2.64	(1) RenamingProof (BOTH BOUNDS(ID, ID))
7.43/2.64	Renamed function symbols to avoid clashes with predefined symbol.
7.43/2.64	----------------------------------------
7.43/2.64	
7.43/2.64	(2)
7.43/2.64	Obligation:
7.43/2.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
7.43/2.64	
7.43/2.64	
7.43/2.64	The TRS R consists of the following rules:
7.43/2.64	
7.43/2.64	   f(s(x)) -> s(f(f(p(s(x)))))
7.43/2.64	   f(0') -> 0'
7.43/2.64	   p(s(x)) -> x
7.43/2.64	
7.43/2.64	S is empty.
7.43/2.64	Rewrite Strategy: INNERMOST
7.43/2.64	----------------------------------------
7.43/2.64	
7.43/2.64	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
7.43/2.64	Infered types.
7.43/2.64	----------------------------------------
7.43/2.64	
7.43/2.64	(4)
7.43/2.64	Obligation:
7.43/2.64	Innermost TRS:
7.43/2.64	Rules:
7.43/2.64	f(s(x)) -> s(f(f(p(s(x)))))
7.43/2.64	f(0') -> 0'
7.43/2.64	p(s(x)) -> x
7.43/2.64	
7.43/2.64	Types:
7.43/2.64	f :: s:0' -> s:0'
7.43/2.64	s :: s:0' -> s:0'
7.43/2.64	p :: s:0' -> s:0'
7.43/2.64	0' :: s:0'
7.43/2.64	hole_s:0'1_0 :: s:0'
7.43/2.64	gen_s:0'2_0 :: Nat -> s:0'
7.43/2.64	
7.43/2.64	----------------------------------------
7.43/2.64	
7.43/2.64	(5) OrderProof (LOWER BOUND(ID))
7.43/2.64	Heuristically decided to analyse the following defined symbols:
7.43/2.64	f
7.43/2.64	----------------------------------------
7.43/2.64	
7.43/2.64	(6)
7.43/2.64	Obligation:
7.43/2.64	Innermost TRS:
7.43/2.64	Rules:
7.43/2.64	f(s(x)) -> s(f(f(p(s(x)))))
7.43/2.64	f(0') -> 0'
7.43/2.64	p(s(x)) -> x
7.43/2.64	
7.43/2.64	Types:
7.43/2.64	f :: s:0' -> s:0'
7.43/2.64	s :: s:0' -> s:0'
7.43/2.64	p :: s:0' -> s:0'
7.43/2.64	0' :: s:0'
7.43/2.64	hole_s:0'1_0 :: s:0'
7.43/2.64	gen_s:0'2_0 :: Nat -> s:0'
7.43/2.64	
7.43/2.64	
7.43/2.64	Generator Equations:
7.43/2.64	gen_s:0'2_0(0) <=> 0'
7.43/2.64	gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))
7.43/2.64	
7.43/2.64	
7.43/2.64	The following defined symbols remain to be analysed:
7.43/2.64	f
7.43/2.64	----------------------------------------
7.43/2.64	
7.43/2.64	(7) RewriteLemmaProof (FINISHED)
7.43/2.64	Proved the following rewrite lemma:
7.43/2.64	f(gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(n4_0), rt in Omega(EXP)
7.43/2.64	
7.43/2.64	Induction Base:
7.43/2.64	f(gen_s:0'2_0(0)) ->_R^Omega(1)
7.43/2.64	0'
7.43/2.64	
7.43/2.64	Induction Step:
7.43/2.64	f(gen_s:0'2_0(+(n4_0, 1))) ->_R^Omega(1)
7.43/2.64	s(f(f(p(s(gen_s:0'2_0(n4_0)))))) ->_R^Omega(1)
7.43/2.64	s(f(f(gen_s:0'2_0(n4_0)))) ->_IH
7.43/2.64	s(f(gen_s:0'2_0(c5_0))) ->_IH
7.43/2.64	s(gen_s:0'2_0(c5_0))
7.43/2.64	
7.43/2.64	We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP
7.43/2.64	----------------------------------------
7.43/2.64	
7.43/2.64	(8)
7.43/2.64	BOUNDS(EXP, INF)
7.43/2.69	EOF