7.23/2.61	WORST_CASE(NON_POLY, ?)
7.23/2.62	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.23/2.62	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.23/2.62	
7.23/2.62	
7.23/2.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
7.23/2.62	
7.23/2.62	(0) CpxTRS
7.23/2.62	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
7.23/2.62	(2) CpxTRS
7.23/2.62	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
7.23/2.62	(4) typed CpxTrs
7.23/2.62	(5) OrderProof [LOWER BOUND(ID), 4 ms]
7.23/2.62	(6) typed CpxTrs
7.23/2.62	(7) RewriteLemmaProof [FINISHED, 466 ms]
7.23/2.62	(8) BOUNDS(EXP, INF)
7.23/2.62	
7.23/2.62	
7.23/2.62	----------------------------------------
7.23/2.62	
7.23/2.62	(0)
7.23/2.62	Obligation:
7.23/2.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
7.23/2.62	
7.23/2.62	
7.23/2.62	The TRS R consists of the following rules:
7.23/2.62	
7.23/2.62	   f(s(x)) -> s(f(f(p(s(x)))))
7.23/2.62	   f(0) -> 0
7.23/2.62	   p(s(x)) -> x
7.23/2.62	
7.23/2.62	S is empty.
7.23/2.62	Rewrite Strategy: FULL
7.23/2.62	----------------------------------------
7.23/2.62	
7.23/2.62	(1) RenamingProof (BOTH BOUNDS(ID, ID))
7.23/2.62	Renamed function symbols to avoid clashes with predefined symbol.
7.23/2.62	----------------------------------------
7.23/2.62	
7.23/2.62	(2)
7.23/2.62	Obligation:
7.23/2.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
7.23/2.62	
7.23/2.62	
7.23/2.62	The TRS R consists of the following rules:
7.23/2.62	
7.23/2.62	   f(s(x)) -> s(f(f(p(s(x)))))
7.23/2.62	   f(0') -> 0'
7.23/2.62	   p(s(x)) -> x
7.23/2.62	
7.23/2.62	S is empty.
7.23/2.62	Rewrite Strategy: FULL
7.23/2.62	----------------------------------------
7.23/2.62	
7.23/2.62	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
7.23/2.62	Infered types.
7.23/2.62	----------------------------------------
7.23/2.62	
7.23/2.62	(4)
7.23/2.62	Obligation:
7.23/2.62	TRS:
7.23/2.62	Rules:
7.23/2.62	f(s(x)) -> s(f(f(p(s(x)))))
7.23/2.62	f(0') -> 0'
7.23/2.62	p(s(x)) -> x
7.23/2.62	
7.23/2.62	Types:
7.23/2.62	f :: s:0' -> s:0'
7.23/2.62	s :: s:0' -> s:0'
7.23/2.62	p :: s:0' -> s:0'
7.23/2.62	0' :: s:0'
7.23/2.62	hole_s:0'1_0 :: s:0'
7.23/2.62	gen_s:0'2_0 :: Nat -> s:0'
7.23/2.62	
7.23/2.62	----------------------------------------
7.23/2.62	
7.23/2.62	(5) OrderProof (LOWER BOUND(ID))
7.23/2.62	Heuristically decided to analyse the following defined symbols:
7.23/2.62	f
7.23/2.62	----------------------------------------
7.23/2.62	
7.23/2.62	(6)
7.23/2.62	Obligation:
7.23/2.62	TRS:
7.23/2.62	Rules:
7.23/2.62	f(s(x)) -> s(f(f(p(s(x)))))
7.23/2.62	f(0') -> 0'
7.23/2.62	p(s(x)) -> x
7.23/2.62	
7.23/2.62	Types:
7.23/2.62	f :: s:0' -> s:0'
7.23/2.62	s :: s:0' -> s:0'
7.23/2.62	p :: s:0' -> s:0'
7.23/2.62	0' :: s:0'
7.23/2.62	hole_s:0'1_0 :: s:0'
7.23/2.62	gen_s:0'2_0 :: Nat -> s:0'
7.23/2.62	
7.23/2.62	
7.23/2.62	Generator Equations:
7.23/2.62	gen_s:0'2_0(0) <=> 0'
7.23/2.62	gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))
7.23/2.62	
7.23/2.62	
7.23/2.62	The following defined symbols remain to be analysed:
7.23/2.62	f
7.23/2.62	----------------------------------------
7.23/2.62	
7.23/2.62	(7) RewriteLemmaProof (FINISHED)
7.23/2.62	Proved the following rewrite lemma:
7.23/2.62	f(gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(n4_0), rt in Omega(EXP)
7.23/2.62	
7.23/2.62	Induction Base:
7.23/2.62	f(gen_s:0'2_0(0)) ->_R^Omega(1)
7.23/2.62	0'
7.23/2.62	
7.23/2.62	Induction Step:
7.23/2.62	f(gen_s:0'2_0(+(n4_0, 1))) ->_R^Omega(1)
7.23/2.62	s(f(f(p(s(gen_s:0'2_0(n4_0)))))) ->_R^Omega(1)
7.23/2.62	s(f(f(gen_s:0'2_0(n4_0)))) ->_IH
7.23/2.62	s(f(gen_s:0'2_0(c5_0))) ->_IH
7.23/2.62	s(gen_s:0'2_0(c5_0))
7.23/2.62	
7.23/2.62	We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP
7.23/2.62	----------------------------------------
7.23/2.62	
7.23/2.62	(8)
7.23/2.62	BOUNDS(EXP, INF)
7.43/2.66	EOF