9.84/3.36	WORST_CASE(NON_POLY, ?)
9.84/3.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
9.84/3.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.84/3.37	
9.84/3.37	
9.84/3.37	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
9.84/3.37	
9.84/3.37	(0) CpxTRS
9.84/3.37	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
9.84/3.37	(2) CpxTRS
9.84/3.37	(3) SlicingProof [LOWER BOUND(ID), 0 ms]
9.84/3.37	(4) CpxTRS
9.84/3.37	(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
9.84/3.37	(6) typed CpxTrs
9.84/3.37	(7) OrderProof [LOWER BOUND(ID), 0 ms]
9.84/3.37	(8) typed CpxTrs
9.84/3.37	(9) RewriteLemmaProof [FINISHED, 836 ms]
9.84/3.37	(10) BOUNDS(EXP, INF)
9.84/3.37	
9.84/3.37	
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(0)
9.84/3.37	Obligation:
9.84/3.37	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
9.84/3.37	
9.84/3.37	
9.84/3.37	The TRS R consists of the following rules:
9.84/3.37	
9.84/3.37	   f(g(X)) -> g(f(f(X)))
9.84/3.37	   f(h(X)) -> h(g(X))
9.84/3.37	
9.84/3.37	S is empty.
9.84/3.37	Rewrite Strategy: INNERMOST
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(1) RenamingProof (BOTH BOUNDS(ID, ID))
9.84/3.37	Renamed function symbols to avoid clashes with predefined symbol.
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(2)
9.84/3.37	Obligation:
9.84/3.37	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
9.84/3.37	
9.84/3.37	
9.84/3.37	The TRS R consists of the following rules:
9.84/3.37	
9.84/3.37	   f(g(X)) -> g(f(f(X)))
9.84/3.37	   f(h(X)) -> h(g(X))
9.84/3.37	
9.84/3.37	S is empty.
9.84/3.37	Rewrite Strategy: INNERMOST
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(3) SlicingProof (LOWER BOUND(ID))
9.84/3.37	Sliced the following arguments:
9.84/3.37	h/0
9.84/3.37	
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(4)
9.84/3.37	Obligation:
9.84/3.37	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
9.84/3.37	
9.84/3.37	
9.84/3.37	The TRS R consists of the following rules:
9.84/3.37	
9.84/3.37	   f(g(X)) -> g(f(f(X)))
9.84/3.37	   f(h) -> h
9.84/3.37	
9.84/3.37	S is empty.
9.84/3.37	Rewrite Strategy: INNERMOST
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
9.84/3.37	Infered types.
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(6)
9.84/3.37	Obligation:
9.84/3.37	Innermost TRS:
9.84/3.37	Rules:
9.84/3.37	f(g(X)) -> g(f(f(X)))
9.84/3.37	f(h) -> h
9.84/3.37	
9.84/3.37	Types:
9.84/3.37	f :: g:h -> g:h
9.84/3.37	g :: g:h -> g:h
9.84/3.37	h :: g:h
9.84/3.37	hole_g:h1_0 :: g:h
9.84/3.37	gen_g:h2_0 :: Nat -> g:h
9.84/3.37	
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(7) OrderProof (LOWER BOUND(ID))
9.84/3.37	Heuristically decided to analyse the following defined symbols:
9.84/3.37	f
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(8)
9.84/3.37	Obligation:
9.84/3.37	Innermost TRS:
9.84/3.37	Rules:
9.84/3.37	f(g(X)) -> g(f(f(X)))
9.84/3.37	f(h) -> h
9.84/3.37	
9.84/3.37	Types:
9.84/3.37	f :: g:h -> g:h
9.84/3.37	g :: g:h -> g:h
9.84/3.37	h :: g:h
9.84/3.37	hole_g:h1_0 :: g:h
9.84/3.37	gen_g:h2_0 :: Nat -> g:h
9.84/3.37	
9.84/3.37	
9.84/3.37	Generator Equations:
9.84/3.37	gen_g:h2_0(0) <=> h
9.84/3.37	gen_g:h2_0(+(x, 1)) <=> g(gen_g:h2_0(x))
9.84/3.37	
9.84/3.37	
9.84/3.37	The following defined symbols remain to be analysed:
9.84/3.37	f
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(9) RewriteLemmaProof (FINISHED)
9.84/3.37	Proved the following rewrite lemma:
9.84/3.37	f(gen_g:h2_0(n4_0)) -> gen_g:h2_0(n4_0), rt in Omega(EXP)
9.84/3.37	
9.84/3.37	Induction Base:
9.84/3.37	f(gen_g:h2_0(0)) ->_R^Omega(1)
9.84/3.37	h
9.84/3.37	
9.84/3.37	Induction Step:
9.84/3.37	f(gen_g:h2_0(+(n4_0, 1))) ->_R^Omega(1)
9.84/3.37	g(f(f(gen_g:h2_0(n4_0)))) ->_IH
9.84/3.37	g(f(gen_g:h2_0(c5_0))) ->_IH
9.84/3.37	g(gen_g:h2_0(c5_0))
9.84/3.37	
9.84/3.37	We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP
9.84/3.37	----------------------------------------
9.84/3.37	
9.84/3.37	(10)
9.84/3.37	BOUNDS(EXP, INF)
10.06/3.43	EOF