779.70/291.48	WORST_CASE(Omega(n^1), ?)
779.70/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
779.70/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
779.70/291.49	
779.70/291.49	
779.70/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
779.70/291.49	
779.70/291.49	(0) CpxTRS
779.70/291.49	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
779.70/291.49	(2) CpxTRS
779.70/291.49	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
779.70/291.49	(4) typed CpxTrs
779.70/291.49	(5) OrderProof [LOWER BOUND(ID), 0 ms]
779.70/291.49	(6) typed CpxTrs
779.70/291.49	(7) RewriteLemmaProof [LOWER BOUND(ID), 463 ms]
779.70/291.49	(8) BEST
779.70/291.49	    (9) proven lower bound
779.70/291.49	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
779.70/291.49	        (11) BOUNDS(n^1, INF)
779.70/291.49	    (12) typed CpxTrs
779.70/291.49	
779.70/291.49	
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(0)
779.70/291.49	Obligation:
779.70/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
779.70/291.49	
779.70/291.49	
779.70/291.49	The TRS R consists of the following rules:
779.70/291.49	
779.70/291.49	   f(g(x)) -> g(f(f(x)))
779.70/291.49	   f(h(x)) -> h(g(x))
779.70/291.49	   f'(s(x), y, y) -> f'(y, x, s(x))
779.70/291.49	
779.70/291.49	S is empty.
779.70/291.49	Rewrite Strategy: FULL
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(1) RenamingProof (BOTH BOUNDS(ID, ID))
779.70/291.49	Renamed function symbols to avoid clashes with predefined symbol.
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(2)
779.70/291.49	Obligation:
779.70/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
779.70/291.49	
779.70/291.49	
779.70/291.49	The TRS R consists of the following rules:
779.70/291.49	
779.70/291.49	   f(g(x)) -> g(f(f(x)))
779.70/291.49	   f(h(x)) -> h(g(x))
779.70/291.49	   f'(s(x), y, y) -> f'(y, x, s(x))
779.70/291.49	
779.70/291.49	S is empty.
779.70/291.49	Rewrite Strategy: FULL
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
779.70/291.49	Infered types.
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(4)
779.70/291.49	Obligation:
779.70/291.49	TRS:
779.70/291.49	Rules:
779.70/291.49	f(g(x)) -> g(f(f(x)))
779.70/291.49	f(h(x)) -> h(g(x))
779.70/291.49	f'(s(x), y, y) -> f'(y, x, s(x))
779.70/291.49	
779.70/291.49	Types:
779.70/291.49	f :: g:h -> g:h
779.70/291.49	g :: g:h -> g:h
779.70/291.49	h :: g:h -> g:h
779.70/291.49	f' :: s -> s -> s -> f'
779.70/291.49	s :: s -> s
779.70/291.49	hole_g:h1_0 :: g:h
779.70/291.49	hole_f'2_0 :: f'
779.70/291.49	hole_s3_0 :: s
779.70/291.49	gen_g:h4_0 :: Nat -> g:h
779.70/291.49	gen_s5_0 :: Nat -> s
779.70/291.49	
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(5) OrderProof (LOWER BOUND(ID))
779.70/291.49	Heuristically decided to analyse the following defined symbols:
779.70/291.49	f, f'
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(6)
779.70/291.49	Obligation:
779.70/291.49	TRS:
779.70/291.49	Rules:
779.70/291.49	f(g(x)) -> g(f(f(x)))
779.70/291.49	f(h(x)) -> h(g(x))
779.70/291.49	f'(s(x), y, y) -> f'(y, x, s(x))
779.70/291.49	
779.70/291.49	Types:
779.70/291.49	f :: g:h -> g:h
779.70/291.49	g :: g:h -> g:h
779.70/291.49	h :: g:h -> g:h
779.70/291.49	f' :: s -> s -> s -> f'
779.70/291.49	s :: s -> s
779.70/291.49	hole_g:h1_0 :: g:h
779.70/291.49	hole_f'2_0 :: f'
779.70/291.49	hole_s3_0 :: s
779.70/291.49	gen_g:h4_0 :: Nat -> g:h
779.70/291.49	gen_s5_0 :: Nat -> s
779.70/291.49	
779.70/291.49	
779.70/291.49	Generator Equations:
779.70/291.49	gen_g:h4_0(0) <=> hole_g:h1_0
779.70/291.49	gen_g:h4_0(+(x, 1)) <=> g(gen_g:h4_0(x))
779.70/291.49	gen_s5_0(0) <=> hole_s3_0
779.70/291.49	gen_s5_0(+(x, 1)) <=> s(gen_s5_0(x))
779.70/291.49	
779.70/291.49	
779.70/291.49	The following defined symbols remain to be analysed:
779.70/291.49	f, f'
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(7) RewriteLemmaProof (LOWER BOUND(ID))
779.70/291.49	Proved the following rewrite lemma:
779.70/291.49	f(gen_g:h4_0(+(1, n7_0))) -> *6_0, rt in Omega(n7_0)
779.70/291.49	
779.70/291.49	Induction Base:
779.70/291.49	f(gen_g:h4_0(+(1, 0)))
779.70/291.49	
779.70/291.49	Induction Step:
779.70/291.49	f(gen_g:h4_0(+(1, +(n7_0, 1)))) ->_R^Omega(1)
779.70/291.49	g(f(f(gen_g:h4_0(+(1, n7_0))))) ->_IH
779.70/291.49	g(f(*6_0))
779.70/291.49	
779.70/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(8)
779.70/291.49	Complex Obligation (BEST)
779.70/291.49	
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(9)
779.70/291.49	Obligation:
779.70/291.49	Proved the lower bound n^1 for the following obligation:
779.70/291.49	
779.70/291.49	TRS:
779.70/291.49	Rules:
779.70/291.49	f(g(x)) -> g(f(f(x)))
779.70/291.49	f(h(x)) -> h(g(x))
779.70/291.49	f'(s(x), y, y) -> f'(y, x, s(x))
779.70/291.49	
779.70/291.49	Types:
779.70/291.49	f :: g:h -> g:h
779.70/291.49	g :: g:h -> g:h
779.70/291.49	h :: g:h -> g:h
779.70/291.49	f' :: s -> s -> s -> f'
779.70/291.49	s :: s -> s
779.70/291.49	hole_g:h1_0 :: g:h
779.70/291.49	hole_f'2_0 :: f'
779.70/291.49	hole_s3_0 :: s
779.70/291.49	gen_g:h4_0 :: Nat -> g:h
779.70/291.49	gen_s5_0 :: Nat -> s
779.70/291.49	
779.70/291.49	
779.70/291.49	Generator Equations:
779.70/291.49	gen_g:h4_0(0) <=> hole_g:h1_0
779.70/291.49	gen_g:h4_0(+(x, 1)) <=> g(gen_g:h4_0(x))
779.70/291.49	gen_s5_0(0) <=> hole_s3_0
779.70/291.49	gen_s5_0(+(x, 1)) <=> s(gen_s5_0(x))
779.70/291.49	
779.70/291.49	
779.70/291.49	The following defined symbols remain to be analysed:
779.70/291.49	f, f'
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(10) LowerBoundPropagationProof (FINISHED)
779.70/291.49	Propagated lower bound.
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(11)
779.70/291.49	BOUNDS(n^1, INF)
779.70/291.49	
779.70/291.49	----------------------------------------
779.70/291.49	
779.70/291.49	(12)
779.70/291.49	Obligation:
779.70/291.49	TRS:
779.70/291.49	Rules:
779.70/291.49	f(g(x)) -> g(f(f(x)))
779.70/291.49	f(h(x)) -> h(g(x))
779.70/291.49	f'(s(x), y, y) -> f'(y, x, s(x))
779.70/291.49	
779.70/291.49	Types:
779.70/291.49	f :: g:h -> g:h
779.70/291.49	g :: g:h -> g:h
779.70/291.49	h :: g:h -> g:h
779.70/291.49	f' :: s -> s -> s -> f'
779.70/291.49	s :: s -> s
779.70/291.49	hole_g:h1_0 :: g:h
779.70/291.49	hole_f'2_0 :: f'
779.70/291.49	hole_s3_0 :: s
779.70/291.49	gen_g:h4_0 :: Nat -> g:h
779.70/291.49	gen_s5_0 :: Nat -> s
779.70/291.49	
779.70/291.49	
779.70/291.49	Lemmas:
779.70/291.49	f(gen_g:h4_0(+(1, n7_0))) -> *6_0, rt in Omega(n7_0)
779.70/291.49	
779.70/291.49	
779.70/291.49	Generator Equations:
779.70/291.49	gen_g:h4_0(0) <=> hole_g:h1_0
779.70/291.49	gen_g:h4_0(+(x, 1)) <=> g(gen_g:h4_0(x))
779.70/291.49	gen_s5_0(0) <=> hole_s3_0
779.70/291.49	gen_s5_0(+(x, 1)) <=> s(gen_s5_0(x))
779.70/291.49	
779.70/291.49	
779.70/291.49	The following defined symbols remain to be analysed:
779.70/291.49	f'
779.77/291.52	EOF