1118.92/292.15	WORST_CASE(Omega(n^1), ?)
1119.09/292.18	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1119.09/292.18	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1119.09/292.18	
1119.09/292.18	
1119.09/292.18	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1119.09/292.18	
1119.09/292.18	(0) CpxTRS
1119.09/292.18	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1119.09/292.18	(2) CpxTRS
1119.09/292.18	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1119.09/292.18	(4) typed CpxTrs
1119.09/292.18	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1119.09/292.18	(6) typed CpxTrs
1119.09/292.18	(7) RewriteLemmaProof [LOWER BOUND(ID), 647 ms]
1119.09/292.18	(8) proven lower bound
1119.09/292.18	(9) LowerBoundPropagationProof [FINISHED, 0 ms]
1119.09/292.18	(10) BOUNDS(n^1, INF)
1119.09/292.18	
1119.09/292.18	
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(0)
1119.09/292.18	Obligation:
1119.09/292.18	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1119.09/292.18	
1119.09/292.18	
1119.09/292.18	The TRS R consists of the following rules:
1119.09/292.18	
1119.09/292.18	   f(x, a(b(y))) -> f(a(b(x)), y)
1119.09/292.18	   f(x, b(c(y))) -> f(b(c(x)), y)
1119.09/292.18	   f(x, c(a(y))) -> f(c(a(x)), y)
1119.09/292.18	   f(a(x), y) -> f(x, a(y))
1119.09/292.18	   f(b(x), y) -> f(x, b(y))
1119.09/292.18	   f(c(x), y) -> f(x, c(y))
1119.09/292.18	
1119.09/292.18	S is empty.
1119.09/292.18	Rewrite Strategy: INNERMOST
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1119.09/292.18	Renamed function symbols to avoid clashes with predefined symbol.
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(2)
1119.09/292.18	Obligation:
1119.09/292.18	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1119.09/292.18	
1119.09/292.18	
1119.09/292.18	The TRS R consists of the following rules:
1119.09/292.18	
1119.09/292.18	   f(x, a(b(y))) -> f(a(b(x)), y)
1119.09/292.18	   f(x, b(c(y))) -> f(b(c(x)), y)
1119.09/292.18	   f(x, c(a(y))) -> f(c(a(x)), y)
1119.09/292.18	   f(a(x), y) -> f(x, a(y))
1119.09/292.18	   f(b(x), y) -> f(x, b(y))
1119.09/292.18	   f(c(x), y) -> f(x, c(y))
1119.09/292.18	
1119.09/292.18	S is empty.
1119.09/292.18	Rewrite Strategy: INNERMOST
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1119.09/292.18	Infered types.
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(4)
1119.09/292.18	Obligation:
1119.09/292.18	Innermost TRS:
1119.09/292.18	Rules:
1119.09/292.18	f(x, a(b(y))) -> f(a(b(x)), y)
1119.09/292.18	f(x, b(c(y))) -> f(b(c(x)), y)
1119.09/292.18	f(x, c(a(y))) -> f(c(a(x)), y)
1119.09/292.18	f(a(x), y) -> f(x, a(y))
1119.09/292.18	f(b(x), y) -> f(x, b(y))
1119.09/292.18	f(c(x), y) -> f(x, c(y))
1119.09/292.18	
1119.09/292.18	Types:
1119.09/292.18	f :: b:a:c -> b:a:c -> f
1119.09/292.18	a :: b:a:c -> b:a:c
1119.09/292.18	b :: b:a:c -> b:a:c
1119.09/292.18	c :: b:a:c -> b:a:c
1119.09/292.18	hole_f1_0 :: f
1119.09/292.18	hole_b:a:c2_0 :: b:a:c
1119.09/292.18	gen_b:a:c3_0 :: Nat -> b:a:c
1119.09/292.18	
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(5) OrderProof (LOWER BOUND(ID))
1119.09/292.18	Heuristically decided to analyse the following defined symbols:
1119.09/292.18	f
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(6)
1119.09/292.18	Obligation:
1119.09/292.18	Innermost TRS:
1119.09/292.18	Rules:
1119.09/292.18	f(x, a(b(y))) -> f(a(b(x)), y)
1119.09/292.18	f(x, b(c(y))) -> f(b(c(x)), y)
1119.09/292.18	f(x, c(a(y))) -> f(c(a(x)), y)
1119.09/292.18	f(a(x), y) -> f(x, a(y))
1119.09/292.18	f(b(x), y) -> f(x, b(y))
1119.09/292.18	f(c(x), y) -> f(x, c(y))
1119.09/292.18	
1119.09/292.18	Types:
1119.09/292.18	f :: b:a:c -> b:a:c -> f
1119.09/292.18	a :: b:a:c -> b:a:c
1119.09/292.18	b :: b:a:c -> b:a:c
1119.09/292.18	c :: b:a:c -> b:a:c
1119.09/292.18	hole_f1_0 :: f
1119.09/292.18	hole_b:a:c2_0 :: b:a:c
1119.09/292.18	gen_b:a:c3_0 :: Nat -> b:a:c
1119.09/292.18	
1119.09/292.18	
1119.09/292.18	Generator Equations:
1119.09/292.18	gen_b:a:c3_0(0) <=> hole_b:a:c2_0
1119.09/292.18	gen_b:a:c3_0(+(x, 1)) <=> a(gen_b:a:c3_0(x))
1119.09/292.18	
1119.09/292.18	
1119.09/292.18	The following defined symbols remain to be analysed:
1119.09/292.18	f
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(7) RewriteLemmaProof (LOWER BOUND(ID))
1119.09/292.18	Proved the following rewrite lemma:
1119.09/292.18	f(gen_b:a:c3_0(+(1, n5_0)), gen_b:a:c3_0(b)) -> *4_0, rt in Omega(n5_0)
1119.09/292.18	
1119.09/292.18	Induction Base:
1119.09/292.18	f(gen_b:a:c3_0(+(1, 0)), gen_b:a:c3_0(b))
1119.09/292.18	
1119.09/292.18	Induction Step:
1119.09/292.18	f(gen_b:a:c3_0(+(1, +(n5_0, 1))), gen_b:a:c3_0(b)) ->_R^Omega(1)
1119.09/292.18	f(gen_b:a:c3_0(+(1, n5_0)), a(gen_b:a:c3_0(b))) ->_IH
1119.09/292.18	*4_0
1119.09/292.18	
1119.09/292.18	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(8)
1119.09/292.18	Obligation:
1119.09/292.18	Proved the lower bound n^1 for the following obligation:
1119.09/292.18	
1119.09/292.18	Innermost TRS:
1119.09/292.18	Rules:
1119.09/292.18	f(x, a(b(y))) -> f(a(b(x)), y)
1119.09/292.18	f(x, b(c(y))) -> f(b(c(x)), y)
1119.09/292.18	f(x, c(a(y))) -> f(c(a(x)), y)
1119.09/292.18	f(a(x), y) -> f(x, a(y))
1119.09/292.18	f(b(x), y) -> f(x, b(y))
1119.09/292.18	f(c(x), y) -> f(x, c(y))
1119.09/292.18	
1119.09/292.18	Types:
1119.09/292.18	f :: b:a:c -> b:a:c -> f
1119.09/292.18	a :: b:a:c -> b:a:c
1119.09/292.18	b :: b:a:c -> b:a:c
1119.09/292.18	c :: b:a:c -> b:a:c
1119.09/292.18	hole_f1_0 :: f
1119.09/292.18	hole_b:a:c2_0 :: b:a:c
1119.09/292.18	gen_b:a:c3_0 :: Nat -> b:a:c
1119.09/292.18	
1119.09/292.18	
1119.09/292.18	Generator Equations:
1119.09/292.18	gen_b:a:c3_0(0) <=> hole_b:a:c2_0
1119.09/292.18	gen_b:a:c3_0(+(x, 1)) <=> a(gen_b:a:c3_0(x))
1119.09/292.18	
1119.09/292.18	
1119.09/292.18	The following defined symbols remain to be analysed:
1119.09/292.18	f
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(9) LowerBoundPropagationProof (FINISHED)
1119.09/292.18	Propagated lower bound.
1119.09/292.18	----------------------------------------
1119.09/292.18	
1119.09/292.18	(10)
1119.09/292.18	BOUNDS(n^1, INF)
1119.09/292.24	EOF