6.42/2.60	WORST_CASE(Omega(n^1), O(n^1))
6.75/2.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.75/2.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.75/2.67	
6.75/2.67	
6.75/2.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.75/2.67	
6.75/2.67	(0) CpxTRS
6.75/2.67	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
6.75/2.67	(2) CpxTRS
6.75/2.67	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
6.75/2.67	    (4) BOUNDS(1, n^1)
6.75/2.67	(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
6.75/2.67	(6) CpxTRS
6.75/2.67	    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
6.75/2.67	    (8) typed CpxTrs
6.75/2.67	    (9) OrderProof [LOWER BOUND(ID), 0 ms]
6.75/2.67	    (10) typed CpxTrs
6.75/2.67	    (11) RewriteLemmaProof [LOWER BOUND(ID), 432 ms]
6.75/2.67	    (12) proven lower bound
6.75/2.67	    (13) LowerBoundPropagationProof [FINISHED, 0 ms]
6.75/2.67	    (14) BOUNDS(n^1, INF)
6.75/2.67	
6.75/2.67	
6.75/2.67	----------------------------------------
6.75/2.67	
6.75/2.67	(0)
6.75/2.67	Obligation:
6.75/2.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.75/2.67	
6.75/2.67	
6.75/2.67	The TRS R consists of the following rules:
6.75/2.67	
6.75/2.67	   f(a) -> f(c(a))
6.75/2.67	   f(c(X)) -> X
6.75/2.67	   f(c(a)) -> f(d(b))
6.75/2.67	   f(a) -> f(d(a))
6.75/2.67	   f(d(X)) -> X
6.75/2.67	   f(c(b)) -> f(d(a))
6.75/2.67	   e(g(X)) -> e(X)
6.75/2.67	
6.75/2.67	S is empty.
6.75/2.67	Rewrite Strategy: FULL
6.75/2.67	----------------------------------------
6.75/2.67	
6.75/2.67	(1) RelTrsToTrsProof (UPPER BOUND(ID))
6.75/2.67	transformed relative TRS to TRS
6.75/2.67	----------------------------------------
6.75/2.67	
6.75/2.67	(2)
6.75/2.67	Obligation:
6.75/2.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
6.75/2.67	
6.75/2.67	
6.75/2.67	The TRS R consists of the following rules:
6.75/2.67	
6.75/2.67	   f(a) -> f(c(a))
6.75/2.67	   f(c(X)) -> X
6.75/2.67	   f(c(a)) -> f(d(b))
6.75/2.67	   f(a) -> f(d(a))
6.75/2.67	   f(d(X)) -> X
6.75/2.67	   f(c(b)) -> f(d(a))
6.75/2.67	   e(g(X)) -> e(X)
6.75/2.67	
6.75/2.67	S is empty.
6.75/2.67	Rewrite Strategy: FULL
6.75/2.67	----------------------------------------
6.75/2.67	
6.75/2.67	(3) CpxTrsMatchBoundsProof (FINISHED)
6.75/2.67	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. 
6.75/2.67	The certificate found is represented by the following graph.
6.75/2.67	
6.75/2.67	"[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
6.75/2.67	{(1,2,[f_1|0, e_1|0, a|1, c_1|1, d_1|1, b|1, g_1|1, e_1|1, a|2, b|2, b|3]), (1,3,[f_1|1]), (1,5,[f_1|1]), (1,7,[f_1|1]), (1,9,[f_1|2]), (2,2,[a|0, c_1|0, d_1|0, b|0, g_1|0]), (3,4,[c_1|1]), (4,2,[a|1]), (5,6,[d_1|1]), (6,2,[a|1]), (7,8,[d_1|1]), (8,2,[b|1]), (9,10,[d_1|2]), (10,2,[b|2])}"
6.75/2.67	----------------------------------------
6.75/2.67	
6.75/2.67	(4)
6.75/2.67	BOUNDS(1, n^1)
6.75/2.67	
6.75/2.67	----------------------------------------
6.75/2.67	
6.75/2.67	(5) RenamingProof (BOTH BOUNDS(ID, ID))
6.75/2.67	Renamed function symbols to avoid clashes with predefined symbol.
6.75/2.67	----------------------------------------
6.75/2.67	
6.75/2.67	(6)
6.75/2.67	Obligation:
6.75/2.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
6.75/2.67	
6.75/2.67	
6.75/2.67	The TRS R consists of the following rules:
6.75/2.67	
6.75/2.67	   f(a) -> f(c(a))
6.75/2.67	   f(c(X)) -> X
6.75/2.67	   f(c(a)) -> f(d(b))
6.75/2.67	   f(a) -> f(d(a))
6.75/2.67	   f(d(X)) -> X
6.75/2.67	   f(c(b)) -> f(d(a))
6.75/2.67	   e(g(X)) -> e(X)
6.75/2.67	
6.75/2.67	S is empty.
6.75/2.67	Rewrite Strategy: FULL
6.75/2.67	----------------------------------------
6.75/2.67	
6.75/2.67	(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
6.75/2.67	Infered types.
6.75/2.67	----------------------------------------
6.75/2.67	
6.75/2.67	(8)
6.75/2.67	Obligation:
6.75/2.67	TRS:
6.75/2.67	Rules:
6.75/2.67	f(a) -> f(c(a))
6.75/2.67	f(c(X)) -> X
6.75/2.67	f(c(a)) -> f(d(b))
6.75/2.67	f(a) -> f(d(a))
6.75/2.67	f(d(X)) -> X
6.75/2.67	f(c(b)) -> f(d(a))
6.75/2.67	e(g(X)) -> e(X)
6.75/2.67	
6.75/2.67	Types:
6.75/2.67	f :: a:c:b:d -> a:c:b:d
6.75/2.67	a :: a:c:b:d
6.75/2.68	c :: a:c:b:d -> a:c:b:d
6.75/2.68	d :: a:c:b:d -> a:c:b:d
6.75/2.68	b :: a:c:b:d
6.75/2.68	e :: g -> e
6.75/2.68	g :: g -> g
6.75/2.68	hole_a:c:b:d1_0 :: a:c:b:d
6.75/2.68	hole_e2_0 :: e
6.75/2.68	hole_g3_0 :: g
6.75/2.68	gen_a:c:b:d4_0 :: Nat -> a:c:b:d
6.75/2.68	gen_g5_0 :: Nat -> g
6.75/2.68	
6.75/2.68	----------------------------------------
6.75/2.68	
6.75/2.68	(9) OrderProof (LOWER BOUND(ID))
6.75/2.68	Heuristically decided to analyse the following defined symbols:
6.75/2.68	f, e
6.75/2.68	----------------------------------------
6.75/2.68	
6.75/2.68	(10)
6.75/2.68	Obligation:
6.75/2.68	TRS:
6.75/2.68	Rules:
6.75/2.68	f(a) -> f(c(a))
6.75/2.68	f(c(X)) -> X
6.75/2.68	f(c(a)) -> f(d(b))
6.75/2.68	f(a) -> f(d(a))
6.75/2.68	f(d(X)) -> X
6.75/2.68	f(c(b)) -> f(d(a))
6.75/2.68	e(g(X)) -> e(X)
6.75/2.68	
6.75/2.68	Types:
6.75/2.68	f :: a:c:b:d -> a:c:b:d
6.75/2.68	a :: a:c:b:d
6.75/2.68	c :: a:c:b:d -> a:c:b:d
6.75/2.68	d :: a:c:b:d -> a:c:b:d
6.75/2.68	b :: a:c:b:d
6.75/2.68	e :: g -> e
6.75/2.68	g :: g -> g
6.75/2.68	hole_a:c:b:d1_0 :: a:c:b:d
6.75/2.68	hole_e2_0 :: e
6.75/2.68	hole_g3_0 :: g
6.75/2.68	gen_a:c:b:d4_0 :: Nat -> a:c:b:d
6.75/2.68	gen_g5_0 :: Nat -> g
6.75/2.68	
6.75/2.68	
6.75/2.68	Generator Equations:
6.75/2.68	gen_a:c:b:d4_0(0) <=> b
6.75/2.68	gen_a:c:b:d4_0(+(x, 1)) <=> c(gen_a:c:b:d4_0(x))
6.75/2.68	gen_g5_0(0) <=> hole_g3_0
6.75/2.68	gen_g5_0(+(x, 1)) <=> g(gen_g5_0(x))
6.75/2.68	
6.75/2.68	
6.75/2.68	The following defined symbols remain to be analysed:
6.75/2.68	f, e
6.75/2.68	----------------------------------------
6.75/2.68	
6.75/2.68	(11) RewriteLemmaProof (LOWER BOUND(ID))
6.75/2.68	Proved the following rewrite lemma:
6.75/2.68	e(gen_g5_0(+(1, n43_0))) -> *6_0, rt in Omega(n43_0)
6.75/2.68	
6.75/2.68	Induction Base:
6.75/2.68	e(gen_g5_0(+(1, 0)))
6.75/2.68	
6.75/2.68	Induction Step:
6.75/2.68	e(gen_g5_0(+(1, +(n43_0, 1)))) ->_R^Omega(1)
6.75/2.68	e(gen_g5_0(+(1, n43_0))) ->_IH
6.75/2.68	*6_0
6.75/2.68	
6.75/2.68	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
6.75/2.68	----------------------------------------
6.75/2.68	
6.75/2.68	(12)
6.75/2.68	Obligation:
6.75/2.68	Proved the lower bound n^1 for the following obligation:
6.75/2.68	
6.75/2.68	TRS:
6.75/2.68	Rules:
6.75/2.68	f(a) -> f(c(a))
6.75/2.68	f(c(X)) -> X
6.75/2.68	f(c(a)) -> f(d(b))
6.75/2.68	f(a) -> f(d(a))
6.75/2.68	f(d(X)) -> X
6.75/2.68	f(c(b)) -> f(d(a))
6.75/2.68	e(g(X)) -> e(X)
6.75/2.68	
6.75/2.68	Types:
6.75/2.68	f :: a:c:b:d -> a:c:b:d
6.75/2.68	a :: a:c:b:d
6.75/2.68	c :: a:c:b:d -> a:c:b:d
6.75/2.68	d :: a:c:b:d -> a:c:b:d
6.75/2.68	b :: a:c:b:d
6.75/2.68	e :: g -> e
6.75/2.68	g :: g -> g
6.75/2.68	hole_a:c:b:d1_0 :: a:c:b:d
6.75/2.68	hole_e2_0 :: e
6.75/2.68	hole_g3_0 :: g
6.75/2.68	gen_a:c:b:d4_0 :: Nat -> a:c:b:d
6.75/2.68	gen_g5_0 :: Nat -> g
6.75/2.68	
6.75/2.68	
6.75/2.68	Generator Equations:
6.75/2.68	gen_a:c:b:d4_0(0) <=> b
6.75/2.68	gen_a:c:b:d4_0(+(x, 1)) <=> c(gen_a:c:b:d4_0(x))
6.75/2.68	gen_g5_0(0) <=> hole_g3_0
6.75/2.68	gen_g5_0(+(x, 1)) <=> g(gen_g5_0(x))
6.75/2.68	
6.75/2.68	
6.75/2.68	The following defined symbols remain to be analysed:
6.75/2.68	e
6.75/2.68	----------------------------------------
6.75/2.68	
6.75/2.68	(13) LowerBoundPropagationProof (FINISHED)
6.75/2.68	Propagated lower bound.
6.75/2.68	----------------------------------------
6.75/2.68	
6.75/2.68	(14)
6.75/2.68	BOUNDS(n^1, INF)
6.92/2.71	EOF