1114.48/291.51	WORST_CASE(Omega(n^2), ?)
1115.16/291.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1115.16/291.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1115.16/291.75	
1115.16/291.75	
1115.16/291.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1115.16/291.75	
1115.16/291.75	(0) CpxTRS
1115.16/291.75	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1115.16/291.75	(2) CpxTRS
1115.16/291.75	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1115.16/291.75	(4) typed CpxTrs
1115.16/291.75	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1115.16/291.75	(6) typed CpxTrs
1115.16/291.75	(7) RewriteLemmaProof [LOWER BOUND(ID), 620 ms]
1115.16/291.75	(8) BEST
1115.16/291.75	    (9) proven lower bound
1115.16/291.75	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1115.16/291.75	        (11) BOUNDS(n^1, INF)
1115.16/291.75	    (12) typed CpxTrs
1115.16/291.75	        (13) RewriteLemmaProof [LOWER BOUND(ID), 120 ms]
1115.16/291.75	        (14) BEST
1115.16/291.75	            (15) proven lower bound
1115.16/291.75	                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
1115.16/291.75	                (17) BOUNDS(n^2, INF)
1115.16/291.75	            (18) typed CpxTrs
1115.16/291.75	                (19) RewriteLemmaProof [LOWER BOUND(ID), 109 ms]
1115.16/291.75	                (20) typed CpxTrs
1115.16/291.75	                (21) RewriteLemmaProof [LOWER BOUND(ID), 70 ms]
1115.16/291.75	                (22) typed CpxTrs
1115.16/291.75	                (23) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
1115.16/291.75	                (24) typed CpxTrs
1115.16/291.75	                (25) RewriteLemmaProof [LOWER BOUND(ID), 128 ms]
1115.16/291.75	                (26) typed CpxTrs
1115.16/291.75	                (27) RewriteLemmaProof [LOWER BOUND(ID), 81 ms]
1115.16/291.75	                (28) BOUNDS(1, INF)
1115.16/291.75	
1115.16/291.75	
1115.16/291.75	----------------------------------------
1115.16/291.75	
1115.16/291.75	(0)
1115.16/291.75	Obligation:
1115.16/291.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1115.16/291.75	
1115.16/291.75	
1115.16/291.75	The TRS R consists of the following rules:
1115.16/291.75	
1115.16/291.75	   a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.16/291.75	   a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.16/291.75	   a__plus(N, 0) -> mark(N)
1115.16/291.75	   a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.16/291.75	   mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.16/291.75	   mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.16/291.75	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.16/291.75	   mark(tt) -> tt
1115.16/291.75	   mark(s(X)) -> s(mark(X))
1115.16/291.75	   mark(0) -> 0
1115.16/291.75	   a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.16/291.75	   a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.16/291.75	   a__plus(X1, X2) -> plus(X1, X2)
1115.16/291.75	
1115.16/291.75	S is empty.
1115.16/291.75	Rewrite Strategy: FULL
1115.16/291.75	----------------------------------------
1115.16/291.75	
1115.16/291.75	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1115.16/291.75	Renamed function symbols to avoid clashes with predefined symbol.
1115.16/291.75	----------------------------------------
1115.16/291.75	
1115.16/291.75	(2)
1115.16/291.75	Obligation:
1115.16/291.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1115.16/291.75	
1115.16/291.75	
1115.16/291.75	The TRS R consists of the following rules:
1115.16/291.75	
1115.16/291.75	   a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.16/291.75	   a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.16/291.75	   a__plus(N, 0') -> mark(N)
1115.16/291.75	   a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.16/291.75	   mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.16/291.75	   mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.16/291.75	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.16/291.75	   mark(tt) -> tt
1115.16/291.75	   mark(s(X)) -> s(mark(X))
1115.16/291.75	   mark(0') -> 0'
1115.16/291.75	   a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.16/291.75	   a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.16/291.75	   a__plus(X1, X2) -> plus(X1, X2)
1115.16/291.75	
1115.16/291.75	S is empty.
1115.16/291.75	Rewrite Strategy: FULL
1115.16/291.75	----------------------------------------
1115.16/291.75	
1115.16/291.75	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1115.16/291.75	Infered types.
1115.16/291.75	----------------------------------------
1115.16/291.75	
1115.16/291.75	(4)
1115.16/291.75	Obligation:
1115.16/291.75	TRS:
1115.16/291.75	Rules:
1115.16/291.75	a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.16/291.75	a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.16/291.75	a__plus(N, 0') -> mark(N)
1115.16/291.75	a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.16/291.75	mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.16/291.75	mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.16/291.75	mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.16/291.75	mark(tt) -> tt
1115.16/291.75	mark(s(X)) -> s(mark(X))
1115.16/291.75	mark(0') -> 0'
1115.16/291.75	a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.16/291.75	a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.16/291.75	a__plus(X1, X2) -> plus(X1, X2)
1115.16/291.75	
1115.16/291.75	Types:
1115.16/291.75	a__U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	tt :: tt:s:0':U11:U12:plus
1115.16/291.75	a__U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	s :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	a__plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	mark :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	0' :: tt:s:0':U11:U12:plus
1115.16/291.75	U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
1115.16/291.75	gen_tt:s:0':U11:U12:plus2_0 :: Nat -> tt:s:0':U11:U12:plus
1115.16/291.75	
1115.16/291.75	----------------------------------------
1115.16/291.75	
1115.16/291.75	(5) OrderProof (LOWER BOUND(ID))
1115.16/291.75	Heuristically decided to analyse the following defined symbols:
1115.16/291.75	a__U11, a__U12, a__plus, mark
1115.16/291.75	
1115.16/291.75	They will be analysed ascendingly in the following order:
1115.16/291.75	a__U11 = a__U12
1115.16/291.75	a__U11 = a__plus
1115.16/291.75	a__U11 = mark
1115.16/291.75	a__U12 = a__plus
1115.16/291.75	a__U12 = mark
1115.16/291.75	a__plus = mark
1115.16/291.75	
1115.16/291.75	----------------------------------------
1115.16/291.75	
1115.16/291.75	(6)
1115.16/291.75	Obligation:
1115.16/291.75	TRS:
1115.16/291.75	Rules:
1115.16/291.75	a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.16/291.75	a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.16/291.75	a__plus(N, 0') -> mark(N)
1115.16/291.75	a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.16/291.75	mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.16/291.75	mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.16/291.75	mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.16/291.75	mark(tt) -> tt
1115.16/291.75	mark(s(X)) -> s(mark(X))
1115.16/291.75	mark(0') -> 0'
1115.16/291.75	a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.16/291.75	a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.16/291.75	a__plus(X1, X2) -> plus(X1, X2)
1115.16/291.75	
1115.16/291.75	Types:
1115.16/291.75	a__U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	tt :: tt:s:0':U11:U12:plus
1115.16/291.75	a__U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	s :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	a__plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	mark :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	0' :: tt:s:0':U11:U12:plus
1115.16/291.75	U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.16/291.75	hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
1115.16/291.75	gen_tt:s:0':U11:U12:plus2_0 :: Nat -> tt:s:0':U11:U12:plus
1115.16/291.75	
1115.16/291.75	
1115.16/291.75	Generator Equations:
1115.16/291.75	gen_tt:s:0':U11:U12:plus2_0(0) <=> tt
1115.16/291.75	gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) <=> s(gen_tt:s:0':U11:U12:plus2_0(x))
1115.16/291.75	
1115.16/291.75	
1115.16/291.75	The following defined symbols remain to be analysed:
1115.16/291.75	a__U12, a__U11, a__plus, mark
1115.16/291.75	
1115.16/291.75	They will be analysed ascendingly in the following order:
1115.16/291.75	a__U11 = a__U12
1115.16/291.75	a__U11 = a__plus
1115.16/291.75	a__U11 = mark
1115.16/291.75	a__U12 = a__plus
1115.16/291.75	a__U12 = mark
1115.16/291.75	a__plus = mark
1115.16/291.75	
1115.16/291.75	----------------------------------------
1115.16/291.75	
1115.16/291.75	(7) RewriteLemmaProof (LOWER BOUND(ID))
1115.16/291.75	Proved the following rewrite lemma:
1115.16/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) -> gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt in Omega(1 + n7802_0)
1115.16/291.75	
1115.16/291.75	Induction Base:
1115.16/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(0)) ->_R^Omega(1)
1115.16/291.75	tt
1115.16/291.75	
1115.16/291.75	Induction Step:
1115.16/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(+(n7802_0, 1))) ->_R^Omega(1)
1115.16/291.75	s(mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0))) ->_IH
1115.16/291.75	s(gen_tt:s:0':U11:U12:plus2_0(c7803_0))
1115.16/291.75	
1115.16/291.75	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1115.16/291.75	----------------------------------------
1115.16/291.75	
1115.16/291.75	(8)
1115.16/291.75	Complex Obligation (BEST)
1115.16/291.75	
1115.16/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(9)
1115.51/291.75	Obligation:
1115.51/291.75	Proved the lower bound n^1 for the following obligation:
1115.51/291.75	
1115.51/291.75	TRS:
1115.51/291.75	Rules:
1115.51/291.75	a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.51/291.75	a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.51/291.75	a__plus(N, 0') -> mark(N)
1115.51/291.75	a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.51/291.75	mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.51/291.75	mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.51/291.75	mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.51/291.75	mark(tt) -> tt
1115.51/291.75	mark(s(X)) -> s(mark(X))
1115.51/291.75	mark(0') -> 0'
1115.51/291.75	a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.51/291.75	a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.51/291.75	a__plus(X1, X2) -> plus(X1, X2)
1115.51/291.75	
1115.51/291.75	Types:
1115.51/291.75	a__U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	tt :: tt:s:0':U11:U12:plus
1115.51/291.75	a__U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	s :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	a__plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	mark :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	0' :: tt:s:0':U11:U12:plus
1115.51/291.75	U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0 :: Nat -> tt:s:0':U11:U12:plus
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Generator Equations:
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(0) <=> tt
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) <=> s(gen_tt:s:0':U11:U12:plus2_0(x))
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	The following defined symbols remain to be analysed:
1115.51/291.75	mark, a__U11
1115.51/291.75	
1115.51/291.75	They will be analysed ascendingly in the following order:
1115.51/291.75	a__U11 = a__U12
1115.51/291.75	a__U11 = a__plus
1115.51/291.75	a__U11 = mark
1115.51/291.75	a__U12 = a__plus
1115.51/291.75	a__U12 = mark
1115.51/291.75	a__plus = mark
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(10) LowerBoundPropagationProof (FINISHED)
1115.51/291.75	Propagated lower bound.
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(11)
1115.51/291.75	BOUNDS(n^1, INF)
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(12)
1115.51/291.75	Obligation:
1115.51/291.75	TRS:
1115.51/291.75	Rules:
1115.51/291.75	a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.51/291.75	a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.51/291.75	a__plus(N, 0') -> mark(N)
1115.51/291.75	a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.51/291.75	mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.51/291.75	mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.51/291.75	mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.51/291.75	mark(tt) -> tt
1115.51/291.75	mark(s(X)) -> s(mark(X))
1115.51/291.75	mark(0') -> 0'
1115.51/291.75	a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.51/291.75	a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.51/291.75	a__plus(X1, X2) -> plus(X1, X2)
1115.51/291.75	
1115.51/291.75	Types:
1115.51/291.75	a__U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	tt :: tt:s:0':U11:U12:plus
1115.51/291.75	a__U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	s :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	a__plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	mark :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	0' :: tt:s:0':U11:U12:plus
1115.51/291.75	U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0 :: Nat -> tt:s:0':U11:U12:plus
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Lemmas:
1115.51/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) -> gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt in Omega(1 + n7802_0)
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Generator Equations:
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(0) <=> tt
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) <=> s(gen_tt:s:0':U11:U12:plus2_0(x))
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	The following defined symbols remain to be analysed:
1115.51/291.75	a__U11, a__U12, a__plus
1115.51/291.75	
1115.51/291.75	They will be analysed ascendingly in the following order:
1115.51/291.75	a__U11 = a__U12
1115.51/291.75	a__U11 = a__plus
1115.51/291.75	a__U11 = mark
1115.51/291.75	a__U12 = a__plus
1115.51/291.75	a__U12 = mark
1115.51/291.75	a__plus = mark
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(13) RewriteLemmaProof (LOWER BOUND(ID))
1115.51/291.75	Proved the following rewrite lemma:
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n8498_0 + n8498_0 + n8498_0^2)
1115.51/291.75	
1115.51/291.75	Induction Base:
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))
1115.51/291.75	
1115.51/291.75	Induction Step:
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n8498_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) ->_R^Omega(1)
1115.51/291.75	a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(+(n8498_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) ->_R^Omega(1)
1115.51/291.75	s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n8498_0))))) ->_L^Omega(1 + c)
1115.51/291.75	s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n8498_0))))) ->_L^Omega(2 + n8498_0)
1115.51/291.75	s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n8498_0)))) ->_R^Omega(1)
1115.51/291.75	s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c))) ->_IH
1115.51/291.75	s(*3_0)
1115.51/291.75	
1115.51/291.75	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(14)
1115.51/291.75	Complex Obligation (BEST)
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(15)
1115.51/291.75	Obligation:
1115.51/291.75	Proved the lower bound n^2 for the following obligation:
1115.51/291.75	
1115.51/291.75	TRS:
1115.51/291.75	Rules:
1115.51/291.75	a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.51/291.75	a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.51/291.75	a__plus(N, 0') -> mark(N)
1115.51/291.75	a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.51/291.75	mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.51/291.75	mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.51/291.75	mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.51/291.75	mark(tt) -> tt
1115.51/291.75	mark(s(X)) -> s(mark(X))
1115.51/291.75	mark(0') -> 0'
1115.51/291.75	a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.51/291.75	a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.51/291.75	a__plus(X1, X2) -> plus(X1, X2)
1115.51/291.75	
1115.51/291.75	Types:
1115.51/291.75	a__U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	tt :: tt:s:0':U11:U12:plus
1115.51/291.75	a__U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	s :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	a__plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	mark :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	0' :: tt:s:0':U11:U12:plus
1115.51/291.75	U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0 :: Nat -> tt:s:0':U11:U12:plus
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Lemmas:
1115.51/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) -> gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt in Omega(1 + n7802_0)
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Generator Equations:
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(0) <=> tt
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) <=> s(gen_tt:s:0':U11:U12:plus2_0(x))
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	The following defined symbols remain to be analysed:
1115.51/291.75	a__U11, a__U12, a__plus
1115.51/291.75	
1115.51/291.75	They will be analysed ascendingly in the following order:
1115.51/291.75	a__U11 = a__U12
1115.51/291.75	a__U11 = a__plus
1115.51/291.75	a__U11 = mark
1115.51/291.75	a__U12 = a__plus
1115.51/291.75	a__U12 = mark
1115.51/291.75	a__plus = mark
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(16) LowerBoundPropagationProof (FINISHED)
1115.51/291.75	Propagated lower bound.
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(17)
1115.51/291.75	BOUNDS(n^2, INF)
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(18)
1115.51/291.75	Obligation:
1115.51/291.75	TRS:
1115.51/291.75	Rules:
1115.51/291.75	a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.51/291.75	a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.51/291.75	a__plus(N, 0') -> mark(N)
1115.51/291.75	a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.51/291.75	mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.51/291.75	mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.51/291.75	mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.51/291.75	mark(tt) -> tt
1115.51/291.75	mark(s(X)) -> s(mark(X))
1115.51/291.75	mark(0') -> 0'
1115.51/291.75	a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.51/291.75	a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.51/291.75	a__plus(X1, X2) -> plus(X1, X2)
1115.51/291.75	
1115.51/291.75	Types:
1115.51/291.75	a__U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	tt :: tt:s:0':U11:U12:plus
1115.51/291.75	a__U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	s :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	a__plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	mark :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	0' :: tt:s:0':U11:U12:plus
1115.51/291.75	U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0 :: Nat -> tt:s:0':U11:U12:plus
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Lemmas:
1115.51/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) -> gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt in Omega(1 + n7802_0)
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n8498_0 + n8498_0 + n8498_0^2)
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Generator Equations:
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(0) <=> tt
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) <=> s(gen_tt:s:0':U11:U12:plus2_0(x))
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	The following defined symbols remain to be analysed:
1115.51/291.75	a__U12, a__plus, mark
1115.51/291.75	
1115.51/291.75	They will be analysed ascendingly in the following order:
1115.51/291.75	a__U11 = a__U12
1115.51/291.75	a__U11 = a__plus
1115.51/291.75	a__U11 = mark
1115.51/291.75	a__U12 = a__plus
1115.51/291.75	a__U12 = mark
1115.51/291.75	a__plus = mark
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(19) RewriteLemmaProof (LOWER BOUND(ID))
1115.51/291.75	Proved the following rewrite lemma:
1115.51/291.75	a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n10562_0 + n10562_0 + n10562_0^2)
1115.51/291.75	
1115.51/291.75	Induction Base:
1115.51/291.75	a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))
1115.51/291.75	
1115.51/291.75	Induction Step:
1115.51/291.75	a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n10562_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) ->_R^Omega(1)
1115.51/291.75	s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(n10562_0, 1))))) ->_L^Omega(1 + c)
1115.51/291.75	s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n10562_0))))) ->_L^Omega(2 + n10562_0)
1115.51/291.75	s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n10562_0)))) ->_R^Omega(1)
1115.51/291.75	s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c))) ->_R^Omega(1)
1115.51/291.75	s(a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c))) ->_IH
1115.51/291.75	s(*3_0)
1115.51/291.75	
1115.51/291.75	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(20)
1115.51/291.75	Obligation:
1115.51/291.75	TRS:
1115.51/291.75	Rules:
1115.51/291.75	a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.51/291.75	a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.51/291.75	a__plus(N, 0') -> mark(N)
1115.51/291.75	a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.51/291.75	mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.51/291.75	mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.51/291.75	mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.51/291.75	mark(tt) -> tt
1115.51/291.75	mark(s(X)) -> s(mark(X))
1115.51/291.75	mark(0') -> 0'
1115.51/291.75	a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.51/291.75	a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.51/291.75	a__plus(X1, X2) -> plus(X1, X2)
1115.51/291.75	
1115.51/291.75	Types:
1115.51/291.75	a__U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	tt :: tt:s:0':U11:U12:plus
1115.51/291.75	a__U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	s :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	a__plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	mark :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	0' :: tt:s:0':U11:U12:plus
1115.51/291.75	U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0 :: Nat -> tt:s:0':U11:U12:plus
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Lemmas:
1115.51/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) -> gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt in Omega(1 + n7802_0)
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n8498_0 + n8498_0 + n8498_0^2)
1115.51/291.75	a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n10562_0 + n10562_0 + n10562_0^2)
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Generator Equations:
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(0) <=> tt
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) <=> s(gen_tt:s:0':U11:U12:plus2_0(x))
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	The following defined symbols remain to be analysed:
1115.51/291.75	a__plus, a__U11, mark
1115.51/291.75	
1115.51/291.75	They will be analysed ascendingly in the following order:
1115.51/291.75	a__U11 = a__U12
1115.51/291.75	a__U11 = a__plus
1115.51/291.75	a__U11 = mark
1115.51/291.75	a__U12 = a__plus
1115.51/291.75	a__U12 = mark
1115.51/291.75	a__plus = mark
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(21) RewriteLemmaProof (LOWER BOUND(ID))
1115.51/291.75	Proved the following rewrite lemma:
1115.51/291.75	a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))) -> *3_0, rt in Omega(a*n12888_0 + n12888_0 + n12888_0^2)
1115.51/291.75	
1115.51/291.75	Induction Base:
1115.51/291.75	a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, 0)))
1115.51/291.75	
1115.51/291.75	Induction Step:
1115.51/291.75	a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, +(n12888_0, 1)))) ->_R^Omega(1)
1115.51/291.75	a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0)), gen_tt:s:0':U11:U12:plus2_0(a)) ->_R^Omega(1)
1115.51/291.75	a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0)), gen_tt:s:0':U11:U12:plus2_0(a)) ->_R^Omega(1)
1115.51/291.75	s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(a)), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))))) ->_L^Omega(1 + a)
1115.51/291.75	s(a__plus(gen_tt:s:0':U11:U12:plus2_0(a), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))))) ->_L^Omega(2 + n12888_0)
1115.51/291.75	s(a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0)))) ->_IH
1115.51/291.75	s(*3_0)
1115.51/291.75	
1115.51/291.75	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(22)
1115.51/291.75	Obligation:
1115.51/291.75	TRS:
1115.51/291.75	Rules:
1115.51/291.75	a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.51/291.75	a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.51/291.75	a__plus(N, 0') -> mark(N)
1115.51/291.75	a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.51/291.75	mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.51/291.75	mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.51/291.75	mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.51/291.75	mark(tt) -> tt
1115.51/291.75	mark(s(X)) -> s(mark(X))
1115.51/291.75	mark(0') -> 0'
1115.51/291.75	a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.51/291.75	a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.51/291.75	a__plus(X1, X2) -> plus(X1, X2)
1115.51/291.75	
1115.51/291.75	Types:
1115.51/291.75	a__U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	tt :: tt:s:0':U11:U12:plus
1115.51/291.75	a__U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	s :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	a__plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	mark :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	0' :: tt:s:0':U11:U12:plus
1115.51/291.75	U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0 :: Nat -> tt:s:0':U11:U12:plus
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Lemmas:
1115.51/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) -> gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt in Omega(1 + n7802_0)
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n8498_0 + n8498_0 + n8498_0^2)
1115.51/291.75	a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n10562_0 + n10562_0 + n10562_0^2)
1115.51/291.75	a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))) -> *3_0, rt in Omega(a*n12888_0 + n12888_0 + n12888_0^2)
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Generator Equations:
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(0) <=> tt
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) <=> s(gen_tt:s:0':U11:U12:plus2_0(x))
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	The following defined symbols remain to be analysed:
1115.51/291.75	mark, a__U11, a__U12
1115.51/291.75	
1115.51/291.75	They will be analysed ascendingly in the following order:
1115.51/291.75	a__U11 = a__U12
1115.51/291.75	a__U11 = a__plus
1115.51/291.75	a__U11 = mark
1115.51/291.75	a__U12 = a__plus
1115.51/291.75	a__U12 = mark
1115.51/291.75	a__plus = mark
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(23) RewriteLemmaProof (LOWER BOUND(ID))
1115.51/291.75	Proved the following rewrite lemma:
1115.51/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0)) -> gen_tt:s:0':U11:U12:plus2_0(n15871_0), rt in Omega(1 + n15871_0)
1115.51/291.75	
1115.51/291.75	Induction Base:
1115.51/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(0)) ->_R^Omega(1)
1115.51/291.75	tt
1115.51/291.75	
1115.51/291.75	Induction Step:
1115.51/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(+(n15871_0, 1))) ->_R^Omega(1)
1115.51/291.75	s(mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0))) ->_IH
1115.51/291.75	s(gen_tt:s:0':U11:U12:plus2_0(c15872_0))
1115.51/291.75	
1115.51/291.75	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(24)
1115.51/291.75	Obligation:
1115.51/291.75	TRS:
1115.51/291.75	Rules:
1115.51/291.75	a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.51/291.75	a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.51/291.75	a__plus(N, 0') -> mark(N)
1115.51/291.75	a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.51/291.75	mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.51/291.75	mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.51/291.75	mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.51/291.75	mark(tt) -> tt
1115.51/291.75	mark(s(X)) -> s(mark(X))
1115.51/291.75	mark(0') -> 0'
1115.51/291.75	a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.51/291.75	a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.51/291.75	a__plus(X1, X2) -> plus(X1, X2)
1115.51/291.75	
1115.51/291.75	Types:
1115.51/291.75	a__U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	tt :: tt:s:0':U11:U12:plus
1115.51/291.75	a__U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	s :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	a__plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	mark :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	0' :: tt:s:0':U11:U12:plus
1115.51/291.75	U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0 :: Nat -> tt:s:0':U11:U12:plus
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Lemmas:
1115.51/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0)) -> gen_tt:s:0':U11:U12:plus2_0(n15871_0), rt in Omega(1 + n15871_0)
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n8498_0 + n8498_0 + n8498_0^2)
1115.51/291.75	a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n10562_0 + n10562_0 + n10562_0^2)
1115.51/291.75	a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))) -> *3_0, rt in Omega(a*n12888_0 + n12888_0 + n12888_0^2)
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Generator Equations:
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(0) <=> tt
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) <=> s(gen_tt:s:0':U11:U12:plus2_0(x))
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	The following defined symbols remain to be analysed:
1115.51/291.75	a__U11, a__U12
1115.51/291.75	
1115.51/291.75	They will be analysed ascendingly in the following order:
1115.51/291.75	a__U11 = a__U12
1115.51/291.75	a__U11 = a__plus
1115.51/291.75	a__U11 = mark
1115.51/291.75	a__U12 = a__plus
1115.51/291.75	a__U12 = mark
1115.51/291.75	a__plus = mark
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(25) RewriteLemmaProof (LOWER BOUND(ID))
1115.51/291.75	Proved the following rewrite lemma:
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n16876_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n16876_0 + n16876_0 + n16876_0^2)
1115.51/291.75	
1115.51/291.75	Induction Base:
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))
1115.51/291.75	
1115.51/291.75	Induction Step:
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n16876_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) ->_R^Omega(1)
1115.51/291.75	a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(+(n16876_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) ->_R^Omega(1)
1115.51/291.75	s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n16876_0))))) ->_L^Omega(1 + c)
1115.51/291.75	s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n16876_0))))) ->_L^Omega(2 + n16876_0)
1115.51/291.75	s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n16876_0)))) ->_R^Omega(1)
1115.51/291.75	s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n16876_0), gen_tt:s:0':U11:U12:plus2_0(c))) ->_IH
1115.51/291.75	s(*3_0)
1115.51/291.75	
1115.51/291.75	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(26)
1115.51/291.75	Obligation:
1115.51/291.75	TRS:
1115.51/291.75	Rules:
1115.51/291.75	a__U11(tt, M, N) -> a__U12(tt, M, N)
1115.51/291.75	a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1115.51/291.75	a__plus(N, 0') -> mark(N)
1115.51/291.75	a__plus(N, s(M)) -> a__U11(tt, M, N)
1115.51/291.75	mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
1115.51/291.75	mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
1115.51/291.75	mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1115.51/291.75	mark(tt) -> tt
1115.51/291.75	mark(s(X)) -> s(mark(X))
1115.51/291.75	mark(0') -> 0'
1115.51/291.75	a__U11(X1, X2, X3) -> U11(X1, X2, X3)
1115.51/291.75	a__U12(X1, X2, X3) -> U12(X1, X2, X3)
1115.51/291.75	a__plus(X1, X2) -> plus(X1, X2)
1115.51/291.75	
1115.51/291.75	Types:
1115.51/291.75	a__U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	tt :: tt:s:0':U11:U12:plus
1115.51/291.75	a__U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	s :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	a__plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	mark :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	0' :: tt:s:0':U11:U12:plus
1115.51/291.75	U11 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	U12 :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	plus :: tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus -> tt:s:0':U11:U12:plus
1115.51/291.75	hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0 :: Nat -> tt:s:0':U11:U12:plus
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Lemmas:
1115.51/291.75	mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0)) -> gen_tt:s:0':U11:U12:plus2_0(n15871_0), rt in Omega(1 + n15871_0)
1115.51/291.75	a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n16876_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n16876_0 + n16876_0 + n16876_0^2)
1115.51/291.75	a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n10562_0 + n10562_0 + n10562_0^2)
1115.51/291.75	a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))) -> *3_0, rt in Omega(a*n12888_0 + n12888_0 + n12888_0^2)
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	Generator Equations:
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(0) <=> tt
1115.51/291.75	gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) <=> s(gen_tt:s:0':U11:U12:plus2_0(x))
1115.51/291.75	
1115.51/291.75	
1115.51/291.75	The following defined symbols remain to be analysed:
1115.51/291.75	a__U12
1115.51/291.75	
1115.51/291.75	They will be analysed ascendingly in the following order:
1115.51/291.75	a__U11 = a__U12
1115.51/291.75	a__U11 = a__plus
1115.51/291.75	a__U11 = mark
1115.51/291.75	a__U12 = a__plus
1115.51/291.75	a__U12 = mark
1115.51/291.75	a__plus = mark
1115.51/291.75	
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(27) RewriteLemmaProof (LOWER BOUND(ID))
1115.51/291.75	Proved the following rewrite lemma:
1115.51/291.75	a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n20155_0), gen_tt:s:0':U11:U12:plus2_0(c)) -> *3_0, rt in Omega(c*n20155_0 + n20155_0 + n20155_0^2)
1115.51/291.75	
1115.51/291.75	Induction Base:
1115.51/291.75	a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))
1115.51/291.75	
1115.51/291.75	Induction Step:
1115.51/291.75	a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n20155_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) ->_R^Omega(1)
1115.51/291.75	s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(n20155_0, 1))))) ->_L^Omega(1 + c)
1115.51/291.75	s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n20155_0))))) ->_L^Omega(2 + n20155_0)
1115.51/291.75	s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n20155_0)))) ->_R^Omega(1)
1115.51/291.75	s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n20155_0), gen_tt:s:0':U11:U12:plus2_0(c))) ->_R^Omega(1)
1115.51/291.75	s(a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(n20155_0), gen_tt:s:0':U11:U12:plus2_0(c))) ->_IH
1115.51/291.75	s(*3_0)
1115.51/291.75	
1115.51/291.75	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1115.51/291.75	----------------------------------------
1115.51/291.75	
1115.51/291.75	(28)
1115.51/291.75	BOUNDS(1, INF)
1115.78/291.88	EOF