318.81/291.51	WORST_CASE(Omega(n^1), ?)
318.81/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
318.81/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
318.81/291.52	
318.81/291.52	
318.81/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.81/291.52	
318.81/291.52	(0) CpxTRS
318.81/291.52	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
318.81/291.52	(2) CpxTRS
318.81/291.52	(3) SlicingProof [LOWER BOUND(ID), 0 ms]
318.81/291.52	(4) CpxTRS
318.81/291.52	(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
318.81/291.52	(6) typed CpxTrs
318.81/291.52	(7) OrderProof [LOWER BOUND(ID), 0 ms]
318.81/291.52	(8) typed CpxTrs
318.81/291.52	(9) RewriteLemmaProof [LOWER BOUND(ID), 477 ms]
318.81/291.52	(10) BEST
318.81/291.52	    (11) proven lower bound
318.81/291.52	        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
318.81/291.52	        (13) BOUNDS(n^1, INF)
318.81/291.52	    (14) typed CpxTrs
318.81/291.52	
318.81/291.52	
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(0)
318.81/291.52	Obligation:
318.81/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.81/291.52	
318.81/291.52	
318.81/291.52	The TRS R consists of the following rules:
318.81/291.52	
318.81/291.52	   din(der(plus(X, Y))) -> u21(din(der(X)), X, Y)
318.81/291.52	   u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX)
318.81/291.52	   u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY))
318.81/291.52	   din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
318.81/291.52	   u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
318.81/291.52	   u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
318.81/291.52	   din(der(der(X))) -> u41(din(der(X)), X)
318.81/291.52	   u41(dout(DX), X) -> u42(din(der(DX)), X, DX)
318.81/291.52	   u42(dout(DDX), X, DX) -> dout(DDX)
318.81/291.52	
318.81/291.52	S is empty.
318.81/291.52	Rewrite Strategy: FULL
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(1) RenamingProof (BOTH BOUNDS(ID, ID))
318.81/291.52	Renamed function symbols to avoid clashes with predefined symbol.
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(2)
318.81/291.52	Obligation:
318.81/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.81/291.52	
318.81/291.52	
318.81/291.52	The TRS R consists of the following rules:
318.81/291.52	
318.81/291.52	   din(der(plus(X, Y))) -> u21(din(der(X)), X, Y)
318.81/291.52	   u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX)
318.81/291.52	   u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY))
318.81/291.52	   din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
318.81/291.52	   u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
318.81/291.52	   u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
318.81/291.52	   din(der(der(X))) -> u41(din(der(X)), X)
318.81/291.52	   u41(dout(DX), X) -> u42(din(der(DX)), X, DX)
318.81/291.52	   u42(dout(DDX), X, DX) -> dout(DDX)
318.81/291.52	
318.81/291.52	S is empty.
318.81/291.52	Rewrite Strategy: FULL
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(3) SlicingProof (LOWER BOUND(ID))
318.81/291.52	Sliced the following arguments:
318.81/291.52	u21/1
318.81/291.52	u22/1
318.81/291.52	u22/2
318.81/291.52	u41/1
318.81/291.52	u42/1
318.81/291.52	u42/2
318.81/291.52	
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(4)
318.81/291.52	Obligation:
318.81/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.81/291.52	
318.81/291.52	
318.81/291.52	The TRS R consists of the following rules:
318.81/291.52	
318.81/291.52	   din(der(plus(X, Y))) -> u21(din(der(X)), Y)
318.81/291.52	   u21(dout(DX), Y) -> u22(din(der(Y)), DX)
318.81/291.52	   u22(dout(DY), DX) -> dout(plus(DX, DY))
318.81/291.52	   din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
318.81/291.52	   u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
318.81/291.52	   u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
318.81/291.52	   din(der(der(X))) -> u41(din(der(X)))
318.81/291.52	   u41(dout(DX)) -> u42(din(der(DX)))
318.81/291.52	   u42(dout(DDX)) -> dout(DDX)
318.81/291.52	
318.81/291.52	S is empty.
318.81/291.52	Rewrite Strategy: FULL
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
318.81/291.52	Infered types.
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(6)
318.81/291.52	Obligation:
318.81/291.52	TRS:
318.81/291.52	Rules:
318.81/291.52	din(der(plus(X, Y))) -> u21(din(der(X)), Y)
318.81/291.52	u21(dout(DX), Y) -> u22(din(der(Y)), DX)
318.81/291.52	u22(dout(DY), DX) -> dout(plus(DX, DY))
318.81/291.52	din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
318.81/291.52	u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
318.81/291.52	u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
318.81/291.52	din(der(der(X))) -> u41(din(der(X)))
318.81/291.52	u41(dout(DX)) -> u42(din(der(DX)))
318.81/291.52	u42(dout(DDX)) -> dout(DDX)
318.81/291.52	
318.81/291.52	Types:
318.81/291.52	din :: plus:der:times -> dout
318.81/291.52	der :: plus:der:times -> plus:der:times
318.81/291.52	plus :: plus:der:times -> plus:der:times -> plus:der:times
318.81/291.52	u21 :: dout -> plus:der:times -> dout
318.81/291.52	dout :: plus:der:times -> dout
318.81/291.52	u22 :: dout -> plus:der:times -> dout
318.81/291.52	times :: plus:der:times -> plus:der:times -> plus:der:times
318.81/291.52	u31 :: dout -> plus:der:times -> plus:der:times -> dout
318.81/291.52	u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
318.81/291.52	u41 :: dout -> dout
318.81/291.52	u42 :: dout -> dout
318.81/291.52	hole_dout1_0 :: dout
318.81/291.52	hole_plus:der:times2_0 :: plus:der:times
318.81/291.52	gen_plus:der:times3_0 :: Nat -> plus:der:times
318.81/291.52	
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(7) OrderProof (LOWER BOUND(ID))
318.81/291.52	Heuristically decided to analyse the following defined symbols:
318.81/291.52	din, u41
318.81/291.52	
318.81/291.52	They will be analysed ascendingly in the following order:
318.81/291.52	din = u41
318.81/291.52	
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(8)
318.81/291.52	Obligation:
318.81/291.52	TRS:
318.81/291.52	Rules:
318.81/291.52	din(der(plus(X, Y))) -> u21(din(der(X)), Y)
318.81/291.52	u21(dout(DX), Y) -> u22(din(der(Y)), DX)
318.81/291.52	u22(dout(DY), DX) -> dout(plus(DX, DY))
318.81/291.52	din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
318.81/291.52	u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
318.81/291.52	u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
318.81/291.52	din(der(der(X))) -> u41(din(der(X)))
318.81/291.52	u41(dout(DX)) -> u42(din(der(DX)))
318.81/291.52	u42(dout(DDX)) -> dout(DDX)
318.81/291.52	
318.81/291.52	Types:
318.81/291.52	din :: plus:der:times -> dout
318.81/291.52	der :: plus:der:times -> plus:der:times
318.81/291.52	plus :: plus:der:times -> plus:der:times -> plus:der:times
318.81/291.52	u21 :: dout -> plus:der:times -> dout
318.81/291.52	dout :: plus:der:times -> dout
318.81/291.52	u22 :: dout -> plus:der:times -> dout
318.81/291.52	times :: plus:der:times -> plus:der:times -> plus:der:times
318.81/291.52	u31 :: dout -> plus:der:times -> plus:der:times -> dout
318.81/291.52	u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
318.81/291.52	u41 :: dout -> dout
318.81/291.52	u42 :: dout -> dout
318.81/291.52	hole_dout1_0 :: dout
318.81/291.52	hole_plus:der:times2_0 :: plus:der:times
318.81/291.52	gen_plus:der:times3_0 :: Nat -> plus:der:times
318.81/291.52	
318.81/291.52	
318.81/291.52	Generator Equations:
318.81/291.52	gen_plus:der:times3_0(0) <=> hole_plus:der:times2_0
318.81/291.52	gen_plus:der:times3_0(+(x, 1)) <=> der(gen_plus:der:times3_0(x))
318.81/291.52	
318.81/291.52	
318.81/291.52	The following defined symbols remain to be analysed:
318.81/291.52	u41, din
318.81/291.52	
318.81/291.52	They will be analysed ascendingly in the following order:
318.81/291.52	din = u41
318.81/291.52	
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(9) RewriteLemmaProof (LOWER BOUND(ID))
318.81/291.52	Proved the following rewrite lemma:
318.81/291.52	din(gen_plus:der:times3_0(+(2, n29_0))) -> *4_0, rt in Omega(n29_0)
318.81/291.52	
318.81/291.52	Induction Base:
318.81/291.52	din(gen_plus:der:times3_0(+(2, 0)))
318.81/291.52	
318.81/291.52	Induction Step:
318.81/291.52	din(gen_plus:der:times3_0(+(2, +(n29_0, 1)))) ->_R^Omega(1)
318.81/291.52	u41(din(der(gen_plus:der:times3_0(+(1, n29_0))))) ->_IH
318.81/291.52	u41(*4_0)
318.81/291.52	
318.81/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(10)
318.81/291.52	Complex Obligation (BEST)
318.81/291.52	
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(11)
318.81/291.52	Obligation:
318.81/291.52	Proved the lower bound n^1 for the following obligation:
318.81/291.52	
318.81/291.52	TRS:
318.81/291.52	Rules:
318.81/291.52	din(der(plus(X, Y))) -> u21(din(der(X)), Y)
318.81/291.52	u21(dout(DX), Y) -> u22(din(der(Y)), DX)
318.81/291.52	u22(dout(DY), DX) -> dout(plus(DX, DY))
318.81/291.52	din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
318.81/291.52	u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
318.81/291.52	u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
318.81/291.52	din(der(der(X))) -> u41(din(der(X)))
318.81/291.52	u41(dout(DX)) -> u42(din(der(DX)))
318.81/291.52	u42(dout(DDX)) -> dout(DDX)
318.81/291.52	
318.81/291.52	Types:
318.81/291.52	din :: plus:der:times -> dout
318.81/291.52	der :: plus:der:times -> plus:der:times
318.81/291.52	plus :: plus:der:times -> plus:der:times -> plus:der:times
318.81/291.52	u21 :: dout -> plus:der:times -> dout
318.81/291.52	dout :: plus:der:times -> dout
318.81/291.52	u22 :: dout -> plus:der:times -> dout
318.81/291.52	times :: plus:der:times -> plus:der:times -> plus:der:times
318.81/291.52	u31 :: dout -> plus:der:times -> plus:der:times -> dout
318.81/291.52	u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
318.81/291.52	u41 :: dout -> dout
318.81/291.52	u42 :: dout -> dout
318.81/291.52	hole_dout1_0 :: dout
318.81/291.52	hole_plus:der:times2_0 :: plus:der:times
318.81/291.52	gen_plus:der:times3_0 :: Nat -> plus:der:times
318.81/291.52	
318.81/291.52	
318.81/291.52	Generator Equations:
318.81/291.52	gen_plus:der:times3_0(0) <=> hole_plus:der:times2_0
318.81/291.52	gen_plus:der:times3_0(+(x, 1)) <=> der(gen_plus:der:times3_0(x))
318.81/291.52	
318.81/291.52	
318.81/291.52	The following defined symbols remain to be analysed:
318.81/291.52	din
318.81/291.52	
318.81/291.52	They will be analysed ascendingly in the following order:
318.81/291.52	din = u41
318.81/291.52	
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(12) LowerBoundPropagationProof (FINISHED)
318.81/291.52	Propagated lower bound.
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(13)
318.81/291.52	BOUNDS(n^1, INF)
318.81/291.52	
318.81/291.52	----------------------------------------
318.81/291.52	
318.81/291.52	(14)
318.81/291.52	Obligation:
318.81/291.52	TRS:
318.81/291.52	Rules:
318.81/291.52	din(der(plus(X, Y))) -> u21(din(der(X)), Y)
318.81/291.52	u21(dout(DX), Y) -> u22(din(der(Y)), DX)
318.81/291.52	u22(dout(DY), DX) -> dout(plus(DX, DY))
318.81/291.52	din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
318.81/291.52	u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
318.81/291.52	u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
318.81/291.52	din(der(der(X))) -> u41(din(der(X)))
318.81/291.52	u41(dout(DX)) -> u42(din(der(DX)))
318.81/291.52	u42(dout(DDX)) -> dout(DDX)
318.81/291.52	
318.81/291.52	Types:
318.81/291.52	din :: plus:der:times -> dout
318.81/291.52	der :: plus:der:times -> plus:der:times
318.81/291.52	plus :: plus:der:times -> plus:der:times -> plus:der:times
318.81/291.52	u21 :: dout -> plus:der:times -> dout
318.81/291.52	dout :: plus:der:times -> dout
318.81/291.52	u22 :: dout -> plus:der:times -> dout
318.81/291.52	times :: plus:der:times -> plus:der:times -> plus:der:times
318.81/291.52	u31 :: dout -> plus:der:times -> plus:der:times -> dout
318.81/291.52	u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
318.81/291.52	u41 :: dout -> dout
318.81/291.52	u42 :: dout -> dout
318.81/291.52	hole_dout1_0 :: dout
318.81/291.52	hole_plus:der:times2_0 :: plus:der:times
318.81/291.52	gen_plus:der:times3_0 :: Nat -> plus:der:times
318.81/291.52	
318.81/291.52	
318.81/291.52	Lemmas:
318.81/291.52	din(gen_plus:der:times3_0(+(2, n29_0))) -> *4_0, rt in Omega(n29_0)
318.81/291.52	
318.81/291.52	
318.81/291.52	Generator Equations:
318.81/291.52	gen_plus:der:times3_0(0) <=> hole_plus:der:times2_0
318.81/291.52	gen_plus:der:times3_0(+(x, 1)) <=> der(gen_plus:der:times3_0(x))
318.81/291.52	
318.81/291.52	
318.81/291.52	The following defined symbols remain to be analysed:
318.81/291.52	u41
318.81/291.52	
318.81/291.52	They will be analysed ascendingly in the following order:
318.81/291.52	din = u41
318.81/291.56	EOF