/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 450 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   din(der(plus(X, Y))) -> u21(din(der(X)), X, Y)
   u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX)
   u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY))
   din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
   u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
   u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
   din(der(der(X))) -> u41(din(der(X)), X)
   u41(dout(DX), X) -> u42(din(der(DX)), X, DX)
   u42(dout(DDX), X, DX) -> dout(DDX)

S is empty.
Rewrite Strategy: FULL
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   din(der(plus(X, Y))) -> u21(din(der(X)), X, Y)
   u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX)
   u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY))
   din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
   u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
   u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
   din(der(der(X))) -> u41(din(der(X)), X)
   u41(dout(DX), X) -> u42(din(der(DX)), X, DX)
   u42(dout(DDX), X, DX) -> dout(DDX)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
u21/1
u22/1
u22/2
u41/1
u42/1
u42/2

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(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   din(der(plus(X, Y))) -> u21(din(der(X)), Y)
   u21(dout(DX), Y) -> u22(din(der(Y)), DX)
   u22(dout(DY), DX) -> dout(plus(DX, DY))
   din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
   u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
   u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
   din(der(der(X))) -> u41(din(der(X)))
   u41(dout(DX)) -> u42(din(der(DX)))
   u42(dout(DDX)) -> dout(DDX)

S is empty.
Rewrite Strategy: FULL
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
TRS:
Rules:
din(der(plus(X, Y))) -> u21(din(der(X)), Y)
u21(dout(DX), Y) -> u22(din(der(Y)), DX)
u22(dout(DY), DX) -> dout(plus(DX, DY))
din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) -> u41(din(der(X)))
u41(dout(DX)) -> u42(din(der(DX)))
u42(dout(DDX)) -> dout(DDX)

Types:
din :: plus:der:times -> dout
der :: plus:der:times -> plus:der:times
plus :: plus:der:times -> plus:der:times -> plus:der:times
u21 :: dout -> plus:der:times -> dout
dout :: plus:der:times -> dout
u22 :: dout -> plus:der:times -> dout
times :: plus:der:times -> plus:der:times -> plus:der:times
u31 :: dout -> plus:der:times -> plus:der:times -> dout
u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
u41 :: dout -> dout
u42 :: dout -> dout
hole_dout1_0 :: dout
hole_plus:der:times2_0 :: plus:der:times
gen_plus:der:times3_0 :: Nat -> plus:der:times

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(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
din, u41

They will be analysed ascendingly in the following order:
din = u41

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(8)
Obligation:
TRS:
Rules:
din(der(plus(X, Y))) -> u21(din(der(X)), Y)
u21(dout(DX), Y) -> u22(din(der(Y)), DX)
u22(dout(DY), DX) -> dout(plus(DX, DY))
din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) -> u41(din(der(X)))
u41(dout(DX)) -> u42(din(der(DX)))
u42(dout(DDX)) -> dout(DDX)

Types:
din :: plus:der:times -> dout
der :: plus:der:times -> plus:der:times
plus :: plus:der:times -> plus:der:times -> plus:der:times
u21 :: dout -> plus:der:times -> dout
dout :: plus:der:times -> dout
u22 :: dout -> plus:der:times -> dout
times :: plus:der:times -> plus:der:times -> plus:der:times
u31 :: dout -> plus:der:times -> plus:der:times -> dout
u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
u41 :: dout -> dout
u42 :: dout -> dout
hole_dout1_0 :: dout
hole_plus:der:times2_0 :: plus:der:times
gen_plus:der:times3_0 :: Nat -> plus:der:times


Generator Equations:
gen_plus:der:times3_0(0) <=> hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) <=> der(gen_plus:der:times3_0(x))


The following defined symbols remain to be analysed:
u41, din

They will be analysed ascendingly in the following order:
din = u41

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
din(gen_plus:der:times3_0(+(2, n29_0))) -> *4_0, rt in Omega(n29_0)

Induction Base:
din(gen_plus:der:times3_0(+(2, 0)))

Induction Step:
din(gen_plus:der:times3_0(+(2, +(n29_0, 1)))) ->_R^Omega(1)
u41(din(der(gen_plus:der:times3_0(+(1, n29_0))))) ->_IH
u41(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(10)
Complex Obligation (BEST)

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(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
din(der(plus(X, Y))) -> u21(din(der(X)), Y)
u21(dout(DX), Y) -> u22(din(der(Y)), DX)
u22(dout(DY), DX) -> dout(plus(DX, DY))
din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) -> u41(din(der(X)))
u41(dout(DX)) -> u42(din(der(DX)))
u42(dout(DDX)) -> dout(DDX)

Types:
din :: plus:der:times -> dout
der :: plus:der:times -> plus:der:times
plus :: plus:der:times -> plus:der:times -> plus:der:times
u21 :: dout -> plus:der:times -> dout
dout :: plus:der:times -> dout
u22 :: dout -> plus:der:times -> dout
times :: plus:der:times -> plus:der:times -> plus:der:times
u31 :: dout -> plus:der:times -> plus:der:times -> dout
u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
u41 :: dout -> dout
u42 :: dout -> dout
hole_dout1_0 :: dout
hole_plus:der:times2_0 :: plus:der:times
gen_plus:der:times3_0 :: Nat -> plus:der:times


Generator Equations:
gen_plus:der:times3_0(0) <=> hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) <=> der(gen_plus:der:times3_0(x))


The following defined symbols remain to be analysed:
din

They will be analysed ascendingly in the following order:
din = u41

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(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(14)
Obligation:
TRS:
Rules:
din(der(plus(X, Y))) -> u21(din(der(X)), Y)
u21(dout(DX), Y) -> u22(din(der(Y)), DX)
u22(dout(DY), DX) -> dout(plus(DX, DY))
din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) -> u41(din(der(X)))
u41(dout(DX)) -> u42(din(der(DX)))
u42(dout(DDX)) -> dout(DDX)

Types:
din :: plus:der:times -> dout
der :: plus:der:times -> plus:der:times
plus :: plus:der:times -> plus:der:times -> plus:der:times
u21 :: dout -> plus:der:times -> dout
dout :: plus:der:times -> dout
u22 :: dout -> plus:der:times -> dout
times :: plus:der:times -> plus:der:times -> plus:der:times
u31 :: dout -> plus:der:times -> plus:der:times -> dout
u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
u41 :: dout -> dout
u42 :: dout -> dout
hole_dout1_0 :: dout
hole_plus:der:times2_0 :: plus:der:times
gen_plus:der:times3_0 :: Nat -> plus:der:times


Lemmas:
din(gen_plus:der:times3_0(+(2, n29_0))) -> *4_0, rt in Omega(n29_0)


Generator Equations:
gen_plus:der:times3_0(0) <=> hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) <=> der(gen_plus:der:times3_0(x))


The following defined symbols remain to be analysed:
u41

They will be analysed ascendingly in the following order:
din = u41