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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 229 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) 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:

   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
   a____(X, nil) -> mark(X)
   a____(nil, X) -> mark(X)
   a__U11(tt) -> a__U12(tt)
   a__U12(tt) -> tt
   a__isNePal(__(I, __(P, I))) -> a__U11(tt)
   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X)) -> a__U11(mark(X))
   mark(U12(X)) -> a__U12(mark(X))
   mark(isNePal(X)) -> a__isNePal(mark(X))
   mark(nil) -> nil
   mark(tt) -> tt
   a____(X1, X2) -> __(X1, X2)
   a__U11(X) -> U11(X)
   a__U12(X) -> U12(X)
   a__isNePal(X) -> isNePal(X)

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.
----------------------------------------

(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:

   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
   a____(X, nil) -> mark(X)
   a____(nil, X) -> mark(X)
   a__U11(tt) -> a__U12(tt)
   a__U12(tt) -> tt
   a__isNePal(__(I, __(P, I))) -> a__U11(tt)
   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X)) -> a__U11(mark(X))
   mark(U12(X)) -> a__U12(mark(X))
   mark(isNePal(X)) -> a__isNePal(mark(X))
   mark(nil) -> nil
   mark(tt) -> tt
   a____(X1, X2) -> __(X1, X2)
   a__U11(X) -> U11(X)
   a__U12(X) -> U12(X)
   a__isNePal(X) -> isNePal(X)

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
TRS:
Rules:
a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) -> mark(X)
a____(nil, X) -> mark(X)
a__U11(tt) -> a__U12(tt)
a__U12(tt) -> tt
a__isNePal(__(I, __(P, I))) -> a__U11(tt)
mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil) -> nil
mark(tt) -> tt
a____(X1, X2) -> __(X1, X2)
a__U11(X) -> U11(X)
a__U12(X) -> U12(X)
a__isNePal(X) -> isNePal(X)

Types:
a____ :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
__ :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
mark :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
nil :: __:nil:tt:U11:U12:isNePal
a__U11 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
tt :: __:nil:tt:U11:U12:isNePal
a__U12 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
a__isNePal :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
U11 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
U12 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
isNePal :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
hole___:nil:tt:U11:U12:isNePal1_0 :: __:nil:tt:U11:U12:isNePal
gen___:nil:tt:U11:U12:isNePal2_0 :: Nat -> __:nil:tt:U11:U12:isNePal

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a____, mark

They will be analysed ascendingly in the following order:
a____ = mark

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(6)
Obligation:
TRS:
Rules:
a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) -> mark(X)
a____(nil, X) -> mark(X)
a__U11(tt) -> a__U12(tt)
a__U12(tt) -> tt
a__isNePal(__(I, __(P, I))) -> a__U11(tt)
mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil) -> nil
mark(tt) -> tt
a____(X1, X2) -> __(X1, X2)
a__U11(X) -> U11(X)
a__U12(X) -> U12(X)
a__isNePal(X) -> isNePal(X)

Types:
a____ :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
__ :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
mark :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
nil :: __:nil:tt:U11:U12:isNePal
a__U11 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
tt :: __:nil:tt:U11:U12:isNePal
a__U12 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
a__isNePal :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
U11 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
U12 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
isNePal :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
hole___:nil:tt:U11:U12:isNePal1_0 :: __:nil:tt:U11:U12:isNePal
gen___:nil:tt:U11:U12:isNePal2_0 :: Nat -> __:nil:tt:U11:U12:isNePal


Generator Equations:
gen___:nil:tt:U11:U12:isNePal2_0(0) <=> nil
gen___:nil:tt:U11:U12:isNePal2_0(+(x, 1)) <=> __(nil, gen___:nil:tt:U11:U12:isNePal2_0(x))


The following defined symbols remain to be analysed:
mark, a____

They will be analysed ascendingly in the following order:
a____ = mark

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen___:nil:tt:U11:U12:isNePal2_0(n4_0)) -> gen___:nil:tt:U11:U12:isNePal2_0(0), rt in Omega(1 + n4_0)

Induction Base:
mark(gen___:nil:tt:U11:U12:isNePal2_0(0)) ->_R^Omega(1)
nil

Induction Step:
mark(gen___:nil:tt:U11:U12:isNePal2_0(+(n4_0, 1))) ->_R^Omega(1)
a____(mark(nil), mark(gen___:nil:tt:U11:U12:isNePal2_0(n4_0))) ->_R^Omega(1)
a____(nil, mark(gen___:nil:tt:U11:U12:isNePal2_0(n4_0))) ->_IH
a____(nil, gen___:nil:tt:U11:U12:isNePal2_0(0)) ->_R^Omega(1)
mark(nil) ->_R^Omega(1)
nil

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(8)
Complex Obligation (BEST)

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(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) -> mark(X)
a____(nil, X) -> mark(X)
a__U11(tt) -> a__U12(tt)
a__U12(tt) -> tt
a__isNePal(__(I, __(P, I))) -> a__U11(tt)
mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil) -> nil
mark(tt) -> tt
a____(X1, X2) -> __(X1, X2)
a__U11(X) -> U11(X)
a__U12(X) -> U12(X)
a__isNePal(X) -> isNePal(X)

Types:
a____ :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
__ :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
mark :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
nil :: __:nil:tt:U11:U12:isNePal
a__U11 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
tt :: __:nil:tt:U11:U12:isNePal
a__U12 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
a__isNePal :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
U11 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
U12 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
isNePal :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
hole___:nil:tt:U11:U12:isNePal1_0 :: __:nil:tt:U11:U12:isNePal
gen___:nil:tt:U11:U12:isNePal2_0 :: Nat -> __:nil:tt:U11:U12:isNePal


Generator Equations:
gen___:nil:tt:U11:U12:isNePal2_0(0) <=> nil
gen___:nil:tt:U11:U12:isNePal2_0(+(x, 1)) <=> __(nil, gen___:nil:tt:U11:U12:isNePal2_0(x))


The following defined symbols remain to be analysed:
mark, a____

They will be analysed ascendingly in the following order:
a____ = mark

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
TRS:
Rules:
a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) -> mark(X)
a____(nil, X) -> mark(X)
a__U11(tt) -> a__U12(tt)
a__U12(tt) -> tt
a__isNePal(__(I, __(P, I))) -> a__U11(tt)
mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil) -> nil
mark(tt) -> tt
a____(X1, X2) -> __(X1, X2)
a__U11(X) -> U11(X)
a__U12(X) -> U12(X)
a__isNePal(X) -> isNePal(X)

Types:
a____ :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
__ :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
mark :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
nil :: __:nil:tt:U11:U12:isNePal
a__U11 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
tt :: __:nil:tt:U11:U12:isNePal
a__U12 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
a__isNePal :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
U11 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
U12 :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
isNePal :: __:nil:tt:U11:U12:isNePal -> __:nil:tt:U11:U12:isNePal
hole___:nil:tt:U11:U12:isNePal1_0 :: __:nil:tt:U11:U12:isNePal
gen___:nil:tt:U11:U12:isNePal2_0 :: Nat -> __:nil:tt:U11:U12:isNePal


Lemmas:
mark(gen___:nil:tt:U11:U12:isNePal2_0(n4_0)) -> gen___:nil:tt:U11:U12:isNePal2_0(0), rt in Omega(1 + n4_0)


Generator Equations:
gen___:nil:tt:U11:U12:isNePal2_0(0) <=> nil
gen___:nil:tt:U11:U12:isNePal2_0(+(x, 1)) <=> __(nil, gen___:nil:tt:U11:U12:isNePal2_0(x))


The following defined symbols remain to be analysed:
a____

They will be analysed ascendingly in the following order:
a____ = mark