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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 40 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 1173 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 89 ms]
        (18) BOUNDS(1, INF)


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(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

S is empty.
Rewrite Strategy: INNERMOST
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(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_dfib(x_1, x_2)) -> dfib(encArg(x_1), encArg(x_2))
   encode_dfib(x_1, x_2) -> dfib(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))


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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

The (relative) TRS S consists of the following rules:

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_dfib(x_1, x_2)) -> dfib(encArg(x_1), encArg(x_2))
   encode_dfib(x_1, x_2) -> dfib(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

The (relative) TRS S consists of the following rules:

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_dfib(x_1, x_2)) -> dfib(encArg(x_1), encArg(x_2))
   encode_dfib(x_1, x_2) -> dfib(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST
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(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

The (relative) TRS S consists of the following rules:

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_dfib(x_1, x_2)) -> dfib(encArg(x_1), encArg(x_2))
   encode_dfib(x_1, x_2) -> dfib(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Innermost TRS:
Rules:
dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_dfib(x_1, x_2)) -> dfib(encArg(x_1), encArg(x_2))
encode_dfib(x_1, x_2) -> dfib(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))

Types:
dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
s :: s:cons_dfib -> s:cons_dfib
encArg :: s:cons_dfib -> s:cons_dfib
cons_dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
encode_dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
encode_s :: s:cons_dfib -> s:cons_dfib
hole_s:cons_dfib1_0 :: s:cons_dfib
gen_s:cons_dfib2_0 :: Nat -> s:cons_dfib

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
dfib, encArg

They will be analysed ascendingly in the following order:
dfib < encArg

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(10)
Obligation:
Innermost TRS:
Rules:
dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_dfib(x_1, x_2)) -> dfib(encArg(x_1), encArg(x_2))
encode_dfib(x_1, x_2) -> dfib(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))

Types:
dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
s :: s:cons_dfib -> s:cons_dfib
encArg :: s:cons_dfib -> s:cons_dfib
cons_dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
encode_dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
encode_s :: s:cons_dfib -> s:cons_dfib
hole_s:cons_dfib1_0 :: s:cons_dfib
gen_s:cons_dfib2_0 :: Nat -> s:cons_dfib


Generator Equations:
gen_s:cons_dfib2_0(0) <=> hole_s:cons_dfib1_0
gen_s:cons_dfib2_0(+(x, 1)) <=> s(gen_s:cons_dfib2_0(x))


The following defined symbols remain to be analysed:
dfib, encArg

They will be analysed ascendingly in the following order:
dfib < encArg

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
dfib(gen_s:cons_dfib2_0(+(2, *(2, n4_0))), gen_s:cons_dfib2_0(b)) -> *3_0, rt in Omega(n4_0)

Induction Base:
dfib(gen_s:cons_dfib2_0(+(2, *(2, 0))), gen_s:cons_dfib2_0(b))

Induction Step:
dfib(gen_s:cons_dfib2_0(+(2, *(2, +(n4_0, 1)))), gen_s:cons_dfib2_0(b)) ->_R^Omega(1)
dfib(s(gen_s:cons_dfib2_0(+(2, *(2, n4_0)))), dfib(gen_s:cons_dfib2_0(+(2, *(2, n4_0))), gen_s:cons_dfib2_0(b))) ->_IH
dfib(s(gen_s:cons_dfib2_0(+(2, *(2, n4_0)))), *3_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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(12)
Complex Obligation (BEST)

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(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_dfib(x_1, x_2)) -> dfib(encArg(x_1), encArg(x_2))
encode_dfib(x_1, x_2) -> dfib(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))

Types:
dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
s :: s:cons_dfib -> s:cons_dfib
encArg :: s:cons_dfib -> s:cons_dfib
cons_dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
encode_dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
encode_s :: s:cons_dfib -> s:cons_dfib
hole_s:cons_dfib1_0 :: s:cons_dfib
gen_s:cons_dfib2_0 :: Nat -> s:cons_dfib


Generator Equations:
gen_s:cons_dfib2_0(0) <=> hole_s:cons_dfib1_0
gen_s:cons_dfib2_0(+(x, 1)) <=> s(gen_s:cons_dfib2_0(x))


The following defined symbols remain to be analysed:
dfib, encArg

They will be analysed ascendingly in the following order:
dfib < encArg

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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Innermost TRS:
Rules:
dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_dfib(x_1, x_2)) -> dfib(encArg(x_1), encArg(x_2))
encode_dfib(x_1, x_2) -> dfib(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))

Types:
dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
s :: s:cons_dfib -> s:cons_dfib
encArg :: s:cons_dfib -> s:cons_dfib
cons_dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
encode_dfib :: s:cons_dfib -> s:cons_dfib -> s:cons_dfib
encode_s :: s:cons_dfib -> s:cons_dfib
hole_s:cons_dfib1_0 :: s:cons_dfib
gen_s:cons_dfib2_0 :: Nat -> s:cons_dfib


Lemmas:
dfib(gen_s:cons_dfib2_0(+(2, *(2, n4_0))), gen_s:cons_dfib2_0(b)) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_s:cons_dfib2_0(0) <=> hole_s:cons_dfib1_0
gen_s:cons_dfib2_0(+(x, 1)) <=> s(gen_s:cons_dfib2_0(x))


The following defined symbols remain to be analysed:
encArg
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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_s:cons_dfib2_0(+(1, n650_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_s:cons_dfib2_0(+(1, 0)))

Induction Step:
encArg(gen_s:cons_dfib2_0(+(1, +(n650_0, 1)))) ->_R^Omega(0)
s(encArg(gen_s:cons_dfib2_0(+(1, n650_0)))) ->_IH
s(*3_0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(18)
BOUNDS(1, INF)