/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 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), 167 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), 277 ms]
(12) proven lower bound
(13) LowerBoundPropagationProof [FINISHED, 0 ms]
(14) BOUNDS(n^1, INF)


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

(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:

   :(x, x) -> e
   :(x, e) -> x
   i(:(x, y)) -> :(y, x)
   :(:(x, y), z) -> :(x, :(z, i(y)))
   :(e, x) -> i(x)
   i(i(x)) -> x
   i(e) -> e
   :(x, :(y, i(x))) -> i(y)
   :(x, :(y, :(i(x), z))) -> :(i(z), y)
   :(i(x), :(y, x)) -> i(y)
   :(i(x), :(y, :(x, z))) -> :(i(z), y)

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

(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(e) -> e
   encArg(cons_:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
   encode_e -> e
   encode_i(x_1) -> i(encArg(x_1))


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

(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:

   :(x, x) -> e
   :(x, e) -> x
   i(:(x, y)) -> :(y, x)
   :(:(x, y), z) -> :(x, :(z, i(y)))
   :(e, x) -> i(x)
   i(i(x)) -> x
   i(e) -> e
   :(x, :(y, i(x))) -> i(y)
   :(x, :(y, :(i(x), z))) -> :(i(z), y)
   :(i(x), :(y, x)) -> i(y)
   :(i(x), :(y, :(x, z))) -> :(i(z), y)

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

   encArg(e) -> e
   encArg(cons_:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
   encode_e -> e
   encode_i(x_1) -> i(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(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:

   :(x, x) -> e
   :(x, e) -> x
   i(:(x, y)) -> :(y, x)
   :(:(x, y), z) -> :(x, :(z, i(y)))
   :(e, x) -> i(x)
   i(i(x)) -> x
   i(e) -> e
   :(x, :(y, i(x))) -> i(y)
   :(x, :(y, :(i(x), z))) -> :(i(z), y)
   :(i(x), :(y, x)) -> i(y)
   :(i(x), :(y, :(x, z))) -> :(i(z), y)

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

   encArg(e) -> e
   encArg(cons_:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
   encode_e -> e
   encode_i(x_1) -> i(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(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:

   :(x, x) -> e
   :(x, e) -> x
   i(:(x, y)) -> :(y, x)
   :(:(x, y), z) -> :(x, :(z, i(y)))
   :(e, x) -> i(x)
   i(i(x)) -> x
   i(e) -> e
   :(x, :(y, i(x))) -> i(y)
   :(x, :(y, :(i(x), z))) -> :(i(z), y)
   :(i(x), :(y, x)) -> i(y)
   :(i(x), :(y, :(x, z))) -> :(i(z), y)

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

   encArg(e) -> e
   encArg(cons_:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
   encode_e -> e
   encode_i(x_1) -> i(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Innermost TRS:
Rules:
:(x, x) -> e
:(x, e) -> x
i(:(x, y)) -> :(y, x)
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
i(i(x)) -> x
i(e) -> e
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
encArg(e) -> e
encArg(cons_:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
encode_e -> e
encode_i(x_1) -> i(encArg(x_1))

Types:
: :: e:cons_::cons_i -> e:cons_::cons_i -> e:cons_::cons_i
e :: e:cons_::cons_i
i :: e:cons_::cons_i -> e:cons_::cons_i
encArg :: e:cons_::cons_i -> e:cons_::cons_i
cons_: :: e:cons_::cons_i -> e:cons_::cons_i -> e:cons_::cons_i
cons_i :: e:cons_::cons_i -> e:cons_::cons_i
encode_: :: e:cons_::cons_i -> e:cons_::cons_i -> e:cons_::cons_i
encode_e :: e:cons_::cons_i
encode_i :: e:cons_::cons_i -> e:cons_::cons_i
hole_e:cons_::cons_i1_0 :: e:cons_::cons_i
gen_e:cons_::cons_i2_0 :: Nat -> e:cons_::cons_i

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

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

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(10)
Obligation:
Innermost TRS:
Rules:
:(x, x) -> e
:(x, e) -> x
i(:(x, y)) -> :(y, x)
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
i(i(x)) -> x
i(e) -> e
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
encArg(e) -> e
encArg(cons_:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
encode_e -> e
encode_i(x_1) -> i(encArg(x_1))

Types:
: :: e:cons_::cons_i -> e:cons_::cons_i -> e:cons_::cons_i
e :: e:cons_::cons_i
i :: e:cons_::cons_i -> e:cons_::cons_i
encArg :: e:cons_::cons_i -> e:cons_::cons_i
cons_: :: e:cons_::cons_i -> e:cons_::cons_i -> e:cons_::cons_i
cons_i :: e:cons_::cons_i -> e:cons_::cons_i
encode_: :: e:cons_::cons_i -> e:cons_::cons_i -> e:cons_::cons_i
encode_e :: e:cons_::cons_i
encode_i :: e:cons_::cons_i -> e:cons_::cons_i
hole_e:cons_::cons_i1_0 :: e:cons_::cons_i
gen_e:cons_::cons_i2_0 :: Nat -> e:cons_::cons_i


Generator Equations:
gen_e:cons_::cons_i2_0(0) <=> e
gen_e:cons_::cons_i2_0(+(x, 1)) <=> cons_:(e, gen_e:cons_::cons_i2_0(x))


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

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

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_e:cons_::cons_i2_0(n317_0)) -> gen_e:cons_::cons_i2_0(0), rt in Omega(n317_0)

Induction Base:
encArg(gen_e:cons_::cons_i2_0(0)) ->_R^Omega(0)
e

Induction Step:
encArg(gen_e:cons_::cons_i2_0(+(n317_0, 1))) ->_R^Omega(0)
:(encArg(e), encArg(gen_e:cons_::cons_i2_0(n317_0))) ->_R^Omega(0)
:(e, encArg(gen_e:cons_::cons_i2_0(n317_0))) ->_IH
:(e, gen_e:cons_::cons_i2_0(0)) ->_R^Omega(1)
e

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(12)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
:(x, x) -> e
:(x, e) -> x
i(:(x, y)) -> :(y, x)
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
i(i(x)) -> x
i(e) -> e
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
encArg(e) -> e
encArg(cons_:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
encode_e -> e
encode_i(x_1) -> i(encArg(x_1))

Types:
: :: e:cons_::cons_i -> e:cons_::cons_i -> e:cons_::cons_i
e :: e:cons_::cons_i
i :: e:cons_::cons_i -> e:cons_::cons_i
encArg :: e:cons_::cons_i -> e:cons_::cons_i
cons_: :: e:cons_::cons_i -> e:cons_::cons_i -> e:cons_::cons_i
cons_i :: e:cons_::cons_i -> e:cons_::cons_i
encode_: :: e:cons_::cons_i -> e:cons_::cons_i -> e:cons_::cons_i
encode_e :: e:cons_::cons_i
encode_i :: e:cons_::cons_i -> e:cons_::cons_i
hole_e:cons_::cons_i1_0 :: e:cons_::cons_i
gen_e:cons_::cons_i2_0 :: Nat -> e:cons_::cons_i


Generator Equations:
gen_e:cons_::cons_i2_0(0) <=> e
gen_e:cons_::cons_i2_0(+(x, 1)) <=> cons_:(e, gen_e:cons_::cons_i2_0(x))


The following defined symbols remain to be analysed:
encArg
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(13) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(14)
BOUNDS(n^1, INF)