/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Derivational Complexity (full) 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), 168 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), 366 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), 51 ms]
        (18) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   f(s(x), x) -> f(s(x), round(s(x)))
   round(0) -> 0
   round(0) -> s(0)
   round(s(0)) -> s(0)
   round(s(s(x))) -> s(s(round(x)))

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

(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(0) -> 0
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_round(x_1)) -> round(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_round(x_1) -> round(encArg(x_1))
   encode_0 -> 0


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

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


The TRS R consists of the following rules:

   f(s(x), x) -> f(s(x), round(s(x)))
   round(0) -> 0
   round(0) -> s(0)
   round(s(0)) -> s(0)
   round(s(s(x))) -> s(s(round(x)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_round(x_1)) -> round(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_round(x_1) -> round(encArg(x_1))
   encode_0 -> 0

Rewrite Strategy: FULL
----------------------------------------

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

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


The TRS R consists of the following rules:

   f(s(x), x) -> f(s(x), round(s(x)))
   round(0) -> 0
   round(0) -> s(0)
   round(s(0)) -> s(0)
   round(s(s(x))) -> s(s(round(x)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_round(x_1)) -> round(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_round(x_1) -> round(encArg(x_1))
   encode_0 -> 0

Rewrite Strategy: FULL
----------------------------------------

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

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(s(x), x) -> f(s(x), round(s(x)))
   round(0') -> 0'
   round(0') -> s(0')
   round(s(0')) -> s(0')
   round(s(s(x))) -> s(s(round(x)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0') -> 0'
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_round(x_1)) -> round(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_round(x_1) -> round(encArg(x_1))
   encode_0 -> 0'

Rewrite Strategy: FULL
----------------------------------------

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

(8)
Obligation:
TRS:
Rules:
f(s(x), x) -> f(s(x), round(s(x)))
round(0') -> 0'
round(0') -> s(0')
round(s(0')) -> s(0')
round(s(s(x))) -> s(s(round(x)))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_round(x_1)) -> round(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_round(x_1) -> round(encArg(x_1))
encode_0 -> 0'

Types:
f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
s :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
0' :: s:0':cons_f:cons_round
encArg :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
cons_f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
cons_round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_s :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_0 :: s:0':cons_f:cons_round
hole_s:0':cons_f:cons_round1_3 :: s:0':cons_f:cons_round
gen_s:0':cons_f:cons_round2_3 :: Nat -> s:0':cons_f:cons_round

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

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

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

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

(10)
Obligation:
TRS:
Rules:
f(s(x), x) -> f(s(x), round(s(x)))
round(0') -> 0'
round(0') -> s(0')
round(s(0')) -> s(0')
round(s(s(x))) -> s(s(round(x)))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_round(x_1)) -> round(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_round(x_1) -> round(encArg(x_1))
encode_0 -> 0'

Types:
f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
s :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
0' :: s:0':cons_f:cons_round
encArg :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
cons_f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
cons_round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_s :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_0 :: s:0':cons_f:cons_round
hole_s:0':cons_f:cons_round1_3 :: s:0':cons_f:cons_round
gen_s:0':cons_f:cons_round2_3 :: Nat -> s:0':cons_f:cons_round


Generator Equations:
gen_s:0':cons_f:cons_round2_3(0) <=> 0'
gen_s:0':cons_f:cons_round2_3(+(x, 1)) <=> s(gen_s:0':cons_f:cons_round2_3(x))


The following defined symbols remain to be analysed:
round, f, encArg

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
round(gen_s:0':cons_f:cons_round2_3(*(2, n4_3))) -> gen_s:0':cons_f:cons_round2_3(*(2, n4_3)), rt in Omega(1 + n4_3)

Induction Base:
round(gen_s:0':cons_f:cons_round2_3(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
round(gen_s:0':cons_f:cons_round2_3(*(2, +(n4_3, 1)))) ->_R^Omega(1)
s(s(round(gen_s:0':cons_f:cons_round2_3(*(2, n4_3))))) ->_IH
s(s(gen_s:0':cons_f:cons_round2_3(*(2, c5_3))))

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

(12)
Complex Obligation (BEST)

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

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

TRS:
Rules:
f(s(x), x) -> f(s(x), round(s(x)))
round(0') -> 0'
round(0') -> s(0')
round(s(0')) -> s(0')
round(s(s(x))) -> s(s(round(x)))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_round(x_1)) -> round(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_round(x_1) -> round(encArg(x_1))
encode_0 -> 0'

Types:
f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
s :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
0' :: s:0':cons_f:cons_round
encArg :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
cons_f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
cons_round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_s :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_0 :: s:0':cons_f:cons_round
hole_s:0':cons_f:cons_round1_3 :: s:0':cons_f:cons_round
gen_s:0':cons_f:cons_round2_3 :: Nat -> s:0':cons_f:cons_round


Generator Equations:
gen_s:0':cons_f:cons_round2_3(0) <=> 0'
gen_s:0':cons_f:cons_round2_3(+(x, 1)) <=> s(gen_s:0':cons_f:cons_round2_3(x))


The following defined symbols remain to be analysed:
round, f, encArg

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

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

(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
TRS:
Rules:
f(s(x), x) -> f(s(x), round(s(x)))
round(0') -> 0'
round(0') -> s(0')
round(s(0')) -> s(0')
round(s(s(x))) -> s(s(round(x)))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_round(x_1)) -> round(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_round(x_1) -> round(encArg(x_1))
encode_0 -> 0'

Types:
f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
s :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
0' :: s:0':cons_f:cons_round
encArg :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
cons_f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
cons_round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_f :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_s :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_round :: s:0':cons_f:cons_round -> s:0':cons_f:cons_round
encode_0 :: s:0':cons_f:cons_round
hole_s:0':cons_f:cons_round1_3 :: s:0':cons_f:cons_round
gen_s:0':cons_f:cons_round2_3 :: Nat -> s:0':cons_f:cons_round


Lemmas:
round(gen_s:0':cons_f:cons_round2_3(*(2, n4_3))) -> gen_s:0':cons_f:cons_round2_3(*(2, n4_3)), rt in Omega(1 + n4_3)


Generator Equations:
gen_s:0':cons_f:cons_round2_3(0) <=> 0'
gen_s:0':cons_f:cons_round2_3(+(x, 1)) <=> s(gen_s:0':cons_f:cons_round2_3(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_s:0':cons_f:cons_round2_3(n587_3)) -> gen_s:0':cons_f:cons_round2_3(n587_3), rt in Omega(0)

Induction Base:
encArg(gen_s:0':cons_f:cons_round2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_s:0':cons_f:cons_round2_3(+(n587_3, 1))) ->_R^Omega(0)
s(encArg(gen_s:0':cons_f:cons_round2_3(n587_3))) ->_IH
s(gen_s:0':cons_f:cons_round2_3(c588_3))

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

(18)
BOUNDS(1, INF)