/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 191 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), 264 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), 34 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(g(x), s(0), y) -> f(g(s(0)), y, g(x))
   g(s(x)) -> s(g(x))
   g(0) -> 0

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, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_s(x_1) -> s(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(g(x), s(0), y) -> f(g(s(0)), y, g(x))
   g(s(x)) -> s(g(x))
   g(0) -> 0

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, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_s(x_1) -> s(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(g(x), s(0), y) -> f(g(s(0)), y, g(x))
   g(s(x)) -> s(g(x))
   g(0) -> 0

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, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_s(x_1) -> s(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(g(x), s(0'), y) -> f(g(s(0')), y, g(x))
   g(s(x)) -> s(g(x))
   g(0') -> 0'

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, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0'

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

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

(8)
Obligation:
TRS:
Rules:
f(g(x), s(0'), y) -> f(g(s(0')), y, g(x))
g(s(x)) -> s(g(x))
g(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1) -> g(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_0 -> 0'

Types:
f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
s :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
0' :: 0':s:cons_f:cons_g
encArg :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
cons_f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
cons_g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_s :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_0 :: 0':s:cons_f:cons_g
hole_0':s:cons_f:cons_g1_4 :: 0':s:cons_f:cons_g
gen_0':s:cons_f:cons_g2_4 :: Nat -> 0':s:cons_f:cons_g

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

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

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

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

(10)
Obligation:
TRS:
Rules:
f(g(x), s(0'), y) -> f(g(s(0')), y, g(x))
g(s(x)) -> s(g(x))
g(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1) -> g(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_0 -> 0'

Types:
f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
s :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
0' :: 0':s:cons_f:cons_g
encArg :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
cons_f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
cons_g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_s :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_0 :: 0':s:cons_f:cons_g
hole_0':s:cons_f:cons_g1_4 :: 0':s:cons_f:cons_g
gen_0':s:cons_f:cons_g2_4 :: Nat -> 0':s:cons_f:cons_g


Generator Equations:
gen_0':s:cons_f:cons_g2_4(0) <=> 0'
gen_0':s:cons_f:cons_g2_4(+(x, 1)) <=> s(gen_0':s:cons_f:cons_g2_4(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':s:cons_f:cons_g2_4(n4_4)) -> gen_0':s:cons_f:cons_g2_4(n4_4), rt in Omega(1 + n4_4)

Induction Base:
g(gen_0':s:cons_f:cons_g2_4(0)) ->_R^Omega(1)
0'

Induction Step:
g(gen_0':s:cons_f:cons_g2_4(+(n4_4, 1))) ->_R^Omega(1)
s(g(gen_0':s:cons_f:cons_g2_4(n4_4))) ->_IH
s(gen_0':s:cons_f:cons_g2_4(c5_4))

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(g(x), s(0'), y) -> f(g(s(0')), y, g(x))
g(s(x)) -> s(g(x))
g(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1) -> g(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_0 -> 0'

Types:
f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
s :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
0' :: 0':s:cons_f:cons_g
encArg :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
cons_f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
cons_g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_s :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_0 :: 0':s:cons_f:cons_g
hole_0':s:cons_f:cons_g1_4 :: 0':s:cons_f:cons_g
gen_0':s:cons_f:cons_g2_4 :: Nat -> 0':s:cons_f:cons_g


Generator Equations:
gen_0':s:cons_f:cons_g2_4(0) <=> 0'
gen_0':s:cons_f:cons_g2_4(+(x, 1)) <=> s(gen_0':s:cons_f:cons_g2_4(x))


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

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
TRS:
Rules:
f(g(x), s(0'), y) -> f(g(s(0')), y, g(x))
g(s(x)) -> s(g(x))
g(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1) -> g(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_0 -> 0'

Types:
f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
s :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
0' :: 0':s:cons_f:cons_g
encArg :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
cons_f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
cons_g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_f :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_g :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_s :: 0':s:cons_f:cons_g -> 0':s:cons_f:cons_g
encode_0 :: 0':s:cons_f:cons_g
hole_0':s:cons_f:cons_g1_4 :: 0':s:cons_f:cons_g
gen_0':s:cons_f:cons_g2_4 :: Nat -> 0':s:cons_f:cons_g


Lemmas:
g(gen_0':s:cons_f:cons_g2_4(n4_4)) -> gen_0':s:cons_f:cons_g2_4(n4_4), rt in Omega(1 + n4_4)


Generator Equations:
gen_0':s:cons_f:cons_g2_4(0) <=> 0'
gen_0':s:cons_f:cons_g2_4(+(x, 1)) <=> s(gen_0':s:cons_f:cons_g2_4(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_0':s:cons_f:cons_g2_4(n190_4)) -> gen_0':s:cons_f:cons_g2_4(n190_4), rt in Omega(0)

Induction Base:
encArg(gen_0':s:cons_f:cons_g2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:cons_f:cons_g2_4(+(n190_4, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:cons_f:cons_g2_4(n190_4))) ->_IH
s(gen_0':s:cons_f:cons_g2_4(c191_4))

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

(18)
BOUNDS(1, INF)