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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 79 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), 462 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), 90 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 91 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 105 ms]
        (22) 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:

   s(b(x1)) -> b(s(s(s(x1))))
   s(b(s(x1))) -> b(t(x1))
   t(b(x1)) -> b(s(x1))
   t(b(s(x1))) -> u(t(b(x1)))
   b(u(x1)) -> b(s(x1))
   t(s(x1)) -> t(t(x1))
   t(u(x1)) -> u(t(x1))
   s(u(x1)) -> s(s(x1))

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(u(x_1)) -> u(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_t(x_1)) -> t(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_t(x_1) -> t(encArg(x_1))
   encode_u(x_1) -> u(encArg(x_1))


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

(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:

   s(b(x1)) -> b(s(s(s(x1))))
   s(b(s(x1))) -> b(t(x1))
   t(b(x1)) -> b(s(x1))
   t(b(s(x1))) -> u(t(b(x1)))
   b(u(x1)) -> b(s(x1))
   t(s(x1)) -> t(t(x1))
   t(u(x1)) -> u(t(x1))
   s(u(x1)) -> s(s(x1))

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

   encArg(u(x_1)) -> u(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_t(x_1)) -> t(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_t(x_1) -> t(encArg(x_1))
   encode_u(x_1) -> u(encArg(x_1))

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:

   s(b(x1)) -> b(s(s(s(x1))))
   s(b(s(x1))) -> b(t(x1))
   t(b(x1)) -> b(s(x1))
   t(b(s(x1))) -> u(t(b(x1)))
   b(u(x1)) -> b(s(x1))
   t(s(x1)) -> t(t(x1))
   t(u(x1)) -> u(t(x1))
   s(u(x1)) -> s(s(x1))

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

   encArg(u(x_1)) -> u(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_t(x_1)) -> t(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_t(x_1) -> t(encArg(x_1))
   encode_u(x_1) -> u(encArg(x_1))

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:

   s(b(x1)) -> b(s(s(s(x1))))
   s(b(s(x1))) -> b(t(x1))
   t(b(x1)) -> b(s(x1))
   t(b(s(x1))) -> u(t(b(x1)))
   b(u(x1)) -> b(s(x1))
   t(s(x1)) -> t(t(x1))
   t(u(x1)) -> u(t(x1))
   s(u(x1)) -> s(s(x1))

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

   encArg(u(x_1)) -> u(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_t(x_1)) -> t(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_t(x_1) -> t(encArg(x_1))
   encode_u(x_1) -> u(encArg(x_1))

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

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

(8)
Obligation:
TRS:
Rules:
s(b(x1)) -> b(s(s(s(x1))))
s(b(s(x1))) -> b(t(x1))
t(b(x1)) -> b(s(x1))
t(b(s(x1))) -> u(t(b(x1)))
b(u(x1)) -> b(s(x1))
t(s(x1)) -> t(t(x1))
t(u(x1)) -> u(t(x1))
s(u(x1)) -> s(s(x1))
encArg(u(x_1)) -> u(encArg(x_1))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_t(x_1)) -> t(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_t(x_1) -> t(encArg(x_1))
encode_u(x_1) -> u(encArg(x_1))

Types:
s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encArg :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
hole_u:cons_s:cons_t:cons_b1_0 :: u:cons_s:cons_t:cons_b
gen_u:cons_s:cons_t:cons_b2_0 :: Nat -> u:cons_s:cons_t:cons_b

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

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

They will be analysed ascendingly in the following order:
s = b
s = t
s < encArg
b = t
b < encArg
t < encArg

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

(10)
Obligation:
TRS:
Rules:
s(b(x1)) -> b(s(s(s(x1))))
s(b(s(x1))) -> b(t(x1))
t(b(x1)) -> b(s(x1))
t(b(s(x1))) -> u(t(b(x1)))
b(u(x1)) -> b(s(x1))
t(s(x1)) -> t(t(x1))
t(u(x1)) -> u(t(x1))
s(u(x1)) -> s(s(x1))
encArg(u(x_1)) -> u(encArg(x_1))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_t(x_1)) -> t(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_t(x_1) -> t(encArg(x_1))
encode_u(x_1) -> u(encArg(x_1))

Types:
s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encArg :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
hole_u:cons_s:cons_t:cons_b1_0 :: u:cons_s:cons_t:cons_b
gen_u:cons_s:cons_t:cons_b2_0 :: Nat -> u:cons_s:cons_t:cons_b


Generator Equations:
gen_u:cons_s:cons_t:cons_b2_0(0) <=> hole_u:cons_s:cons_t:cons_b1_0
gen_u:cons_s:cons_t:cons_b2_0(+(x, 1)) <=> u(gen_u:cons_s:cons_t:cons_b2_0(x))


The following defined symbols remain to be analysed:
b, s, t, encArg

They will be analysed ascendingly in the following order:
s = b
s = t
s < encArg
b = t
b < encArg
t < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_u:cons_s:cons_t:cons_b2_0(+(1, n354_0))) -> *3_0, rt in Omega(n354_0)

Induction Base:
s(gen_u:cons_s:cons_t:cons_b2_0(+(1, 0)))

Induction Step:
s(gen_u:cons_s:cons_t:cons_b2_0(+(1, +(n354_0, 1)))) ->_R^Omega(1)
s(s(gen_u:cons_s:cons_t:cons_b2_0(+(1, n354_0)))) ->_IH
s(*3_0)

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:
s(b(x1)) -> b(s(s(s(x1))))
s(b(s(x1))) -> b(t(x1))
t(b(x1)) -> b(s(x1))
t(b(s(x1))) -> u(t(b(x1)))
b(u(x1)) -> b(s(x1))
t(s(x1)) -> t(t(x1))
t(u(x1)) -> u(t(x1))
s(u(x1)) -> s(s(x1))
encArg(u(x_1)) -> u(encArg(x_1))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_t(x_1)) -> t(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_t(x_1) -> t(encArg(x_1))
encode_u(x_1) -> u(encArg(x_1))

Types:
s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encArg :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
hole_u:cons_s:cons_t:cons_b1_0 :: u:cons_s:cons_t:cons_b
gen_u:cons_s:cons_t:cons_b2_0 :: Nat -> u:cons_s:cons_t:cons_b


Generator Equations:
gen_u:cons_s:cons_t:cons_b2_0(0) <=> hole_u:cons_s:cons_t:cons_b1_0
gen_u:cons_s:cons_t:cons_b2_0(+(x, 1)) <=> u(gen_u:cons_s:cons_t:cons_b2_0(x))


The following defined symbols remain to be analysed:
s, t, encArg

They will be analysed ascendingly in the following order:
s = b
s = t
s < encArg
b = t
b < encArg
t < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
TRS:
Rules:
s(b(x1)) -> b(s(s(s(x1))))
s(b(s(x1))) -> b(t(x1))
t(b(x1)) -> b(s(x1))
t(b(s(x1))) -> u(t(b(x1)))
b(u(x1)) -> b(s(x1))
t(s(x1)) -> t(t(x1))
t(u(x1)) -> u(t(x1))
s(u(x1)) -> s(s(x1))
encArg(u(x_1)) -> u(encArg(x_1))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_t(x_1)) -> t(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_t(x_1) -> t(encArg(x_1))
encode_u(x_1) -> u(encArg(x_1))

Types:
s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encArg :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
hole_u:cons_s:cons_t:cons_b1_0 :: u:cons_s:cons_t:cons_b
gen_u:cons_s:cons_t:cons_b2_0 :: Nat -> u:cons_s:cons_t:cons_b


Lemmas:
s(gen_u:cons_s:cons_t:cons_b2_0(+(1, n354_0))) -> *3_0, rt in Omega(n354_0)


Generator Equations:
gen_u:cons_s:cons_t:cons_b2_0(0) <=> hole_u:cons_s:cons_t:cons_b1_0
gen_u:cons_s:cons_t:cons_b2_0(+(x, 1)) <=> u(gen_u:cons_s:cons_t:cons_b2_0(x))


The following defined symbols remain to be analysed:
t, b, encArg

They will be analysed ascendingly in the following order:
s = b
s = t
s < encArg
b = t
b < encArg
t < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
t(gen_u:cons_s:cons_t:cons_b2_0(+(1, n771_0))) -> *3_0, rt in Omega(n771_0)

Induction Base:
t(gen_u:cons_s:cons_t:cons_b2_0(+(1, 0)))

Induction Step:
t(gen_u:cons_s:cons_t:cons_b2_0(+(1, +(n771_0, 1)))) ->_R^Omega(1)
u(t(gen_u:cons_s:cons_t:cons_b2_0(+(1, n771_0)))) ->_IH
u(*3_0)

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

(18)
Obligation:
TRS:
Rules:
s(b(x1)) -> b(s(s(s(x1))))
s(b(s(x1))) -> b(t(x1))
t(b(x1)) -> b(s(x1))
t(b(s(x1))) -> u(t(b(x1)))
b(u(x1)) -> b(s(x1))
t(s(x1)) -> t(t(x1))
t(u(x1)) -> u(t(x1))
s(u(x1)) -> s(s(x1))
encArg(u(x_1)) -> u(encArg(x_1))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_t(x_1)) -> t(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_t(x_1) -> t(encArg(x_1))
encode_u(x_1) -> u(encArg(x_1))

Types:
s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encArg :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
hole_u:cons_s:cons_t:cons_b1_0 :: u:cons_s:cons_t:cons_b
gen_u:cons_s:cons_t:cons_b2_0 :: Nat -> u:cons_s:cons_t:cons_b


Lemmas:
s(gen_u:cons_s:cons_t:cons_b2_0(+(1, n354_0))) -> *3_0, rt in Omega(n354_0)
t(gen_u:cons_s:cons_t:cons_b2_0(+(1, n771_0))) -> *3_0, rt in Omega(n771_0)


Generator Equations:
gen_u:cons_s:cons_t:cons_b2_0(0) <=> hole_u:cons_s:cons_t:cons_b1_0
gen_u:cons_s:cons_t:cons_b2_0(+(x, 1)) <=> u(gen_u:cons_s:cons_t:cons_b2_0(x))


The following defined symbols remain to be analysed:
b, s, encArg

They will be analysed ascendingly in the following order:
s = b
s = t
s < encArg
b = t
b < encArg
t < encArg

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_u:cons_s:cons_t:cons_b2_0(+(1, n2278_0))) -> *3_0, rt in Omega(n2278_0)

Induction Base:
s(gen_u:cons_s:cons_t:cons_b2_0(+(1, 0)))

Induction Step:
s(gen_u:cons_s:cons_t:cons_b2_0(+(1, +(n2278_0, 1)))) ->_R^Omega(1)
s(s(gen_u:cons_s:cons_t:cons_b2_0(+(1, n2278_0)))) ->_IH
s(*3_0)

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

(20)
Obligation:
TRS:
Rules:
s(b(x1)) -> b(s(s(s(x1))))
s(b(s(x1))) -> b(t(x1))
t(b(x1)) -> b(s(x1))
t(b(s(x1))) -> u(t(b(x1)))
b(u(x1)) -> b(s(x1))
t(s(x1)) -> t(t(x1))
t(u(x1)) -> u(t(x1))
s(u(x1)) -> s(s(x1))
encArg(u(x_1)) -> u(encArg(x_1))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_t(x_1)) -> t(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_t(x_1) -> t(encArg(x_1))
encode_u(x_1) -> u(encArg(x_1))

Types:
s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encArg :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
cons_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_s :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_b :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_t :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
encode_u :: u:cons_s:cons_t:cons_b -> u:cons_s:cons_t:cons_b
hole_u:cons_s:cons_t:cons_b1_0 :: u:cons_s:cons_t:cons_b
gen_u:cons_s:cons_t:cons_b2_0 :: Nat -> u:cons_s:cons_t:cons_b


Lemmas:
s(gen_u:cons_s:cons_t:cons_b2_0(+(1, n2278_0))) -> *3_0, rt in Omega(n2278_0)
t(gen_u:cons_s:cons_t:cons_b2_0(+(1, n771_0))) -> *3_0, rt in Omega(n771_0)


Generator Equations:
gen_u:cons_s:cons_t:cons_b2_0(0) <=> hole_u:cons_s:cons_t:cons_b1_0
gen_u:cons_s:cons_t:cons_b2_0(+(x, 1)) <=> u(gen_u:cons_s:cons_t:cons_b2_0(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_u:cons_s:cons_t:cons_b2_0(+(1, n2899_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_u:cons_s:cons_t:cons_b2_0(+(1, 0)))

Induction Step:
encArg(gen_u:cons_s:cons_t:cons_b2_0(+(1, +(n2899_0, 1)))) ->_R^Omega(0)
u(encArg(gen_u:cons_s:cons_t:cons_b2_0(+(1, n2899_0)))) ->_IH
u(*3_0)

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

(22)
BOUNDS(1, INF)