/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^2), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 232 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), 239 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), 2393 ms]
        (18) BEST
            (19) proven lower bound
                (20) LowerBoundPropagationProof [FINISHED, 0 ms]
                (21) BOUNDS(n^2, INF)
            (22) typed CpxTrs
                (23) RewriteLemmaProof [LOWER BOUND(ID), 38 ms]
                (24) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   fac(s(x)) -> *(fac(p(s(x))), s(x))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))

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(s(x_1)) -> s(encArg(x_1))
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_fac(x_1)) -> fac(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_fac(x_1) -> fac(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0


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

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


The TRS R consists of the following rules:

   fac(s(x)) -> *(fac(p(s(x))), s(x))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_fac(x_1)) -> fac(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_fac(x_1) -> fac(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0

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^2, INF).


The TRS R consists of the following rules:

   fac(s(x)) -> *(fac(p(s(x))), s(x))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_fac(x_1)) -> fac(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_fac(x_1) -> fac(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0

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^2, INF).


The TRS R consists of the following rules:

   fac(s(x)) -> *'(fac(p(s(x))), s(x))
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
   encArg(0') -> 0'
   encArg(cons_fac(x_1)) -> fac(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_fac(x_1) -> fac(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0'

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

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

(8)
Obligation:
Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))), s(x))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(0') -> 0'
encArg(cons_fac(x_1)) -> fac(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_fac(x_1) -> fac(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_0 -> 0'

Types:
fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
*' :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
0' :: s:*':0':cons_fac:cons_p
encArg :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_* :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_0 :: s:*':0':cons_fac:cons_p
hole_s:*':0':cons_fac:cons_p1_3 :: s:*':0':cons_fac:cons_p
gen_s:*':0':cons_fac:cons_p2_3 :: Nat -> s:*':0':cons_fac:cons_p

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))), s(x))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(0') -> 0'
encArg(cons_fac(x_1)) -> fac(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_fac(x_1) -> fac(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_0 -> 0'

Types:
fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
*' :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
0' :: s:*':0':cons_fac:cons_p
encArg :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_* :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_0 :: s:*':0':cons_fac:cons_p
hole_s:*':0':cons_fac:cons_p1_3 :: s:*':0':cons_fac:cons_p
gen_s:*':0':cons_fac:cons_p2_3 :: Nat -> s:*':0':cons_fac:cons_p


Generator Equations:
gen_s:*':0':cons_fac:cons_p2_3(0) <=> 0'
gen_s:*':0':cons_fac:cons_p2_3(+(x, 1)) <=> s(gen_s:*':0':cons_fac:cons_p2_3(x))


The following defined symbols remain to be analysed:
p, fac, encArg

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_s:*':0':cons_fac:cons_p2_3(+(1, n4_3))) -> gen_s:*':0':cons_fac:cons_p2_3(n4_3), rt in Omega(1 + n4_3)

Induction Base:
p(gen_s:*':0':cons_fac:cons_p2_3(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
p(gen_s:*':0':cons_fac:cons_p2_3(+(1, +(n4_3, 1)))) ->_R^Omega(1)
s(p(s(gen_s:*':0':cons_fac:cons_p2_3(n4_3)))) ->_IH
s(gen_s:*':0':cons_fac:cons_p2_3(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:

Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))), s(x))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(0') -> 0'
encArg(cons_fac(x_1)) -> fac(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_fac(x_1) -> fac(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_0 -> 0'

Types:
fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
*' :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
0' :: s:*':0':cons_fac:cons_p
encArg :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_* :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_0 :: s:*':0':cons_fac:cons_p
hole_s:*':0':cons_fac:cons_p1_3 :: s:*':0':cons_fac:cons_p
gen_s:*':0':cons_fac:cons_p2_3 :: Nat -> s:*':0':cons_fac:cons_p


Generator Equations:
gen_s:*':0':cons_fac:cons_p2_3(0) <=> 0'
gen_s:*':0':cons_fac:cons_p2_3(+(x, 1)) <=> s(gen_s:*':0':cons_fac:cons_p2_3(x))


The following defined symbols remain to be analysed:
p, fac, encArg

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))), s(x))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(0') -> 0'
encArg(cons_fac(x_1)) -> fac(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_fac(x_1) -> fac(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_0 -> 0'

Types:
fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
*' :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
0' :: s:*':0':cons_fac:cons_p
encArg :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_* :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_0 :: s:*':0':cons_fac:cons_p
hole_s:*':0':cons_fac:cons_p1_3 :: s:*':0':cons_fac:cons_p
gen_s:*':0':cons_fac:cons_p2_3 :: Nat -> s:*':0':cons_fac:cons_p


Lemmas:
p(gen_s:*':0':cons_fac:cons_p2_3(+(1, n4_3))) -> gen_s:*':0':cons_fac:cons_p2_3(n4_3), rt in Omega(1 + n4_3)


Generator Equations:
gen_s:*':0':cons_fac:cons_p2_3(0) <=> 0'
gen_s:*':0':cons_fac:cons_p2_3(+(x, 1)) <=> s(gen_s:*':0':cons_fac:cons_p2_3(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fac(gen_s:*':0':cons_fac:cons_p2_3(+(1, n261_3))) -> *3_3, rt in Omega(n261_3 + n261_3^2)

Induction Base:
fac(gen_s:*':0':cons_fac:cons_p2_3(+(1, 0)))

Induction Step:
fac(gen_s:*':0':cons_fac:cons_p2_3(+(1, +(n261_3, 1)))) ->_R^Omega(1)
*'(fac(p(s(gen_s:*':0':cons_fac:cons_p2_3(+(1, n261_3))))), s(gen_s:*':0':cons_fac:cons_p2_3(+(1, n261_3)))) ->_L^Omega(2 + n261_3)
*'(fac(gen_s:*':0':cons_fac:cons_p2_3(+(1, n261_3))), s(gen_s:*':0':cons_fac:cons_p2_3(+(1, n261_3)))) ->_IH
*'(*3_3, s(gen_s:*':0':cons_fac:cons_p2_3(+(1, n261_3))))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))), s(x))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(0') -> 0'
encArg(cons_fac(x_1)) -> fac(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_fac(x_1) -> fac(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_0 -> 0'

Types:
fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
*' :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
0' :: s:*':0':cons_fac:cons_p
encArg :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_* :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_0 :: s:*':0':cons_fac:cons_p
hole_s:*':0':cons_fac:cons_p1_3 :: s:*':0':cons_fac:cons_p
gen_s:*':0':cons_fac:cons_p2_3 :: Nat -> s:*':0':cons_fac:cons_p


Lemmas:
p(gen_s:*':0':cons_fac:cons_p2_3(+(1, n4_3))) -> gen_s:*':0':cons_fac:cons_p2_3(n4_3), rt in Omega(1 + n4_3)


Generator Equations:
gen_s:*':0':cons_fac:cons_p2_3(0) <=> 0'
gen_s:*':0':cons_fac:cons_p2_3(+(x, 1)) <=> s(gen_s:*':0':cons_fac:cons_p2_3(x))


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

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

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

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

(21)
BOUNDS(n^2, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))), s(x))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(0') -> 0'
encArg(cons_fac(x_1)) -> fac(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_fac(x_1) -> fac(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_0 -> 0'

Types:
fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
*' :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
0' :: s:*':0':cons_fac:cons_p
encArg :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
cons_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_fac :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_s :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_* :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_p :: s:*':0':cons_fac:cons_p -> s:*':0':cons_fac:cons_p
encode_0 :: s:*':0':cons_fac:cons_p
hole_s:*':0':cons_fac:cons_p1_3 :: s:*':0':cons_fac:cons_p
gen_s:*':0':cons_fac:cons_p2_3 :: Nat -> s:*':0':cons_fac:cons_p


Lemmas:
p(gen_s:*':0':cons_fac:cons_p2_3(+(1, n4_3))) -> gen_s:*':0':cons_fac:cons_p2_3(n4_3), rt in Omega(1 + n4_3)
fac(gen_s:*':0':cons_fac:cons_p2_3(+(1, n261_3))) -> *3_3, rt in Omega(n261_3 + n261_3^2)


Generator Equations:
gen_s:*':0':cons_fac:cons_p2_3(0) <=> 0'
gen_s:*':0':cons_fac:cons_p2_3(+(x, 1)) <=> s(gen_s:*':0':cons_fac:cons_p2_3(x))


The following defined symbols remain to be analysed:
encArg
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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_s:*':0':cons_fac:cons_p2_3(n918_3)) -> gen_s:*':0':cons_fac:cons_p2_3(n918_3), rt in Omega(0)

Induction Base:
encArg(gen_s:*':0':cons_fac:cons_p2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_s:*':0':cons_fac:cons_p2_3(+(n918_3, 1))) ->_R^Omega(0)
s(encArg(gen_s:*':0':cons_fac:cons_p2_3(n918_3))) ->_IH
s(gen_s:*':0':cons_fac:cons_p2_3(c919_3))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(24)
BOUNDS(1, INF)