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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 158 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), 31.1 s]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 26 ms]
        (18) BOUNDS(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:

   -(0, y) -> 0
   -(x, 0) -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   p(0) -> 0
   p(s(x)) -> 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(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(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:

   -(0, y) -> 0
   -(x, 0) -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   p(0) -> 0
   p(s(x)) -> x

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(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:

   -(0, y) -> 0
   -(x, 0) -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   p(0) -> 0
   p(s(x)) -> x

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(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:

   -(0', y) -> 0'
   -(x, 0') -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0')
   p(0') -> 0'
   p(s(x)) -> x

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

   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
-(0', y) -> 0'
-(x, 0') -> x
-(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0')
p(0') -> 0'
p(s(x)) -> x
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2))
encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))

Types:
- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
0' :: 0':s:greater:if:cons_-:cons_p
s :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
if :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
greater :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encArg :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
cons_- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
cons_p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_0 :: 0':s:greater:if:cons_-:cons_p
encode_s :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_if :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_greater :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
hole_0':s:greater:if:cons_-:cons_p1_4 :: 0':s:greater:if:cons_-:cons_p
gen_0':s:greater:if:cons_-:cons_p2_4 :: Nat -> 0':s:greater:if:cons_-:cons_p

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
-(0', y) -> 0'
-(x, 0') -> x
-(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0')
p(0') -> 0'
p(s(x)) -> x
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2))
encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))

Types:
- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
0' :: 0':s:greater:if:cons_-:cons_p
s :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
if :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
greater :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encArg :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
cons_- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
cons_p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_0 :: 0':s:greater:if:cons_-:cons_p
encode_s :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_if :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_greater :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
hole_0':s:greater:if:cons_-:cons_p1_4 :: 0':s:greater:if:cons_-:cons_p
gen_0':s:greater:if:cons_-:cons_p2_4 :: Nat -> 0':s:greater:if:cons_-:cons_p


Generator Equations:
gen_0':s:greater:if:cons_-:cons_p2_4(0) <=> 0'
gen_0':s:greater:if:cons_-:cons_p2_4(+(x, 1)) <=> s(gen_0':s:greater:if:cons_-:cons_p2_4(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s:greater:if:cons_-:cons_p2_4(a), gen_0':s:greater:if:cons_-:cons_p2_4(n4_4)) -> *3_4, rt in Omega(n4_4)

Induction Base:
-(gen_0':s:greater:if:cons_-:cons_p2_4(a), gen_0':s:greater:if:cons_-:cons_p2_4(0))

Induction Step:
-(gen_0':s:greater:if:cons_-:cons_p2_4(a), gen_0':s:greater:if:cons_-:cons_p2_4(+(n4_4, 1))) ->_R^Omega(1)
if(greater(gen_0':s:greater:if:cons_-:cons_p2_4(a), s(gen_0':s:greater:if:cons_-:cons_p2_4(n4_4))), s(-(gen_0':s:greater:if:cons_-:cons_p2_4(a), p(s(gen_0':s:greater:if:cons_-:cons_p2_4(n4_4))))), 0') ->_R^Omega(1)
if(greater(gen_0':s:greater:if:cons_-:cons_p2_4(a), s(gen_0':s:greater:if:cons_-:cons_p2_4(n4_4))), s(-(gen_0':s:greater:if:cons_-:cons_p2_4(a), gen_0':s:greater:if:cons_-:cons_p2_4(n4_4))), 0') ->_IH
if(greater(gen_0':s:greater:if:cons_-:cons_p2_4(a), s(gen_0':s:greater:if:cons_-:cons_p2_4(n4_4))), s(*3_4), 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:

Innermost TRS:
Rules:
-(0', y) -> 0'
-(x, 0') -> x
-(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0')
p(0') -> 0'
p(s(x)) -> x
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2))
encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))

Types:
- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
0' :: 0':s:greater:if:cons_-:cons_p
s :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
if :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
greater :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encArg :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
cons_- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
cons_p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_0 :: 0':s:greater:if:cons_-:cons_p
encode_s :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_if :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_greater :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
hole_0':s:greater:if:cons_-:cons_p1_4 :: 0':s:greater:if:cons_-:cons_p
gen_0':s:greater:if:cons_-:cons_p2_4 :: Nat -> 0':s:greater:if:cons_-:cons_p


Generator Equations:
gen_0':s:greater:if:cons_-:cons_p2_4(0) <=> 0'
gen_0':s:greater:if:cons_-:cons_p2_4(+(x, 1)) <=> s(gen_0':s:greater:if:cons_-:cons_p2_4(x))


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

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
-(0', y) -> 0'
-(x, 0') -> x
-(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0')
p(0') -> 0'
p(s(x)) -> x
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2))
encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))

Types:
- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
0' :: 0':s:greater:if:cons_-:cons_p
s :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
if :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
greater :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encArg :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
cons_- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
cons_p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_- :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_0 :: 0':s:greater:if:cons_-:cons_p
encode_s :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_if :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_greater :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
encode_p :: 0':s:greater:if:cons_-:cons_p -> 0':s:greater:if:cons_-:cons_p
hole_0':s:greater:if:cons_-:cons_p1_4 :: 0':s:greater:if:cons_-:cons_p
gen_0':s:greater:if:cons_-:cons_p2_4 :: Nat -> 0':s:greater:if:cons_-:cons_p


Lemmas:
-(gen_0':s:greater:if:cons_-:cons_p2_4(a), gen_0':s:greater:if:cons_-:cons_p2_4(n4_4)) -> *3_4, rt in Omega(n4_4)


Generator Equations:
gen_0':s:greater:if:cons_-:cons_p2_4(0) <=> 0'
gen_0':s:greater:if:cons_-:cons_p2_4(+(x, 1)) <=> s(gen_0':s:greater:if:cons_-:cons_p2_4(x))


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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:greater:if:cons_-:cons_p2_4(n300598_4)) -> gen_0':s:greater:if:cons_-:cons_p2_4(n300598_4), rt in Omega(0)

Induction Base:
encArg(gen_0':s:greater:if:cons_-:cons_p2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:greater:if:cons_-:cons_p2_4(+(n300598_4, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:greater:if:cons_-:cons_p2_4(n300598_4))) ->_IH
s(gen_0':s:greater:if:cons_-:cons_p2_4(c300599_4))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(18)
BOUNDS(1, INF)