WORST_CASE(Omega(n^1),?)
proof of input_qKKK4OxfEI.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 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), 14 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 354 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), 98 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 379 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 69 ms]
        (22) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   first(0, Z) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   sel(0, cons(X, Z)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

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

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, from(s(z0)))
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   FIRST(0, z0) -> c1
   FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2))
   SEL(0, cons(z0, z1)) -> c3
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   FIRST(0, z0) -> c1
   FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2))
   SEL(0, cons(z0, z1)) -> c3
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
K tuples:none
Defined Rule Symbols:   from_1, first_2, sel_2

Defined Pair Symbols:   FROM_1, FIRST_2, SEL_2

Compound Symbols:   c_1, c1, c2_1, c3, c4_1


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

(3) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to 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:

   FROM(z0) -> c(FROM(s(z0)))
   FIRST(0, z0) -> c1
   FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2))
   SEL(0, cons(z0, z1)) -> c3
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))

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

   from(z0) -> cons(z0, from(s(z0)))
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

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:

   FROM(z0) -> c(FROM(s(z0)))
   FIRST(0', z0) -> c1
   FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2))
   SEL(0', cons(z0, z1)) -> c3
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))

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

   from(z0) -> cons(z0, from(s(z0)))
   first(0', z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
   sel(0', cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

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

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

(8)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
FIRST(0', z0) -> c1
FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
FIRST :: s:0' -> cons:nil -> c1:c2
0' :: s:0'
c1 :: c1:c2
cons :: s:0' -> cons:nil -> cons:nil
c2 :: c1:c2 -> c1:c2
SEL :: s:0' -> cons:nil -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
from :: s:0' -> cons:nil
first :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_cons:nil4_5 :: cons:nil
hole_c3:c45_5 :: c3:c4
gen_c6_5 :: Nat -> c
gen_s:0'7_5 :: Nat -> s:0'
gen_c1:c28_5 :: Nat -> c1:c2
gen_cons:nil9_5 :: Nat -> cons:nil
gen_c3:c410_5 :: Nat -> c3:c4

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
FROM, FIRST, SEL, from, first, sel
----------------------------------------

(10)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
FIRST(0', z0) -> c1
FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
FIRST :: s:0' -> cons:nil -> c1:c2
0' :: s:0'
c1 :: c1:c2
cons :: s:0' -> cons:nil -> cons:nil
c2 :: c1:c2 -> c1:c2
SEL :: s:0' -> cons:nil -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
from :: s:0' -> cons:nil
first :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_cons:nil4_5 :: cons:nil
hole_c3:c45_5 :: c3:c4
gen_c6_5 :: Nat -> c
gen_s:0'7_5 :: Nat -> s:0'
gen_c1:c28_5 :: Nat -> c1:c2
gen_cons:nil9_5 :: Nat -> cons:nil
gen_c3:c410_5 :: Nat -> c3:c4


Generator Equations:
gen_c6_5(0) <=> hole_c1_5
gen_c6_5(+(x, 1)) <=> c(gen_c6_5(x))
gen_s:0'7_5(0) <=> 0'
gen_s:0'7_5(+(x, 1)) <=> s(gen_s:0'7_5(x))
gen_c1:c28_5(0) <=> c1
gen_c1:c28_5(+(x, 1)) <=> c2(gen_c1:c28_5(x))
gen_cons:nil9_5(0) <=> nil
gen_cons:nil9_5(+(x, 1)) <=> cons(0', gen_cons:nil9_5(x))
gen_c3:c410_5(0) <=> c3
gen_c3:c410_5(+(x, 1)) <=> c4(gen_c3:c410_5(x))


The following defined symbols remain to be analysed:
FROM, FIRST, SEL, from, first, sel
----------------------------------------

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
FIRST(gen_s:0'7_5(n137_5), gen_cons:nil9_5(n137_5)) -> gen_c1:c28_5(n137_5), rt in Omega(1 + n137_5)

Induction Base:
FIRST(gen_s:0'7_5(0), gen_cons:nil9_5(0)) ->_R^Omega(1)
c1

Induction Step:
FIRST(gen_s:0'7_5(+(n137_5, 1)), gen_cons:nil9_5(+(n137_5, 1))) ->_R^Omega(1)
c2(FIRST(gen_s:0'7_5(n137_5), gen_cons:nil9_5(n137_5))) ->_IH
c2(gen_c1:c28_5(c138_5))

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:
FROM(z0) -> c(FROM(s(z0)))
FIRST(0', z0) -> c1
FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
FIRST :: s:0' -> cons:nil -> c1:c2
0' :: s:0'
c1 :: c1:c2
cons :: s:0' -> cons:nil -> cons:nil
c2 :: c1:c2 -> c1:c2
SEL :: s:0' -> cons:nil -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
from :: s:0' -> cons:nil
first :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_cons:nil4_5 :: cons:nil
hole_c3:c45_5 :: c3:c4
gen_c6_5 :: Nat -> c
gen_s:0'7_5 :: Nat -> s:0'
gen_c1:c28_5 :: Nat -> c1:c2
gen_cons:nil9_5 :: Nat -> cons:nil
gen_c3:c410_5 :: Nat -> c3:c4


Generator Equations:
gen_c6_5(0) <=> hole_c1_5
gen_c6_5(+(x, 1)) <=> c(gen_c6_5(x))
gen_s:0'7_5(0) <=> 0'
gen_s:0'7_5(+(x, 1)) <=> s(gen_s:0'7_5(x))
gen_c1:c28_5(0) <=> c1
gen_c1:c28_5(+(x, 1)) <=> c2(gen_c1:c28_5(x))
gen_cons:nil9_5(0) <=> nil
gen_cons:nil9_5(+(x, 1)) <=> cons(0', gen_cons:nil9_5(x))
gen_c3:c410_5(0) <=> c3
gen_c3:c410_5(+(x, 1)) <=> c4(gen_c3:c410_5(x))


The following defined symbols remain to be analysed:
FIRST, SEL, from, first, sel
----------------------------------------

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
FIRST(0', z0) -> c1
FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
FIRST :: s:0' -> cons:nil -> c1:c2
0' :: s:0'
c1 :: c1:c2
cons :: s:0' -> cons:nil -> cons:nil
c2 :: c1:c2 -> c1:c2
SEL :: s:0' -> cons:nil -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
from :: s:0' -> cons:nil
first :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_cons:nil4_5 :: cons:nil
hole_c3:c45_5 :: c3:c4
gen_c6_5 :: Nat -> c
gen_s:0'7_5 :: Nat -> s:0'
gen_c1:c28_5 :: Nat -> c1:c2
gen_cons:nil9_5 :: Nat -> cons:nil
gen_c3:c410_5 :: Nat -> c3:c4


Lemmas:
FIRST(gen_s:0'7_5(n137_5), gen_cons:nil9_5(n137_5)) -> gen_c1:c28_5(n137_5), rt in Omega(1 + n137_5)


Generator Equations:
gen_c6_5(0) <=> hole_c1_5
gen_c6_5(+(x, 1)) <=> c(gen_c6_5(x))
gen_s:0'7_5(0) <=> 0'
gen_s:0'7_5(+(x, 1)) <=> s(gen_s:0'7_5(x))
gen_c1:c28_5(0) <=> c1
gen_c1:c28_5(+(x, 1)) <=> c2(gen_c1:c28_5(x))
gen_cons:nil9_5(0) <=> nil
gen_cons:nil9_5(+(x, 1)) <=> cons(0', gen_cons:nil9_5(x))
gen_c3:c410_5(0) <=> c3
gen_c3:c410_5(+(x, 1)) <=> c4(gen_c3:c410_5(x))


The following defined symbols remain to be analysed:
SEL, from, first, sel
----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SEL(gen_s:0'7_5(n539_5), gen_cons:nil9_5(+(1, n539_5))) -> gen_c3:c410_5(n539_5), rt in Omega(1 + n539_5)

Induction Base:
SEL(gen_s:0'7_5(0), gen_cons:nil9_5(+(1, 0))) ->_R^Omega(1)
c3

Induction Step:
SEL(gen_s:0'7_5(+(n539_5, 1)), gen_cons:nil9_5(+(1, +(n539_5, 1)))) ->_R^Omega(1)
c4(SEL(gen_s:0'7_5(n539_5), gen_cons:nil9_5(+(1, n539_5)))) ->_IH
c4(gen_c3:c410_5(c540_5))

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

(18)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
FIRST(0', z0) -> c1
FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
FIRST :: s:0' -> cons:nil -> c1:c2
0' :: s:0'
c1 :: c1:c2
cons :: s:0' -> cons:nil -> cons:nil
c2 :: c1:c2 -> c1:c2
SEL :: s:0' -> cons:nil -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
from :: s:0' -> cons:nil
first :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_cons:nil4_5 :: cons:nil
hole_c3:c45_5 :: c3:c4
gen_c6_5 :: Nat -> c
gen_s:0'7_5 :: Nat -> s:0'
gen_c1:c28_5 :: Nat -> c1:c2
gen_cons:nil9_5 :: Nat -> cons:nil
gen_c3:c410_5 :: Nat -> c3:c4


Lemmas:
FIRST(gen_s:0'7_5(n137_5), gen_cons:nil9_5(n137_5)) -> gen_c1:c28_5(n137_5), rt in Omega(1 + n137_5)
SEL(gen_s:0'7_5(n539_5), gen_cons:nil9_5(+(1, n539_5))) -> gen_c3:c410_5(n539_5), rt in Omega(1 + n539_5)


Generator Equations:
gen_c6_5(0) <=> hole_c1_5
gen_c6_5(+(x, 1)) <=> c(gen_c6_5(x))
gen_s:0'7_5(0) <=> 0'
gen_s:0'7_5(+(x, 1)) <=> s(gen_s:0'7_5(x))
gen_c1:c28_5(0) <=> c1
gen_c1:c28_5(+(x, 1)) <=> c2(gen_c1:c28_5(x))
gen_cons:nil9_5(0) <=> nil
gen_cons:nil9_5(+(x, 1)) <=> cons(0', gen_cons:nil9_5(x))
gen_c3:c410_5(0) <=> c3
gen_c3:c410_5(+(x, 1)) <=> c4(gen_c3:c410_5(x))


The following defined symbols remain to be analysed:
from, first, sel
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
first(gen_s:0'7_5(n1189_5), gen_cons:nil9_5(n1189_5)) -> gen_cons:nil9_5(n1189_5), rt in Omega(0)

Induction Base:
first(gen_s:0'7_5(0), gen_cons:nil9_5(0)) ->_R^Omega(0)
nil

Induction Step:
first(gen_s:0'7_5(+(n1189_5, 1)), gen_cons:nil9_5(+(n1189_5, 1))) ->_R^Omega(0)
cons(0', first(gen_s:0'7_5(n1189_5), gen_cons:nil9_5(n1189_5))) ->_IH
cons(0', gen_cons:nil9_5(c1190_5))

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

(20)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
FIRST(0', z0) -> c1
FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
FIRST :: s:0' -> cons:nil -> c1:c2
0' :: s:0'
c1 :: c1:c2
cons :: s:0' -> cons:nil -> cons:nil
c2 :: c1:c2 -> c1:c2
SEL :: s:0' -> cons:nil -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
from :: s:0' -> cons:nil
first :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_cons:nil4_5 :: cons:nil
hole_c3:c45_5 :: c3:c4
gen_c6_5 :: Nat -> c
gen_s:0'7_5 :: Nat -> s:0'
gen_c1:c28_5 :: Nat -> c1:c2
gen_cons:nil9_5 :: Nat -> cons:nil
gen_c3:c410_5 :: Nat -> c3:c4


Lemmas:
FIRST(gen_s:0'7_5(n137_5), gen_cons:nil9_5(n137_5)) -> gen_c1:c28_5(n137_5), rt in Omega(1 + n137_5)
SEL(gen_s:0'7_5(n539_5), gen_cons:nil9_5(+(1, n539_5))) -> gen_c3:c410_5(n539_5), rt in Omega(1 + n539_5)
first(gen_s:0'7_5(n1189_5), gen_cons:nil9_5(n1189_5)) -> gen_cons:nil9_5(n1189_5), rt in Omega(0)


Generator Equations:
gen_c6_5(0) <=> hole_c1_5
gen_c6_5(+(x, 1)) <=> c(gen_c6_5(x))
gen_s:0'7_5(0) <=> 0'
gen_s:0'7_5(+(x, 1)) <=> s(gen_s:0'7_5(x))
gen_c1:c28_5(0) <=> c1
gen_c1:c28_5(+(x, 1)) <=> c2(gen_c1:c28_5(x))
gen_cons:nil9_5(0) <=> nil
gen_cons:nil9_5(+(x, 1)) <=> cons(0', gen_cons:nil9_5(x))
gen_c3:c410_5(0) <=> c3
gen_c3:c410_5(+(x, 1)) <=> c4(gen_c3:c410_5(x))


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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sel(gen_s:0'7_5(n1640_5), gen_cons:nil9_5(+(1, n1640_5))) -> gen_s:0'7_5(0), rt in Omega(0)

Induction Base:
sel(gen_s:0'7_5(0), gen_cons:nil9_5(+(1, 0))) ->_R^Omega(0)
0'

Induction Step:
sel(gen_s:0'7_5(+(n1640_5, 1)), gen_cons:nil9_5(+(1, +(n1640_5, 1)))) ->_R^Omega(0)
sel(gen_s:0'7_5(n1640_5), gen_cons:nil9_5(+(1, n1640_5))) ->_IH
gen_s:0'7_5(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(22)
BOUNDS(1, INF)