WORST_CASE(Omega(n^1),?)
proof of input_2RlOkhbZgv.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 24 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 403 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), 400 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 40 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)))
   head(cons(X, XS)) -> X
   2nd(cons(X, XS)) -> head(XS)
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

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)))
   head(cons(z0, z1)) -> z0
   2nd(cons(z0, z1)) -> head(z1)
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   HEAD(cons(z0, z1)) -> c1
   2ND(cons(z0, z1)) -> c2(HEAD(z1))
   TAKE(0, z0) -> c3
   TAKE(s(z0), cons(z1, z2)) -> c4(TAKE(z0, z2))
   SEL(0, cons(z0, z1)) -> c5
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   HEAD(cons(z0, z1)) -> c1
   2ND(cons(z0, z1)) -> c2(HEAD(z1))
   TAKE(0, z0) -> c3
   TAKE(s(z0), cons(z1, z2)) -> c4(TAKE(z0, z2))
   SEL(0, cons(z0, z1)) -> c5
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
K tuples:none
Defined Rule Symbols:   from_1, head_1, 2nd_1, take_2, sel_2

Defined Pair Symbols:   FROM_1, HEAD_1, 2ND_1, TAKE_2, SEL_2

Compound Symbols:   c_1, c1, c2_1, c3, c4_1, c5, c6_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)))
   HEAD(cons(z0, z1)) -> c1
   2ND(cons(z0, z1)) -> c2(HEAD(z1))
   TAKE(0, z0) -> c3
   TAKE(s(z0), cons(z1, z2)) -> c4(TAKE(z0, z2))
   SEL(0, cons(z0, z1)) -> c5
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))

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

   from(z0) -> cons(z0, from(s(z0)))
   head(cons(z0, z1)) -> z0
   2nd(cons(z0, z1)) -> head(z1)
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(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)))
   HEAD(cons(z0, z1)) -> c1
   2ND(cons(z0, z1)) -> c2(HEAD(z1))
   TAKE(0', z0) -> c3
   TAKE(s(z0), cons(z1, z2)) -> c4(TAKE(z0, z2))
   SEL(0', cons(z0, z1)) -> c5
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))

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

   from(z0) -> cons(z0, from(s(z0)))
   head(cons(z0, z1)) -> z0
   2nd(cons(z0, z1)) -> head(z1)
   take(0', z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(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)))
HEAD(cons(z0, z1)) -> c1
2ND(cons(z0, z1)) -> c2(HEAD(z1))
TAKE(0', z0) -> c3
TAKE(s(z0), cons(z1, z2)) -> c4(TAKE(z0, z2))
SEL(0', cons(z0, z1)) -> c5
SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
head(cons(z0, z1)) -> z0
2nd(cons(z0, z1)) -> head(z1)
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(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'
HEAD :: cons:nil -> c1
cons :: s:0' -> cons:nil -> cons:nil
c1 :: c1
2ND :: cons:nil -> c2
c2 :: c1 -> c2
TAKE :: s:0' -> cons:nil -> c3:c4
0' :: s:0'
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
SEL :: s:0' -> cons:nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
from :: s:0' -> cons:nil
head :: cons:nil -> s:0'
2nd :: cons:nil -> s:0'
take :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_7 :: c
hole_s:0'2_7 :: s:0'
hole_c13_7 :: c1
hole_cons:nil4_7 :: cons:nil
hole_c25_7 :: c2
hole_c3:c46_7 :: c3:c4
hole_c5:c67_7 :: c5:c6
gen_c8_7 :: Nat -> c
gen_s:0'9_7 :: Nat -> s:0'
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c3:c411_7 :: Nat -> c3:c4
gen_c5:c612_7 :: Nat -> c5:c6

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

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

(10)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
HEAD(cons(z0, z1)) -> c1
2ND(cons(z0, z1)) -> c2(HEAD(z1))
TAKE(0', z0) -> c3
TAKE(s(z0), cons(z1, z2)) -> c4(TAKE(z0, z2))
SEL(0', cons(z0, z1)) -> c5
SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
head(cons(z0, z1)) -> z0
2nd(cons(z0, z1)) -> head(z1)
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(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'
HEAD :: cons:nil -> c1
cons :: s:0' -> cons:nil -> cons:nil
c1 :: c1
2ND :: cons:nil -> c2
c2 :: c1 -> c2
TAKE :: s:0' -> cons:nil -> c3:c4
0' :: s:0'
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
SEL :: s:0' -> cons:nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
from :: s:0' -> cons:nil
head :: cons:nil -> s:0'
2nd :: cons:nil -> s:0'
take :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_7 :: c
hole_s:0'2_7 :: s:0'
hole_c13_7 :: c1
hole_cons:nil4_7 :: cons:nil
hole_c25_7 :: c2
hole_c3:c46_7 :: c3:c4
hole_c5:c67_7 :: c5:c6
gen_c8_7 :: Nat -> c
gen_s:0'9_7 :: Nat -> s:0'
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c3:c411_7 :: Nat -> c3:c4
gen_c5:c612_7 :: Nat -> c5:c6


Generator Equations:
gen_c8_7(0) <=> hole_c1_7
gen_c8_7(+(x, 1)) <=> c(gen_c8_7(x))
gen_s:0'9_7(0) <=> 0'
gen_s:0'9_7(+(x, 1)) <=> s(gen_s:0'9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c3:c411_7(0) <=> c3
gen_c3:c411_7(+(x, 1)) <=> c4(gen_c3:c411_7(x))
gen_c5:c612_7(0) <=> c5
gen_c5:c612_7(+(x, 1)) <=> c6(gen_c5:c612_7(x))


The following defined symbols remain to be analysed:
FROM, TAKE, SEL, from, take, sel
----------------------------------------

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
TAKE(gen_s:0'9_7(n139_7), gen_cons:nil10_7(n139_7)) -> gen_c3:c411_7(n139_7), rt in Omega(1 + n139_7)

Induction Base:
TAKE(gen_s:0'9_7(0), gen_cons:nil10_7(0)) ->_R^Omega(1)
c3

Induction Step:
TAKE(gen_s:0'9_7(+(n139_7, 1)), gen_cons:nil10_7(+(n139_7, 1))) ->_R^Omega(1)
c4(TAKE(gen_s:0'9_7(n139_7), gen_cons:nil10_7(n139_7))) ->_IH
c4(gen_c3:c411_7(c140_7))

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)))
HEAD(cons(z0, z1)) -> c1
2ND(cons(z0, z1)) -> c2(HEAD(z1))
TAKE(0', z0) -> c3
TAKE(s(z0), cons(z1, z2)) -> c4(TAKE(z0, z2))
SEL(0', cons(z0, z1)) -> c5
SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
head(cons(z0, z1)) -> z0
2nd(cons(z0, z1)) -> head(z1)
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(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'
HEAD :: cons:nil -> c1
cons :: s:0' -> cons:nil -> cons:nil
c1 :: c1
2ND :: cons:nil -> c2
c2 :: c1 -> c2
TAKE :: s:0' -> cons:nil -> c3:c4
0' :: s:0'
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
SEL :: s:0' -> cons:nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
from :: s:0' -> cons:nil
head :: cons:nil -> s:0'
2nd :: cons:nil -> s:0'
take :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_7 :: c
hole_s:0'2_7 :: s:0'
hole_c13_7 :: c1
hole_cons:nil4_7 :: cons:nil
hole_c25_7 :: c2
hole_c3:c46_7 :: c3:c4
hole_c5:c67_7 :: c5:c6
gen_c8_7 :: Nat -> c
gen_s:0'9_7 :: Nat -> s:0'
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c3:c411_7 :: Nat -> c3:c4
gen_c5:c612_7 :: Nat -> c5:c6


Generator Equations:
gen_c8_7(0) <=> hole_c1_7
gen_c8_7(+(x, 1)) <=> c(gen_c8_7(x))
gen_s:0'9_7(0) <=> 0'
gen_s:0'9_7(+(x, 1)) <=> s(gen_s:0'9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c3:c411_7(0) <=> c3
gen_c3:c411_7(+(x, 1)) <=> c4(gen_c3:c411_7(x))
gen_c5:c612_7(0) <=> c5
gen_c5:c612_7(+(x, 1)) <=> c6(gen_c5:c612_7(x))


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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
HEAD(cons(z0, z1)) -> c1
2ND(cons(z0, z1)) -> c2(HEAD(z1))
TAKE(0', z0) -> c3
TAKE(s(z0), cons(z1, z2)) -> c4(TAKE(z0, z2))
SEL(0', cons(z0, z1)) -> c5
SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
head(cons(z0, z1)) -> z0
2nd(cons(z0, z1)) -> head(z1)
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(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'
HEAD :: cons:nil -> c1
cons :: s:0' -> cons:nil -> cons:nil
c1 :: c1
2ND :: cons:nil -> c2
c2 :: c1 -> c2
TAKE :: s:0' -> cons:nil -> c3:c4
0' :: s:0'
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
SEL :: s:0' -> cons:nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
from :: s:0' -> cons:nil
head :: cons:nil -> s:0'
2nd :: cons:nil -> s:0'
take :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_7 :: c
hole_s:0'2_7 :: s:0'
hole_c13_7 :: c1
hole_cons:nil4_7 :: cons:nil
hole_c25_7 :: c2
hole_c3:c46_7 :: c3:c4
hole_c5:c67_7 :: c5:c6
gen_c8_7 :: Nat -> c
gen_s:0'9_7 :: Nat -> s:0'
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c3:c411_7 :: Nat -> c3:c4
gen_c5:c612_7 :: Nat -> c5:c6


Lemmas:
TAKE(gen_s:0'9_7(n139_7), gen_cons:nil10_7(n139_7)) -> gen_c3:c411_7(n139_7), rt in Omega(1 + n139_7)


Generator Equations:
gen_c8_7(0) <=> hole_c1_7
gen_c8_7(+(x, 1)) <=> c(gen_c8_7(x))
gen_s:0'9_7(0) <=> 0'
gen_s:0'9_7(+(x, 1)) <=> s(gen_s:0'9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c3:c411_7(0) <=> c3
gen_c3:c411_7(+(x, 1)) <=> c4(gen_c3:c411_7(x))
gen_c5:c612_7(0) <=> c5
gen_c5:c612_7(+(x, 1)) <=> c6(gen_c5:c612_7(x))


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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SEL(gen_s:0'9_7(n609_7), gen_cons:nil10_7(+(1, n609_7))) -> gen_c5:c612_7(n609_7), rt in Omega(1 + n609_7)

Induction Base:
SEL(gen_s:0'9_7(0), gen_cons:nil10_7(+(1, 0))) ->_R^Omega(1)
c5

Induction Step:
SEL(gen_s:0'9_7(+(n609_7, 1)), gen_cons:nil10_7(+(1, +(n609_7, 1)))) ->_R^Omega(1)
c6(SEL(gen_s:0'9_7(n609_7), gen_cons:nil10_7(+(1, n609_7)))) ->_IH
c6(gen_c5:c612_7(c610_7))

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)))
HEAD(cons(z0, z1)) -> c1
2ND(cons(z0, z1)) -> c2(HEAD(z1))
TAKE(0', z0) -> c3
TAKE(s(z0), cons(z1, z2)) -> c4(TAKE(z0, z2))
SEL(0', cons(z0, z1)) -> c5
SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
head(cons(z0, z1)) -> z0
2nd(cons(z0, z1)) -> head(z1)
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(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'
HEAD :: cons:nil -> c1
cons :: s:0' -> cons:nil -> cons:nil
c1 :: c1
2ND :: cons:nil -> c2
c2 :: c1 -> c2
TAKE :: s:0' -> cons:nil -> c3:c4
0' :: s:0'
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
SEL :: s:0' -> cons:nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
from :: s:0' -> cons:nil
head :: cons:nil -> s:0'
2nd :: cons:nil -> s:0'
take :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_7 :: c
hole_s:0'2_7 :: s:0'
hole_c13_7 :: c1
hole_cons:nil4_7 :: cons:nil
hole_c25_7 :: c2
hole_c3:c46_7 :: c3:c4
hole_c5:c67_7 :: c5:c6
gen_c8_7 :: Nat -> c
gen_s:0'9_7 :: Nat -> s:0'
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c3:c411_7 :: Nat -> c3:c4
gen_c5:c612_7 :: Nat -> c5:c6


Lemmas:
TAKE(gen_s:0'9_7(n139_7), gen_cons:nil10_7(n139_7)) -> gen_c3:c411_7(n139_7), rt in Omega(1 + n139_7)
SEL(gen_s:0'9_7(n609_7), gen_cons:nil10_7(+(1, n609_7))) -> gen_c5:c612_7(n609_7), rt in Omega(1 + n609_7)


Generator Equations:
gen_c8_7(0) <=> hole_c1_7
gen_c8_7(+(x, 1)) <=> c(gen_c8_7(x))
gen_s:0'9_7(0) <=> 0'
gen_s:0'9_7(+(x, 1)) <=> s(gen_s:0'9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c3:c411_7(0) <=> c3
gen_c3:c411_7(+(x, 1)) <=> c4(gen_c3:c411_7(x))
gen_c5:c612_7(0) <=> c5
gen_c5:c612_7(+(x, 1)) <=> c6(gen_c5:c612_7(x))


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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
take(gen_s:0'9_7(n1291_7), gen_cons:nil10_7(n1291_7)) -> gen_cons:nil10_7(n1291_7), rt in Omega(0)

Induction Base:
take(gen_s:0'9_7(0), gen_cons:nil10_7(0)) ->_R^Omega(0)
nil

Induction Step:
take(gen_s:0'9_7(+(n1291_7, 1)), gen_cons:nil10_7(+(n1291_7, 1))) ->_R^Omega(0)
cons(0', take(gen_s:0'9_7(n1291_7), gen_cons:nil10_7(n1291_7))) ->_IH
cons(0', gen_cons:nil10_7(c1292_7))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(20)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
HEAD(cons(z0, z1)) -> c1
2ND(cons(z0, z1)) -> c2(HEAD(z1))
TAKE(0', z0) -> c3
TAKE(s(z0), cons(z1, z2)) -> c4(TAKE(z0, z2))
SEL(0', cons(z0, z1)) -> c5
SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
head(cons(z0, z1)) -> z0
2nd(cons(z0, z1)) -> head(z1)
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(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'
HEAD :: cons:nil -> c1
cons :: s:0' -> cons:nil -> cons:nil
c1 :: c1
2ND :: cons:nil -> c2
c2 :: c1 -> c2
TAKE :: s:0' -> cons:nil -> c3:c4
0' :: s:0'
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
SEL :: s:0' -> cons:nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
from :: s:0' -> cons:nil
head :: cons:nil -> s:0'
2nd :: cons:nil -> s:0'
take :: s:0' -> cons:nil -> cons:nil
nil :: cons:nil
sel :: s:0' -> cons:nil -> s:0'
hole_c1_7 :: c
hole_s:0'2_7 :: s:0'
hole_c13_7 :: c1
hole_cons:nil4_7 :: cons:nil
hole_c25_7 :: c2
hole_c3:c46_7 :: c3:c4
hole_c5:c67_7 :: c5:c6
gen_c8_7 :: Nat -> c
gen_s:0'9_7 :: Nat -> s:0'
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c3:c411_7 :: Nat -> c3:c4
gen_c5:c612_7 :: Nat -> c5:c6


Lemmas:
TAKE(gen_s:0'9_7(n139_7), gen_cons:nil10_7(n139_7)) -> gen_c3:c411_7(n139_7), rt in Omega(1 + n139_7)
SEL(gen_s:0'9_7(n609_7), gen_cons:nil10_7(+(1, n609_7))) -> gen_c5:c612_7(n609_7), rt in Omega(1 + n609_7)
take(gen_s:0'9_7(n1291_7), gen_cons:nil10_7(n1291_7)) -> gen_cons:nil10_7(n1291_7), rt in Omega(0)


Generator Equations:
gen_c8_7(0) <=> hole_c1_7
gen_c8_7(+(x, 1)) <=> c(gen_c8_7(x))
gen_s:0'9_7(0) <=> 0'
gen_s:0'9_7(+(x, 1)) <=> s(gen_s:0'9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c3:c411_7(0) <=> c3
gen_c3:c411_7(+(x, 1)) <=> c4(gen_c3:c411_7(x))
gen_c5:c612_7(0) <=> c5
gen_c5:c612_7(+(x, 1)) <=> c6(gen_c5:c612_7(x))


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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sel(gen_s:0'9_7(n1742_7), gen_cons:nil10_7(+(1, n1742_7))) -> gen_s:0'9_7(0), rt in Omega(0)

Induction Base:
sel(gen_s:0'9_7(0), gen_cons:nil10_7(+(1, 0))) ->_R^Omega(0)
0'

Induction Step:
sel(gen_s:0'9_7(+(n1742_7, 1)), gen_cons:nil10_7(+(1, +(n1742_7, 1)))) ->_R^Omega(0)
sel(gen_s:0'9_7(n1742_7), gen_cons:nil10_7(+(1, n1742_7))) ->_IH
gen_s:0'9_7(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)