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


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 280 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 40 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 33 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 34 ms]
        (18) typed CpxTrs


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

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


The TRS R consists of the following rules:

   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   length(nil) -> 0
   length(cons(x, y)) -> s(length(y))
   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   if(true, c, l, ys, zs) -> nil
   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
   take(0, cons(x, xs), ys) -> x
   take(0, nil, cons(y, ys)) -> y
   take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

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

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
   plus(x, 0') -> x
   plus(x, s(y)) -> s(plus(x, y))
   length(nil) -> 0'
   length(cons(x, y)) -> s(length(y))
   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   if(true, c, l, ys, zs) -> nil
   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
   take(0', cons(x, xs), ys) -> x
   take(0', nil, cons(y, ys)) -> y
   take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

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

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

(4)
Obligation:
TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
helpa, plus, length, ge, helpb, take

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb
take < helpb

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

(6)
Obligation:
TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
plus, helpa, length, ge, helpb, take

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb
take < helpb

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s6_0(a), gen_0':s6_0(n8_0)) -> gen_0':s6_0(+(n8_0, a)), rt in Omega(1 + n8_0)

Induction Base:
plus(gen_0':s6_0(a), gen_0':s6_0(0)) ->_R^Omega(1)
gen_0':s6_0(a)

Induction Step:
plus(gen_0':s6_0(a), gen_0':s6_0(+(n8_0, 1))) ->_R^Omega(1)
s(plus(gen_0':s6_0(a), gen_0':s6_0(n8_0))) ->_IH
s(gen_0':s6_0(+(a, c9_0)))

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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
plus, helpa, length, ge, helpb, take

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb
take < helpb

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(a), gen_0':s6_0(n8_0)) -> gen_0':s6_0(+(n8_0, a)), rt in Omega(1 + n8_0)


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
length, helpa, ge, helpb, take

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb
take < helpb

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons:xs5_0(n825_0)) -> gen_0':s6_0(n825_0), rt in Omega(1 + n825_0)

Induction Base:
length(gen_nil:cons:xs5_0(0)) ->_R^Omega(1)
0'

Induction Step:
length(gen_nil:cons:xs5_0(+(n825_0, 1))) ->_R^Omega(1)
s(length(gen_nil:cons:xs5_0(n825_0))) ->_IH
s(gen_0':s6_0(c826_0))

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

(14)
Obligation:
TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(a), gen_0':s6_0(n8_0)) -> gen_0':s6_0(+(n8_0, a)), rt in Omega(1 + n8_0)
length(gen_nil:cons:xs5_0(n825_0)) -> gen_0':s6_0(n825_0), rt in Omega(1 + n825_0)


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
ge, helpa, helpb, take

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb
take < helpb

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s6_0(n1099_0), gen_0':s6_0(n1099_0)) -> true, rt in Omega(1 + n1099_0)

Induction Base:
ge(gen_0':s6_0(0), gen_0':s6_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s6_0(+(n1099_0, 1)), gen_0':s6_0(+(n1099_0, 1))) ->_R^Omega(1)
ge(gen_0':s6_0(n1099_0), gen_0':s6_0(n1099_0)) ->_IH
true

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

(16)
Obligation:
TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(a), gen_0':s6_0(n8_0)) -> gen_0':s6_0(+(n8_0, a)), rt in Omega(1 + n8_0)
length(gen_nil:cons:xs5_0(n825_0)) -> gen_0':s6_0(n825_0), rt in Omega(1 + n825_0)
ge(gen_0':s6_0(n1099_0), gen_0':s6_0(n1099_0)) -> true, rt in Omega(1 + n1099_0)


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
take, helpa, helpb

They will be analysed ascendingly in the following order:
helpa = helpb
take < helpb

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
take(gen_0':s6_0(n1397_0), gen_nil:cons:xs5_0(0), gen_nil:cons:xs5_0(+(1, n1397_0))) -> hole_take3_0, rt in Omega(1 + n1397_0)

Induction Base:
take(gen_0':s6_0(0), gen_nil:cons:xs5_0(0), gen_nil:cons:xs5_0(+(1, 0))) ->_R^Omega(1)
hole_take3_0

Induction Step:
take(gen_0':s6_0(+(n1397_0, 1)), gen_nil:cons:xs5_0(0), gen_nil:cons:xs5_0(+(1, +(n1397_0, 1)))) ->_R^Omega(1)
take(gen_0':s6_0(n1397_0), nil, gen_nil:cons:xs5_0(+(1, n1397_0))) ->_IH
hole_take3_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:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(a), gen_0':s6_0(n8_0)) -> gen_0':s6_0(+(n8_0, a)), rt in Omega(1 + n8_0)
length(gen_nil:cons:xs5_0(n825_0)) -> gen_0':s6_0(n825_0), rt in Omega(1 + n825_0)
ge(gen_0':s6_0(n1099_0), gen_0':s6_0(n1099_0)) -> true, rt in Omega(1 + n1099_0)
take(gen_0':s6_0(n1397_0), gen_nil:cons:xs5_0(0), gen_nil:cons:xs5_0(+(1, n1397_0))) -> hole_take3_0, rt in Omega(1 + n1397_0)


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
helpb, helpa

They will be analysed ascendingly in the following order:
helpa = helpb