WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1000 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) SlicingProof [LOWER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 1 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 13.7 s]
(12) typed CpxTrs
(13) RewriteLemmaProof [LOWER BOUND(ID), 3895 ms]
(14) BEST
    (15) proven lower bound
        (16) LowerBoundPropagationProof [FINISHED, 0 ms]
        (17) BOUNDS(n^1, INF)
    (18) typed CpxTrs


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

(0)
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:

   A__FST(0, z0) -> c
   A__FST(s(z0), cons(z1, z2)) -> c1(MARK(z1))
   A__FST(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   A__ADD(0, z0) -> c5(MARK(z0))
   A__ADD(s(z0), z1) -> c6
   A__ADD(z0, z1) -> c7
   A__LEN(nil) -> c8
   A__LEN(cons(z0, z1)) -> c9
   A__LEN(z0) -> c10
   MARK(fst(z0, z1)) -> c11(A__FST(mark(z0), mark(z1)), MARK(z0))
   MARK(fst(z0, z1)) -> c12(A__FST(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c13(A__FROM(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c14(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c15(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(len(z0)) -> c16(A__LEN(mark(z0)), MARK(z0))
   MARK(0) -> c17
   MARK(s(z0)) -> c18
   MARK(nil) -> c19
   MARK(cons(z0, z1)) -> c20(MARK(z0))

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

   a__fst(0, z0) -> nil
   a__fst(s(z0), cons(z1, z2)) -> cons(mark(z1), fst(z0, z2))
   a__fst(z0, z1) -> fst(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(add(z0, z1))
   a__add(z0, z1) -> add(z0, z1)
   a__len(nil) -> 0
   a__len(cons(z0, z1)) -> s(len(z1))
   a__len(z0) -> len(z0)
   mark(fst(z0, z1)) -> a__fst(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(len(z0)) -> a__len(mark(z0))
   mark(0) -> 0
   mark(s(z0)) -> s(z0)
   mark(nil) -> nil
   mark(cons(z0, z1)) -> cons(mark(z0), z1)

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(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:

   A__FST(0, z0) -> c
   A__FST(s(z0), cons(z1, z2)) -> c1(MARK(z1))
   A__FST(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   A__ADD(0, z0) -> c5(MARK(z0))
   A__ADD(s(z0), z1) -> c6
   A__ADD(z0, z1) -> c7
   A__LEN(nil) -> c8
   A__LEN(cons(z0, z1)) -> c9
   A__LEN(z0) -> c10
   MARK(fst(z0, z1)) -> c11(A__FST(mark(z0), mark(z1)), MARK(z0))
   MARK(fst(z0, z1)) -> c12(A__FST(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c13(A__FROM(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c14(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c15(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(len(z0)) -> c16(A__LEN(mark(z0)), MARK(z0))
   MARK(0) -> c17
   MARK(s(z0)) -> c18
   MARK(nil) -> c19
   MARK(cons(z0, z1)) -> c20(MARK(z0))

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

   a__fst(0, z0) -> nil
   a__fst(s(z0), cons(z1, z2)) -> cons(mark(z1), fst(z0, z2))
   a__fst(z0, z1) -> fst(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(add(z0, z1))
   a__add(z0, z1) -> add(z0, z1)
   a__len(nil) -> 0
   a__len(cons(z0, z1)) -> s(len(z1))
   a__len(z0) -> len(z0)
   mark(fst(z0, z1)) -> a__fst(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(len(z0)) -> a__len(mark(z0))
   mark(0) -> 0
   mark(s(z0)) -> s(z0)
   mark(nil) -> nil
   mark(cons(z0, z1)) -> cons(mark(z0), z1)

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

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

(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:

   A__FST(0', z0) -> c
   A__FST(s(z0), cons(z1, z2)) -> c1(MARK(z1))
   A__FST(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   A__ADD(0', z0) -> c5(MARK(z0))
   A__ADD(s(z0), z1) -> c6
   A__ADD(z0, z1) -> c7
   A__LEN(nil) -> c8
   A__LEN(cons(z0, z1)) -> c9
   A__LEN(z0) -> c10
   MARK(fst(z0, z1)) -> c11(A__FST(mark(z0), mark(z1)), MARK(z0))
   MARK(fst(z0, z1)) -> c12(A__FST(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c13(A__FROM(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c14(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c15(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(len(z0)) -> c16(A__LEN(mark(z0)), MARK(z0))
   MARK(0') -> c17
   MARK(s(z0)) -> c18
   MARK(nil) -> c19
   MARK(cons(z0, z1)) -> c20(MARK(z0))

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

   a__fst(0', z0) -> nil
   a__fst(s(z0), cons(z1, z2)) -> cons(mark(z1), fst(z0, z2))
   a__fst(z0, z1) -> fst(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__add(0', z0) -> mark(z0)
   a__add(s(z0), z1) -> s(add(z0, z1))
   a__add(z0, z1) -> add(z0, z1)
   a__len(nil) -> 0'
   a__len(cons(z0, z1)) -> s(len(z1))
   a__len(z0) -> len(z0)
   mark(fst(z0, z1)) -> a__fst(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(len(z0)) -> a__len(mark(z0))
   mark(0') -> 0'
   mark(s(z0)) -> s(z0)
   mark(nil) -> nil
   mark(cons(z0, z1)) -> cons(mark(z0), z1)

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

(5) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
s/0
cons/1

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

(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:

   A__FST(0', z0) -> c
   A__FST(s, cons(z1)) -> c1(MARK(z1))
   A__FST(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   A__ADD(0', z0) -> c5(MARK(z0))
   A__ADD(s, z1) -> c6
   A__ADD(z0, z1) -> c7
   A__LEN(nil) -> c8
   A__LEN(cons(z0)) -> c9
   A__LEN(z0) -> c10
   MARK(fst(z0, z1)) -> c11(A__FST(mark(z0), mark(z1)), MARK(z0))
   MARK(fst(z0, z1)) -> c12(A__FST(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c13(A__FROM(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c14(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c15(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(len(z0)) -> c16(A__LEN(mark(z0)), MARK(z0))
   MARK(0') -> c17
   MARK(s) -> c18
   MARK(nil) -> c19
   MARK(cons(z0)) -> c20(MARK(z0))

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

   a__fst(0', z0) -> nil
   a__fst(s, cons(z1)) -> cons(mark(z1))
   a__fst(z0, z1) -> fst(z0, z1)
   a__from(z0) -> cons(mark(z0))
   a__from(z0) -> from(z0)
   a__add(0', z0) -> mark(z0)
   a__add(s, z1) -> s
   a__add(z0, z1) -> add(z0, z1)
   a__len(nil) -> 0'
   a__len(cons(z0)) -> s
   a__len(z0) -> len(z0)
   mark(fst(z0, z1)) -> a__fst(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(len(z0)) -> a__len(mark(z0))
   mark(0') -> 0'
   mark(s) -> s
   mark(nil) -> nil
   mark(cons(z0)) -> cons(mark(z0))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
A__FST(0', z0) -> c
A__FST(s, cons(z1)) -> c1(MARK(z1))
A__FST(z0, z1) -> c2
A__FROM(z0) -> c3(MARK(z0))
A__FROM(z0) -> c4
A__ADD(0', z0) -> c5(MARK(z0))
A__ADD(s, z1) -> c6
A__ADD(z0, z1) -> c7
A__LEN(nil) -> c8
A__LEN(cons(z0)) -> c9
A__LEN(z0) -> c10
MARK(fst(z0, z1)) -> c11(A__FST(mark(z0), mark(z1)), MARK(z0))
MARK(fst(z0, z1)) -> c12(A__FST(mark(z0), mark(z1)), MARK(z1))
MARK(from(z0)) -> c13(A__FROM(mark(z0)), MARK(z0))
MARK(add(z0, z1)) -> c14(A__ADD(mark(z0), mark(z1)), MARK(z0))
MARK(add(z0, z1)) -> c15(A__ADD(mark(z0), mark(z1)), MARK(z1))
MARK(len(z0)) -> c16(A__LEN(mark(z0)), MARK(z0))
MARK(0') -> c17
MARK(s) -> c18
MARK(nil) -> c19
MARK(cons(z0)) -> c20(MARK(z0))
a__fst(0', z0) -> nil
a__fst(s, cons(z1)) -> cons(mark(z1))
a__fst(z0, z1) -> fst(z0, z1)
a__from(z0) -> cons(mark(z0))
a__from(z0) -> from(z0)
a__add(0', z0) -> mark(z0)
a__add(s, z1) -> s
a__add(z0, z1) -> add(z0, z1)
a__len(nil) -> 0'
a__len(cons(z0)) -> s
a__len(z0) -> len(z0)
mark(fst(z0, z1)) -> a__fst(mark(z0), mark(z1))
mark(from(z0)) -> a__from(mark(z0))
mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
mark(len(z0)) -> a__len(mark(z0))
mark(0') -> 0'
mark(s) -> s
mark(nil) -> nil
mark(cons(z0)) -> cons(mark(z0))

Types:
A__FST :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> c:c1:c2
0' :: 0':s:cons:nil:fst:from:add:len
c :: c:c1:c2
s :: 0':s:cons:nil:fst:from:add:len
cons :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c1 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
MARK :: 0':s:cons:nil:fst:from:add:len -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c2 :: c:c1:c2
A__FROM :: 0':s:cons:nil:fst:from:add:len -> c3:c4
c3 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c3:c4
c4 :: c3:c4
A__ADD :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> c5:c6:c7
c5 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7
A__LEN :: 0':s:cons:nil:fst:from:add:len -> c8:c9:c10
nil :: 0':s:cons:nil:fst:from:add:len
c8 :: c8:c9:c10
c9 :: c8:c9:c10
c10 :: c8:c9:c10
fst :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c11 :: c:c1:c2 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
mark :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c12 :: c:c1:c2 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
from :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c13 :: c3:c4 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
add :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c14 :: c5:c6:c7 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c15 :: c5:c6:c7 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
len :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c16 :: c8:c9:c10 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c17 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c18 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c19 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c20 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
a__fst :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__from :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__add :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__len :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
hole_c:c1:c21_21 :: c:c1:c2
hole_0':s:cons:nil:fst:from:add:len2_21 :: 0':s:cons:nil:fst:from:add:len
hole_c11:c12:c13:c14:c15:c16:c17:c18:c19:c203_21 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
hole_c3:c44_21 :: c3:c4
hole_c5:c6:c75_21 :: c5:c6:c7
hole_c8:c9:c106_21 :: c8:c9:c10
gen_0':s:cons:nil:fst:from:add:len7_21 :: Nat -> 0':s:cons:nil:fst:from:add:len
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21 :: Nat -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MARK, A__FROM, mark, a__from

They will be analysed ascendingly in the following order:
MARK = A__FROM
mark < MARK
mark = a__from

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

(10)
Obligation:
Innermost TRS:
Rules:
A__FST(0', z0) -> c
A__FST(s, cons(z1)) -> c1(MARK(z1))
A__FST(z0, z1) -> c2
A__FROM(z0) -> c3(MARK(z0))
A__FROM(z0) -> c4
A__ADD(0', z0) -> c5(MARK(z0))
A__ADD(s, z1) -> c6
A__ADD(z0, z1) -> c7
A__LEN(nil) -> c8
A__LEN(cons(z0)) -> c9
A__LEN(z0) -> c10
MARK(fst(z0, z1)) -> c11(A__FST(mark(z0), mark(z1)), MARK(z0))
MARK(fst(z0, z1)) -> c12(A__FST(mark(z0), mark(z1)), MARK(z1))
MARK(from(z0)) -> c13(A__FROM(mark(z0)), MARK(z0))
MARK(add(z0, z1)) -> c14(A__ADD(mark(z0), mark(z1)), MARK(z0))
MARK(add(z0, z1)) -> c15(A__ADD(mark(z0), mark(z1)), MARK(z1))
MARK(len(z0)) -> c16(A__LEN(mark(z0)), MARK(z0))
MARK(0') -> c17
MARK(s) -> c18
MARK(nil) -> c19
MARK(cons(z0)) -> c20(MARK(z0))
a__fst(0', z0) -> nil
a__fst(s, cons(z1)) -> cons(mark(z1))
a__fst(z0, z1) -> fst(z0, z1)
a__from(z0) -> cons(mark(z0))
a__from(z0) -> from(z0)
a__add(0', z0) -> mark(z0)
a__add(s, z1) -> s
a__add(z0, z1) -> add(z0, z1)
a__len(nil) -> 0'
a__len(cons(z0)) -> s
a__len(z0) -> len(z0)
mark(fst(z0, z1)) -> a__fst(mark(z0), mark(z1))
mark(from(z0)) -> a__from(mark(z0))
mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
mark(len(z0)) -> a__len(mark(z0))
mark(0') -> 0'
mark(s) -> s
mark(nil) -> nil
mark(cons(z0)) -> cons(mark(z0))

Types:
A__FST :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> c:c1:c2
0' :: 0':s:cons:nil:fst:from:add:len
c :: c:c1:c2
s :: 0':s:cons:nil:fst:from:add:len
cons :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c1 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
MARK :: 0':s:cons:nil:fst:from:add:len -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c2 :: c:c1:c2
A__FROM :: 0':s:cons:nil:fst:from:add:len -> c3:c4
c3 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c3:c4
c4 :: c3:c4
A__ADD :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> c5:c6:c7
c5 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7
A__LEN :: 0':s:cons:nil:fst:from:add:len -> c8:c9:c10
nil :: 0':s:cons:nil:fst:from:add:len
c8 :: c8:c9:c10
c9 :: c8:c9:c10
c10 :: c8:c9:c10
fst :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c11 :: c:c1:c2 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
mark :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c12 :: c:c1:c2 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
from :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c13 :: c3:c4 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
add :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c14 :: c5:c6:c7 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c15 :: c5:c6:c7 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
len :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c16 :: c8:c9:c10 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c17 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c18 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c19 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c20 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
a__fst :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__from :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__add :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__len :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
hole_c:c1:c21_21 :: c:c1:c2
hole_0':s:cons:nil:fst:from:add:len2_21 :: 0':s:cons:nil:fst:from:add:len
hole_c11:c12:c13:c14:c15:c16:c17:c18:c19:c203_21 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
hole_c3:c44_21 :: c3:c4
hole_c5:c6:c75_21 :: c5:c6:c7
hole_c8:c9:c106_21 :: c8:c9:c10
gen_0':s:cons:nil:fst:from:add:len7_21 :: Nat -> 0':s:cons:nil:fst:from:add:len
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21 :: Nat -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20


Generator Equations:
gen_0':s:cons:nil:fst:from:add:len7_21(0) <=> 0'
gen_0':s:cons:nil:fst:from:add:len7_21(+(x, 1)) <=> cons(gen_0':s:cons:nil:fst:from:add:len7_21(x))
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(0) <=> c17
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(+(x, 1)) <=> c11(c, gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(x))


The following defined symbols remain to be analysed:
a__from, MARK, A__FROM, mark

They will be analysed ascendingly in the following order:
MARK = A__FROM
mark < MARK
mark = a__from

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':s:cons:nil:fst:from:add:len7_21(n27540_21)) -> gen_0':s:cons:nil:fst:from:add:len7_21(n27540_21), rt in Omega(0)

Induction Base:
mark(gen_0':s:cons:nil:fst:from:add:len7_21(0)) ->_R^Omega(0)
0'

Induction Step:
mark(gen_0':s:cons:nil:fst:from:add:len7_21(+(n27540_21, 1))) ->_R^Omega(0)
cons(mark(gen_0':s:cons:nil:fst:from:add:len7_21(n27540_21))) ->_IH
cons(gen_0':s:cons:nil:fst:from:add:len7_21(c27541_21))

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

(12)
Obligation:
Innermost TRS:
Rules:
A__FST(0', z0) -> c
A__FST(s, cons(z1)) -> c1(MARK(z1))
A__FST(z0, z1) -> c2
A__FROM(z0) -> c3(MARK(z0))
A__FROM(z0) -> c4
A__ADD(0', z0) -> c5(MARK(z0))
A__ADD(s, z1) -> c6
A__ADD(z0, z1) -> c7
A__LEN(nil) -> c8
A__LEN(cons(z0)) -> c9
A__LEN(z0) -> c10
MARK(fst(z0, z1)) -> c11(A__FST(mark(z0), mark(z1)), MARK(z0))
MARK(fst(z0, z1)) -> c12(A__FST(mark(z0), mark(z1)), MARK(z1))
MARK(from(z0)) -> c13(A__FROM(mark(z0)), MARK(z0))
MARK(add(z0, z1)) -> c14(A__ADD(mark(z0), mark(z1)), MARK(z0))
MARK(add(z0, z1)) -> c15(A__ADD(mark(z0), mark(z1)), MARK(z1))
MARK(len(z0)) -> c16(A__LEN(mark(z0)), MARK(z0))
MARK(0') -> c17
MARK(s) -> c18
MARK(nil) -> c19
MARK(cons(z0)) -> c20(MARK(z0))
a__fst(0', z0) -> nil
a__fst(s, cons(z1)) -> cons(mark(z1))
a__fst(z0, z1) -> fst(z0, z1)
a__from(z0) -> cons(mark(z0))
a__from(z0) -> from(z0)
a__add(0', z0) -> mark(z0)
a__add(s, z1) -> s
a__add(z0, z1) -> add(z0, z1)
a__len(nil) -> 0'
a__len(cons(z0)) -> s
a__len(z0) -> len(z0)
mark(fst(z0, z1)) -> a__fst(mark(z0), mark(z1))
mark(from(z0)) -> a__from(mark(z0))
mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
mark(len(z0)) -> a__len(mark(z0))
mark(0') -> 0'
mark(s) -> s
mark(nil) -> nil
mark(cons(z0)) -> cons(mark(z0))

Types:
A__FST :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> c:c1:c2
0' :: 0':s:cons:nil:fst:from:add:len
c :: c:c1:c2
s :: 0':s:cons:nil:fst:from:add:len
cons :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c1 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
MARK :: 0':s:cons:nil:fst:from:add:len -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c2 :: c:c1:c2
A__FROM :: 0':s:cons:nil:fst:from:add:len -> c3:c4
c3 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c3:c4
c4 :: c3:c4
A__ADD :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> c5:c6:c7
c5 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7
A__LEN :: 0':s:cons:nil:fst:from:add:len -> c8:c9:c10
nil :: 0':s:cons:nil:fst:from:add:len
c8 :: c8:c9:c10
c9 :: c8:c9:c10
c10 :: c8:c9:c10
fst :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c11 :: c:c1:c2 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
mark :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c12 :: c:c1:c2 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
from :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c13 :: c3:c4 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
add :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c14 :: c5:c6:c7 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c15 :: c5:c6:c7 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
len :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c16 :: c8:c9:c10 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c17 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c18 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c19 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c20 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
a__fst :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__from :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__add :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__len :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
hole_c:c1:c21_21 :: c:c1:c2
hole_0':s:cons:nil:fst:from:add:len2_21 :: 0':s:cons:nil:fst:from:add:len
hole_c11:c12:c13:c14:c15:c16:c17:c18:c19:c203_21 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
hole_c3:c44_21 :: c3:c4
hole_c5:c6:c75_21 :: c5:c6:c7
hole_c8:c9:c106_21 :: c8:c9:c10
gen_0':s:cons:nil:fst:from:add:len7_21 :: Nat -> 0':s:cons:nil:fst:from:add:len
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21 :: Nat -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20


Lemmas:
mark(gen_0':s:cons:nil:fst:from:add:len7_21(n27540_21)) -> gen_0':s:cons:nil:fst:from:add:len7_21(n27540_21), rt in Omega(0)


Generator Equations:
gen_0':s:cons:nil:fst:from:add:len7_21(0) <=> 0'
gen_0':s:cons:nil:fst:from:add:len7_21(+(x, 1)) <=> cons(gen_0':s:cons:nil:fst:from:add:len7_21(x))
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(0) <=> c17
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(+(x, 1)) <=> c11(c, gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(x))


The following defined symbols remain to be analysed:
a__from, MARK, A__FROM

They will be analysed ascendingly in the following order:
MARK = A__FROM
mark < MARK
mark = a__from

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MARK(gen_0':s:cons:nil:fst:from:add:len7_21(+(1, n91594_21))) -> *9_21, rt in Omega(n91594_21)

Induction Base:
MARK(gen_0':s:cons:nil:fst:from:add:len7_21(+(1, 0)))

Induction Step:
MARK(gen_0':s:cons:nil:fst:from:add:len7_21(+(1, +(n91594_21, 1)))) ->_R^Omega(1)
c20(MARK(gen_0':s:cons:nil:fst:from:add:len7_21(+(1, n91594_21)))) ->_IH
c20(*9_21)

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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
A__FST(0', z0) -> c
A__FST(s, cons(z1)) -> c1(MARK(z1))
A__FST(z0, z1) -> c2
A__FROM(z0) -> c3(MARK(z0))
A__FROM(z0) -> c4
A__ADD(0', z0) -> c5(MARK(z0))
A__ADD(s, z1) -> c6
A__ADD(z0, z1) -> c7
A__LEN(nil) -> c8
A__LEN(cons(z0)) -> c9
A__LEN(z0) -> c10
MARK(fst(z0, z1)) -> c11(A__FST(mark(z0), mark(z1)), MARK(z0))
MARK(fst(z0, z1)) -> c12(A__FST(mark(z0), mark(z1)), MARK(z1))
MARK(from(z0)) -> c13(A__FROM(mark(z0)), MARK(z0))
MARK(add(z0, z1)) -> c14(A__ADD(mark(z0), mark(z1)), MARK(z0))
MARK(add(z0, z1)) -> c15(A__ADD(mark(z0), mark(z1)), MARK(z1))
MARK(len(z0)) -> c16(A__LEN(mark(z0)), MARK(z0))
MARK(0') -> c17
MARK(s) -> c18
MARK(nil) -> c19
MARK(cons(z0)) -> c20(MARK(z0))
a__fst(0', z0) -> nil
a__fst(s, cons(z1)) -> cons(mark(z1))
a__fst(z0, z1) -> fst(z0, z1)
a__from(z0) -> cons(mark(z0))
a__from(z0) -> from(z0)
a__add(0', z0) -> mark(z0)
a__add(s, z1) -> s
a__add(z0, z1) -> add(z0, z1)
a__len(nil) -> 0'
a__len(cons(z0)) -> s
a__len(z0) -> len(z0)
mark(fst(z0, z1)) -> a__fst(mark(z0), mark(z1))
mark(from(z0)) -> a__from(mark(z0))
mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
mark(len(z0)) -> a__len(mark(z0))
mark(0') -> 0'
mark(s) -> s
mark(nil) -> nil
mark(cons(z0)) -> cons(mark(z0))

Types:
A__FST :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> c:c1:c2
0' :: 0':s:cons:nil:fst:from:add:len
c :: c:c1:c2
s :: 0':s:cons:nil:fst:from:add:len
cons :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c1 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
MARK :: 0':s:cons:nil:fst:from:add:len -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c2 :: c:c1:c2
A__FROM :: 0':s:cons:nil:fst:from:add:len -> c3:c4
c3 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c3:c4
c4 :: c3:c4
A__ADD :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> c5:c6:c7
c5 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7
A__LEN :: 0':s:cons:nil:fst:from:add:len -> c8:c9:c10
nil :: 0':s:cons:nil:fst:from:add:len
c8 :: c8:c9:c10
c9 :: c8:c9:c10
c10 :: c8:c9:c10
fst :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c11 :: c:c1:c2 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
mark :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c12 :: c:c1:c2 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
from :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c13 :: c3:c4 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
add :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c14 :: c5:c6:c7 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c15 :: c5:c6:c7 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
len :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c16 :: c8:c9:c10 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c17 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c18 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c19 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c20 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
a__fst :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__from :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__add :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__len :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
hole_c:c1:c21_21 :: c:c1:c2
hole_0':s:cons:nil:fst:from:add:len2_21 :: 0':s:cons:nil:fst:from:add:len
hole_c11:c12:c13:c14:c15:c16:c17:c18:c19:c203_21 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
hole_c3:c44_21 :: c3:c4
hole_c5:c6:c75_21 :: c5:c6:c7
hole_c8:c9:c106_21 :: c8:c9:c10
gen_0':s:cons:nil:fst:from:add:len7_21 :: Nat -> 0':s:cons:nil:fst:from:add:len
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21 :: Nat -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20


Lemmas:
mark(gen_0':s:cons:nil:fst:from:add:len7_21(n27540_21)) -> gen_0':s:cons:nil:fst:from:add:len7_21(n27540_21), rt in Omega(0)


Generator Equations:
gen_0':s:cons:nil:fst:from:add:len7_21(0) <=> 0'
gen_0':s:cons:nil:fst:from:add:len7_21(+(x, 1)) <=> cons(gen_0':s:cons:nil:fst:from:add:len7_21(x))
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(0) <=> c17
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(+(x, 1)) <=> c11(c, gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(x))


The following defined symbols remain to be analysed:
MARK

They will be analysed ascendingly in the following order:
MARK = A__FROM

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
Innermost TRS:
Rules:
A__FST(0', z0) -> c
A__FST(s, cons(z1)) -> c1(MARK(z1))
A__FST(z0, z1) -> c2
A__FROM(z0) -> c3(MARK(z0))
A__FROM(z0) -> c4
A__ADD(0', z0) -> c5(MARK(z0))
A__ADD(s, z1) -> c6
A__ADD(z0, z1) -> c7
A__LEN(nil) -> c8
A__LEN(cons(z0)) -> c9
A__LEN(z0) -> c10
MARK(fst(z0, z1)) -> c11(A__FST(mark(z0), mark(z1)), MARK(z0))
MARK(fst(z0, z1)) -> c12(A__FST(mark(z0), mark(z1)), MARK(z1))
MARK(from(z0)) -> c13(A__FROM(mark(z0)), MARK(z0))
MARK(add(z0, z1)) -> c14(A__ADD(mark(z0), mark(z1)), MARK(z0))
MARK(add(z0, z1)) -> c15(A__ADD(mark(z0), mark(z1)), MARK(z1))
MARK(len(z0)) -> c16(A__LEN(mark(z0)), MARK(z0))
MARK(0') -> c17
MARK(s) -> c18
MARK(nil) -> c19
MARK(cons(z0)) -> c20(MARK(z0))
a__fst(0', z0) -> nil
a__fst(s, cons(z1)) -> cons(mark(z1))
a__fst(z0, z1) -> fst(z0, z1)
a__from(z0) -> cons(mark(z0))
a__from(z0) -> from(z0)
a__add(0', z0) -> mark(z0)
a__add(s, z1) -> s
a__add(z0, z1) -> add(z0, z1)
a__len(nil) -> 0'
a__len(cons(z0)) -> s
a__len(z0) -> len(z0)
mark(fst(z0, z1)) -> a__fst(mark(z0), mark(z1))
mark(from(z0)) -> a__from(mark(z0))
mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
mark(len(z0)) -> a__len(mark(z0))
mark(0') -> 0'
mark(s) -> s
mark(nil) -> nil
mark(cons(z0)) -> cons(mark(z0))

Types:
A__FST :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> c:c1:c2
0' :: 0':s:cons:nil:fst:from:add:len
c :: c:c1:c2
s :: 0':s:cons:nil:fst:from:add:len
cons :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c1 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
MARK :: 0':s:cons:nil:fst:from:add:len -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c2 :: c:c1:c2
A__FROM :: 0':s:cons:nil:fst:from:add:len -> c3:c4
c3 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c3:c4
c4 :: c3:c4
A__ADD :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> c5:c6:c7
c5 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7
A__LEN :: 0':s:cons:nil:fst:from:add:len -> c8:c9:c10
nil :: 0':s:cons:nil:fst:from:add:len
c8 :: c8:c9:c10
c9 :: c8:c9:c10
c10 :: c8:c9:c10
fst :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c11 :: c:c1:c2 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
mark :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c12 :: c:c1:c2 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
from :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c13 :: c3:c4 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
add :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c14 :: c5:c6:c7 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c15 :: c5:c6:c7 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
len :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
c16 :: c8:c9:c10 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c17 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c18 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c19 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
c20 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
a__fst :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__from :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__add :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
a__len :: 0':s:cons:nil:fst:from:add:len -> 0':s:cons:nil:fst:from:add:len
hole_c:c1:c21_21 :: c:c1:c2
hole_0':s:cons:nil:fst:from:add:len2_21 :: 0':s:cons:nil:fst:from:add:len
hole_c11:c12:c13:c14:c15:c16:c17:c18:c19:c203_21 :: c11:c12:c13:c14:c15:c16:c17:c18:c19:c20
hole_c3:c44_21 :: c3:c4
hole_c5:c6:c75_21 :: c5:c6:c7
hole_c8:c9:c106_21 :: c8:c9:c10
gen_0':s:cons:nil:fst:from:add:len7_21 :: Nat -> 0':s:cons:nil:fst:from:add:len
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21 :: Nat -> c11:c12:c13:c14:c15:c16:c17:c18:c19:c20


Lemmas:
mark(gen_0':s:cons:nil:fst:from:add:len7_21(n27540_21)) -> gen_0':s:cons:nil:fst:from:add:len7_21(n27540_21), rt in Omega(0)
MARK(gen_0':s:cons:nil:fst:from:add:len7_21(+(1, n91594_21))) -> *9_21, rt in Omega(n91594_21)


Generator Equations:
gen_0':s:cons:nil:fst:from:add:len7_21(0) <=> 0'
gen_0':s:cons:nil:fst:from:add:len7_21(+(x, 1)) <=> cons(gen_0':s:cons:nil:fst:from:add:len7_21(x))
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(0) <=> c17
gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(+(x, 1)) <=> c11(c, gen_c11:c12:c13:c14:c15:c16:c17:c18:c19:c208_21(x))


The following defined symbols remain to be analysed:
A__FROM

They will be analysed ascendingly in the following order:
MARK = A__FROM