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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 1189 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), 17 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 21.0 s]
(12) typed CpxTrs
(13) RewriteLemmaProof [LOWER BOUND(ID), 1778 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__APP(nil, z0) -> c(MARK(z0))
   A__APP(cons(z0, z1), z2) -> c1(MARK(z0))
   A__APP(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   A__ZWADR(nil, z0) -> c5
   A__ZWADR(z0, nil) -> c6
   A__ZWADR(cons(z0, z1), cons(z2, z3)) -> c7(A__APP(mark(z2), cons(mark(z0), nil)), MARK(z2))
   A__ZWADR(cons(z0, z1), cons(z2, z3)) -> c8(A__APP(mark(z2), cons(mark(z0), nil)), MARK(z0))
   A__ZWADR(z0, z1) -> c9
   A__PREFIX(z0) -> c10
   A__PREFIX(z0) -> c11
   MARK(app(z0, z1)) -> c12(A__APP(mark(z0), mark(z1)), MARK(z0))
   MARK(app(z0, z1)) -> c13(A__APP(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c14(A__FROM(mark(z0)), MARK(z0))
   MARK(zWadr(z0, z1)) -> c15(A__ZWADR(mark(z0), mark(z1)), MARK(z0))
   MARK(zWadr(z0, z1)) -> c16(A__ZWADR(mark(z0), mark(z1)), MARK(z1))
   MARK(prefix(z0)) -> c17(A__PREFIX(mark(z0)), MARK(z0))
   MARK(nil) -> c18
   MARK(cons(z0, z1)) -> c19(MARK(z0))
   MARK(s(z0)) -> c20(MARK(z0))

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

   a__app(nil, z0) -> mark(z0)
   a__app(cons(z0, z1), z2) -> cons(mark(z0), app(z1, z2))
   a__app(z0, z1) -> app(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__zWadr(nil, z0) -> nil
   a__zWadr(z0, nil) -> nil
   a__zWadr(cons(z0, z1), cons(z2, z3)) -> cons(a__app(mark(z2), cons(mark(z0), nil)), zWadr(z1, z3))
   a__zWadr(z0, z1) -> zWadr(z0, z1)
   a__prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   a__prefix(z0) -> prefix(z0)
   mark(app(z0, z1)) -> a__app(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(zWadr(z0, z1)) -> a__zWadr(mark(z0), mark(z1))
   mark(prefix(z0)) -> a__prefix(mark(z0))
   mark(nil) -> nil
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))

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__APP(nil, z0) -> c(MARK(z0))
   A__APP(cons(z0, z1), z2) -> c1(MARK(z0))
   A__APP(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   A__ZWADR(nil, z0) -> c5
   A__ZWADR(z0, nil) -> c6
   A__ZWADR(cons(z0, z1), cons(z2, z3)) -> c7(A__APP(mark(z2), cons(mark(z0), nil)), MARK(z2))
   A__ZWADR(cons(z0, z1), cons(z2, z3)) -> c8(A__APP(mark(z2), cons(mark(z0), nil)), MARK(z0))
   A__ZWADR(z0, z1) -> c9
   A__PREFIX(z0) -> c10
   A__PREFIX(z0) -> c11
   MARK(app(z0, z1)) -> c12(A__APP(mark(z0), mark(z1)), MARK(z0))
   MARK(app(z0, z1)) -> c13(A__APP(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c14(A__FROM(mark(z0)), MARK(z0))
   MARK(zWadr(z0, z1)) -> c15(A__ZWADR(mark(z0), mark(z1)), MARK(z0))
   MARK(zWadr(z0, z1)) -> c16(A__ZWADR(mark(z0), mark(z1)), MARK(z1))
   MARK(prefix(z0)) -> c17(A__PREFIX(mark(z0)), MARK(z0))
   MARK(nil) -> c18
   MARK(cons(z0, z1)) -> c19(MARK(z0))
   MARK(s(z0)) -> c20(MARK(z0))

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

   a__app(nil, z0) -> mark(z0)
   a__app(cons(z0, z1), z2) -> cons(mark(z0), app(z1, z2))
   a__app(z0, z1) -> app(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__zWadr(nil, z0) -> nil
   a__zWadr(z0, nil) -> nil
   a__zWadr(cons(z0, z1), cons(z2, z3)) -> cons(a__app(mark(z2), cons(mark(z0), nil)), zWadr(z1, z3))
   a__zWadr(z0, z1) -> zWadr(z0, z1)
   a__prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   a__prefix(z0) -> prefix(z0)
   mark(app(z0, z1)) -> a__app(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(zWadr(z0, z1)) -> a__zWadr(mark(z0), mark(z1))
   mark(prefix(z0)) -> a__prefix(mark(z0))
   mark(nil) -> nil
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))

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__APP(nil, z0) -> c(MARK(z0))
   A__APP(cons(z0, z1), z2) -> c1(MARK(z0))
   A__APP(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   A__ZWADR(nil, z0) -> c5
   A__ZWADR(z0, nil) -> c6
   A__ZWADR(cons(z0, z1), cons(z2, z3)) -> c7(A__APP(mark(z2), cons(mark(z0), nil)), MARK(z2))
   A__ZWADR(cons(z0, z1), cons(z2, z3)) -> c8(A__APP(mark(z2), cons(mark(z0), nil)), MARK(z0))
   A__ZWADR(z0, z1) -> c9
   A__PREFIX(z0) -> c10
   A__PREFIX(z0) -> c11
   MARK(app(z0, z1)) -> c12(A__APP(mark(z0), mark(z1)), MARK(z0))
   MARK(app(z0, z1)) -> c13(A__APP(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c14(A__FROM(mark(z0)), MARK(z0))
   MARK(zWadr(z0, z1)) -> c15(A__ZWADR(mark(z0), mark(z1)), MARK(z0))
   MARK(zWadr(z0, z1)) -> c16(A__ZWADR(mark(z0), mark(z1)), MARK(z1))
   MARK(prefix(z0)) -> c17(A__PREFIX(mark(z0)), MARK(z0))
   MARK(nil) -> c18
   MARK(cons(z0, z1)) -> c19(MARK(z0))
   MARK(s(z0)) -> c20(MARK(z0))

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

   a__app(nil, z0) -> mark(z0)
   a__app(cons(z0, z1), z2) -> cons(mark(z0), app(z1, z2))
   a__app(z0, z1) -> app(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__zWadr(nil, z0) -> nil
   a__zWadr(z0, nil) -> nil
   a__zWadr(cons(z0, z1), cons(z2, z3)) -> cons(a__app(mark(z2), cons(mark(z0), nil)), zWadr(z1, z3))
   a__zWadr(z0, z1) -> zWadr(z0, z1)
   a__prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   a__prefix(z0) -> prefix(z0)
   mark(app(z0, z1)) -> a__app(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(zWadr(z0, z1)) -> a__zWadr(mark(z0), mark(z1))
   mark(prefix(z0)) -> a__prefix(mark(z0))
   mark(nil) -> nil
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))

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

(5) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
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__APP(nil, z0) -> c(MARK(z0))
   A__APP(cons(z0), z2) -> c1(MARK(z0))
   A__APP(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   A__ZWADR(nil, z0) -> c5
   A__ZWADR(z0, nil) -> c6
   A__ZWADR(cons(z0), cons(z2)) -> c7(A__APP(mark(z2), cons(mark(z0))), MARK(z2))
   A__ZWADR(cons(z0), cons(z2)) -> c8(A__APP(mark(z2), cons(mark(z0))), MARK(z0))
   A__ZWADR(z0, z1) -> c9
   A__PREFIX(z0) -> c10
   A__PREFIX(z0) -> c11
   MARK(app(z0, z1)) -> c12(A__APP(mark(z0), mark(z1)), MARK(z0))
   MARK(app(z0, z1)) -> c13(A__APP(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c14(A__FROM(mark(z0)), MARK(z0))
   MARK(zWadr(z0, z1)) -> c15(A__ZWADR(mark(z0), mark(z1)), MARK(z0))
   MARK(zWadr(z0, z1)) -> c16(A__ZWADR(mark(z0), mark(z1)), MARK(z1))
   MARK(prefix(z0)) -> c17(A__PREFIX(mark(z0)), MARK(z0))
   MARK(nil) -> c18
   MARK(cons(z0)) -> c19(MARK(z0))
   MARK(s(z0)) -> c20(MARK(z0))

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

   a__app(nil, z0) -> mark(z0)
   a__app(cons(z0), z2) -> cons(mark(z0))
   a__app(z0, z1) -> app(z0, z1)
   a__from(z0) -> cons(mark(z0))
   a__from(z0) -> from(z0)
   a__zWadr(nil, z0) -> nil
   a__zWadr(z0, nil) -> nil
   a__zWadr(cons(z0), cons(z2)) -> cons(a__app(mark(z2), cons(mark(z0))))
   a__zWadr(z0, z1) -> zWadr(z0, z1)
   a__prefix(z0) -> cons(nil)
   a__prefix(z0) -> prefix(z0)
   mark(app(z0, z1)) -> a__app(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(zWadr(z0, z1)) -> a__zWadr(mark(z0), mark(z1))
   mark(prefix(z0)) -> a__prefix(mark(z0))
   mark(nil) -> nil
   mark(cons(z0)) -> cons(mark(z0))
   mark(s(z0)) -> s(mark(z0))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
A__APP(nil, z0) -> c(MARK(z0))
A__APP(cons(z0), z2) -> c1(MARK(z0))
A__APP(z0, z1) -> c2
A__FROM(z0) -> c3(MARK(z0))
A__FROM(z0) -> c4
A__ZWADR(nil, z0) -> c5
A__ZWADR(z0, nil) -> c6
A__ZWADR(cons(z0), cons(z2)) -> c7(A__APP(mark(z2), cons(mark(z0))), MARK(z2))
A__ZWADR(cons(z0), cons(z2)) -> c8(A__APP(mark(z2), cons(mark(z0))), MARK(z0))
A__ZWADR(z0, z1) -> c9
A__PREFIX(z0) -> c10
A__PREFIX(z0) -> c11
MARK(app(z0, z1)) -> c12(A__APP(mark(z0), mark(z1)), MARK(z0))
MARK(app(z0, z1)) -> c13(A__APP(mark(z0), mark(z1)), MARK(z1))
MARK(from(z0)) -> c14(A__FROM(mark(z0)), MARK(z0))
MARK(zWadr(z0, z1)) -> c15(A__ZWADR(mark(z0), mark(z1)), MARK(z0))
MARK(zWadr(z0, z1)) -> c16(A__ZWADR(mark(z0), mark(z1)), MARK(z1))
MARK(prefix(z0)) -> c17(A__PREFIX(mark(z0)), MARK(z0))
MARK(nil) -> c18
MARK(cons(z0)) -> c19(MARK(z0))
MARK(s(z0)) -> c20(MARK(z0))
a__app(nil, z0) -> mark(z0)
a__app(cons(z0), z2) -> cons(mark(z0))
a__app(z0, z1) -> app(z0, z1)
a__from(z0) -> cons(mark(z0))
a__from(z0) -> from(z0)
a__zWadr(nil, z0) -> nil
a__zWadr(z0, nil) -> nil
a__zWadr(cons(z0), cons(z2)) -> cons(a__app(mark(z2), cons(mark(z0))))
a__zWadr(z0, z1) -> zWadr(z0, z1)
a__prefix(z0) -> cons(nil)
a__prefix(z0) -> prefix(z0)
mark(app(z0, z1)) -> a__app(mark(z0), mark(z1))
mark(from(z0)) -> a__from(mark(z0))
mark(zWadr(z0, z1)) -> a__zWadr(mark(z0), mark(z1))
mark(prefix(z0)) -> a__prefix(mark(z0))
mark(nil) -> nil
mark(cons(z0)) -> cons(mark(z0))
mark(s(z0)) -> s(mark(z0))

Types:
A__APP :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> c:c1:c2
nil :: nil:cons:app:from:zWadr:prefix:s
c :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
MARK :: nil:cons:app:from:zWadr:prefix:s -> c12:c13:c14:c15:c16:c17:c18:c19:c20
cons :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c1 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
c2 :: c:c1:c2
A__FROM :: nil:cons:app:from:zWadr:prefix:s -> c3:c4
c3 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c3:c4
c4 :: c3:c4
A__ZWADR :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> c5:c6:c7:c8:c9
c5 :: c5:c6:c7:c8:c9
c6 :: c5:c6:c7:c8:c9
c7 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7:c8:c9
mark :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c8 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7:c8:c9
c9 :: c5:c6:c7:c8:c9
A__PREFIX :: nil:cons:app:from:zWadr:prefix:s -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
app :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c12 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c13 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
from :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c14 :: c3:c4 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
zWadr :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c15 :: c5:c6:c7:c8:c9 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c16 :: c5:c6:c7:c8:c9 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
prefix :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c17 :: c10:c11 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c18 :: c12:c13:c14:c15:c16:c17:c18:c19:c20
c19 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
s :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c20 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
a__app :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
hole_c:c1:c21_21 :: c:c1:c2
hole_nil:cons:app:from:zWadr:prefix:s2_21 :: nil:cons:app:from:zWadr:prefix:s
hole_c12:c13:c14:c15:c16:c17:c18:c19:c203_21 :: c12:c13:c14:c15:c16:c17:c18:c19:c20
hole_c3:c44_21 :: c3:c4
hole_c5:c6:c7:c8:c95_21 :: c5:c6:c7:c8:c9
hole_c10:c116_21 :: c10:c11
gen_nil:cons:app:from:zWadr:prefix:s7_21 :: Nat -> nil:cons:app:from:zWadr:prefix:s
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21 :: Nat -> c12:c13:c14:c15:c16:c17:c18:c19:c20

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
A__APP(nil, z0) -> c(MARK(z0))
A__APP(cons(z0), z2) -> c1(MARK(z0))
A__APP(z0, z1) -> c2
A__FROM(z0) -> c3(MARK(z0))
A__FROM(z0) -> c4
A__ZWADR(nil, z0) -> c5
A__ZWADR(z0, nil) -> c6
A__ZWADR(cons(z0), cons(z2)) -> c7(A__APP(mark(z2), cons(mark(z0))), MARK(z2))
A__ZWADR(cons(z0), cons(z2)) -> c8(A__APP(mark(z2), cons(mark(z0))), MARK(z0))
A__ZWADR(z0, z1) -> c9
A__PREFIX(z0) -> c10
A__PREFIX(z0) -> c11
MARK(app(z0, z1)) -> c12(A__APP(mark(z0), mark(z1)), MARK(z0))
MARK(app(z0, z1)) -> c13(A__APP(mark(z0), mark(z1)), MARK(z1))
MARK(from(z0)) -> c14(A__FROM(mark(z0)), MARK(z0))
MARK(zWadr(z0, z1)) -> c15(A__ZWADR(mark(z0), mark(z1)), MARK(z0))
MARK(zWadr(z0, z1)) -> c16(A__ZWADR(mark(z0), mark(z1)), MARK(z1))
MARK(prefix(z0)) -> c17(A__PREFIX(mark(z0)), MARK(z0))
MARK(nil) -> c18
MARK(cons(z0)) -> c19(MARK(z0))
MARK(s(z0)) -> c20(MARK(z0))
a__app(nil, z0) -> mark(z0)
a__app(cons(z0), z2) -> cons(mark(z0))
a__app(z0, z1) -> app(z0, z1)
a__from(z0) -> cons(mark(z0))
a__from(z0) -> from(z0)
a__zWadr(nil, z0) -> nil
a__zWadr(z0, nil) -> nil
a__zWadr(cons(z0), cons(z2)) -> cons(a__app(mark(z2), cons(mark(z0))))
a__zWadr(z0, z1) -> zWadr(z0, z1)
a__prefix(z0) -> cons(nil)
a__prefix(z0) -> prefix(z0)
mark(app(z0, z1)) -> a__app(mark(z0), mark(z1))
mark(from(z0)) -> a__from(mark(z0))
mark(zWadr(z0, z1)) -> a__zWadr(mark(z0), mark(z1))
mark(prefix(z0)) -> a__prefix(mark(z0))
mark(nil) -> nil
mark(cons(z0)) -> cons(mark(z0))
mark(s(z0)) -> s(mark(z0))

Types:
A__APP :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> c:c1:c2
nil :: nil:cons:app:from:zWadr:prefix:s
c :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
MARK :: nil:cons:app:from:zWadr:prefix:s -> c12:c13:c14:c15:c16:c17:c18:c19:c20
cons :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c1 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
c2 :: c:c1:c2
A__FROM :: nil:cons:app:from:zWadr:prefix:s -> c3:c4
c3 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c3:c4
c4 :: c3:c4
A__ZWADR :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> c5:c6:c7:c8:c9
c5 :: c5:c6:c7:c8:c9
c6 :: c5:c6:c7:c8:c9
c7 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7:c8:c9
mark :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c8 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7:c8:c9
c9 :: c5:c6:c7:c8:c9
A__PREFIX :: nil:cons:app:from:zWadr:prefix:s -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
app :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c12 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c13 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
from :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c14 :: c3:c4 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
zWadr :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c15 :: c5:c6:c7:c8:c9 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c16 :: c5:c6:c7:c8:c9 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
prefix :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c17 :: c10:c11 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c18 :: c12:c13:c14:c15:c16:c17:c18:c19:c20
c19 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
s :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c20 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
a__app :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
hole_c:c1:c21_21 :: c:c1:c2
hole_nil:cons:app:from:zWadr:prefix:s2_21 :: nil:cons:app:from:zWadr:prefix:s
hole_c12:c13:c14:c15:c16:c17:c18:c19:c203_21 :: c12:c13:c14:c15:c16:c17:c18:c19:c20
hole_c3:c44_21 :: c3:c4
hole_c5:c6:c7:c8:c95_21 :: c5:c6:c7:c8:c9
hole_c10:c116_21 :: c10:c11
gen_nil:cons:app:from:zWadr:prefix:s7_21 :: Nat -> nil:cons:app:from:zWadr:prefix:s
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21 :: Nat -> c12:c13:c14:c15:c16:c17:c18:c19:c20


Generator Equations:
gen_nil:cons:app:from:zWadr:prefix:s7_21(0) <=> nil
gen_nil:cons:app:from:zWadr:prefix:s7_21(+(x, 1)) <=> cons(gen_nil:cons:app:from:zWadr:prefix:s7_21(x))
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(0) <=> c18
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(+(x, 1)) <=> c12(c(c18), gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_nil:cons:app:from:zWadr:prefix:s7_21(n36756_21)) -> gen_nil:cons:app:from:zWadr:prefix:s7_21(n36756_21), rt in Omega(0)

Induction Base:
mark(gen_nil:cons:app:from:zWadr:prefix:s7_21(0)) ->_R^Omega(0)
nil

Induction Step:
mark(gen_nil:cons:app:from:zWadr:prefix:s7_21(+(n36756_21, 1))) ->_R^Omega(0)
cons(mark(gen_nil:cons:app:from:zWadr:prefix:s7_21(n36756_21))) ->_IH
cons(gen_nil:cons:app:from:zWadr:prefix:s7_21(c36757_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__APP(nil, z0) -> c(MARK(z0))
A__APP(cons(z0), z2) -> c1(MARK(z0))
A__APP(z0, z1) -> c2
A__FROM(z0) -> c3(MARK(z0))
A__FROM(z0) -> c4
A__ZWADR(nil, z0) -> c5
A__ZWADR(z0, nil) -> c6
A__ZWADR(cons(z0), cons(z2)) -> c7(A__APP(mark(z2), cons(mark(z0))), MARK(z2))
A__ZWADR(cons(z0), cons(z2)) -> c8(A__APP(mark(z2), cons(mark(z0))), MARK(z0))
A__ZWADR(z0, z1) -> c9
A__PREFIX(z0) -> c10
A__PREFIX(z0) -> c11
MARK(app(z0, z1)) -> c12(A__APP(mark(z0), mark(z1)), MARK(z0))
MARK(app(z0, z1)) -> c13(A__APP(mark(z0), mark(z1)), MARK(z1))
MARK(from(z0)) -> c14(A__FROM(mark(z0)), MARK(z0))
MARK(zWadr(z0, z1)) -> c15(A__ZWADR(mark(z0), mark(z1)), MARK(z0))
MARK(zWadr(z0, z1)) -> c16(A__ZWADR(mark(z0), mark(z1)), MARK(z1))
MARK(prefix(z0)) -> c17(A__PREFIX(mark(z0)), MARK(z0))
MARK(nil) -> c18
MARK(cons(z0)) -> c19(MARK(z0))
MARK(s(z0)) -> c20(MARK(z0))
a__app(nil, z0) -> mark(z0)
a__app(cons(z0), z2) -> cons(mark(z0))
a__app(z0, z1) -> app(z0, z1)
a__from(z0) -> cons(mark(z0))
a__from(z0) -> from(z0)
a__zWadr(nil, z0) -> nil
a__zWadr(z0, nil) -> nil
a__zWadr(cons(z0), cons(z2)) -> cons(a__app(mark(z2), cons(mark(z0))))
a__zWadr(z0, z1) -> zWadr(z0, z1)
a__prefix(z0) -> cons(nil)
a__prefix(z0) -> prefix(z0)
mark(app(z0, z1)) -> a__app(mark(z0), mark(z1))
mark(from(z0)) -> a__from(mark(z0))
mark(zWadr(z0, z1)) -> a__zWadr(mark(z0), mark(z1))
mark(prefix(z0)) -> a__prefix(mark(z0))
mark(nil) -> nil
mark(cons(z0)) -> cons(mark(z0))
mark(s(z0)) -> s(mark(z0))

Types:
A__APP :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> c:c1:c2
nil :: nil:cons:app:from:zWadr:prefix:s
c :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
MARK :: nil:cons:app:from:zWadr:prefix:s -> c12:c13:c14:c15:c16:c17:c18:c19:c20
cons :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c1 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
c2 :: c:c1:c2
A__FROM :: nil:cons:app:from:zWadr:prefix:s -> c3:c4
c3 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c3:c4
c4 :: c3:c4
A__ZWADR :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> c5:c6:c7:c8:c9
c5 :: c5:c6:c7:c8:c9
c6 :: c5:c6:c7:c8:c9
c7 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7:c8:c9
mark :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c8 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7:c8:c9
c9 :: c5:c6:c7:c8:c9
A__PREFIX :: nil:cons:app:from:zWadr:prefix:s -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
app :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c12 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c13 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
from :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c14 :: c3:c4 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
zWadr :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c15 :: c5:c6:c7:c8:c9 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c16 :: c5:c6:c7:c8:c9 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
prefix :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c17 :: c10:c11 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c18 :: c12:c13:c14:c15:c16:c17:c18:c19:c20
c19 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
s :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c20 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
a__app :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
hole_c:c1:c21_21 :: c:c1:c2
hole_nil:cons:app:from:zWadr:prefix:s2_21 :: nil:cons:app:from:zWadr:prefix:s
hole_c12:c13:c14:c15:c16:c17:c18:c19:c203_21 :: c12:c13:c14:c15:c16:c17:c18:c19:c20
hole_c3:c44_21 :: c3:c4
hole_c5:c6:c7:c8:c95_21 :: c5:c6:c7:c8:c9
hole_c10:c116_21 :: c10:c11
gen_nil:cons:app:from:zWadr:prefix:s7_21 :: Nat -> nil:cons:app:from:zWadr:prefix:s
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21 :: Nat -> c12:c13:c14:c15:c16:c17:c18:c19:c20


Lemmas:
mark(gen_nil:cons:app:from:zWadr:prefix:s7_21(n36756_21)) -> gen_nil:cons:app:from:zWadr:prefix:s7_21(n36756_21), rt in Omega(0)


Generator Equations:
gen_nil:cons:app:from:zWadr:prefix:s7_21(0) <=> nil
gen_nil:cons:app:from:zWadr:prefix:s7_21(+(x, 1)) <=> cons(gen_nil:cons:app:from:zWadr:prefix:s7_21(x))
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(0) <=> c18
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(+(x, 1)) <=> c12(c(c18), gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(x))


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MARK(gen_nil:cons:app:from:zWadr:prefix:s7_21(+(1, n40049_21))) -> *9_21, rt in Omega(n40049_21)

Induction Base:
MARK(gen_nil:cons:app:from:zWadr:prefix:s7_21(+(1, 0)))

Induction Step:
MARK(gen_nil:cons:app:from:zWadr:prefix:s7_21(+(1, +(n40049_21, 1)))) ->_R^Omega(1)
c19(MARK(gen_nil:cons:app:from:zWadr:prefix:s7_21(+(1, n40049_21)))) ->_IH
c19(*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__APP(nil, z0) -> c(MARK(z0))
A__APP(cons(z0), z2) -> c1(MARK(z0))
A__APP(z0, z1) -> c2
A__FROM(z0) -> c3(MARK(z0))
A__FROM(z0) -> c4
A__ZWADR(nil, z0) -> c5
A__ZWADR(z0, nil) -> c6
A__ZWADR(cons(z0), cons(z2)) -> c7(A__APP(mark(z2), cons(mark(z0))), MARK(z2))
A__ZWADR(cons(z0), cons(z2)) -> c8(A__APP(mark(z2), cons(mark(z0))), MARK(z0))
A__ZWADR(z0, z1) -> c9
A__PREFIX(z0) -> c10
A__PREFIX(z0) -> c11
MARK(app(z0, z1)) -> c12(A__APP(mark(z0), mark(z1)), MARK(z0))
MARK(app(z0, z1)) -> c13(A__APP(mark(z0), mark(z1)), MARK(z1))
MARK(from(z0)) -> c14(A__FROM(mark(z0)), MARK(z0))
MARK(zWadr(z0, z1)) -> c15(A__ZWADR(mark(z0), mark(z1)), MARK(z0))
MARK(zWadr(z0, z1)) -> c16(A__ZWADR(mark(z0), mark(z1)), MARK(z1))
MARK(prefix(z0)) -> c17(A__PREFIX(mark(z0)), MARK(z0))
MARK(nil) -> c18
MARK(cons(z0)) -> c19(MARK(z0))
MARK(s(z0)) -> c20(MARK(z0))
a__app(nil, z0) -> mark(z0)
a__app(cons(z0), z2) -> cons(mark(z0))
a__app(z0, z1) -> app(z0, z1)
a__from(z0) -> cons(mark(z0))
a__from(z0) -> from(z0)
a__zWadr(nil, z0) -> nil
a__zWadr(z0, nil) -> nil
a__zWadr(cons(z0), cons(z2)) -> cons(a__app(mark(z2), cons(mark(z0))))
a__zWadr(z0, z1) -> zWadr(z0, z1)
a__prefix(z0) -> cons(nil)
a__prefix(z0) -> prefix(z0)
mark(app(z0, z1)) -> a__app(mark(z0), mark(z1))
mark(from(z0)) -> a__from(mark(z0))
mark(zWadr(z0, z1)) -> a__zWadr(mark(z0), mark(z1))
mark(prefix(z0)) -> a__prefix(mark(z0))
mark(nil) -> nil
mark(cons(z0)) -> cons(mark(z0))
mark(s(z0)) -> s(mark(z0))

Types:
A__APP :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> c:c1:c2
nil :: nil:cons:app:from:zWadr:prefix:s
c :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
MARK :: nil:cons:app:from:zWadr:prefix:s -> c12:c13:c14:c15:c16:c17:c18:c19:c20
cons :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c1 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
c2 :: c:c1:c2
A__FROM :: nil:cons:app:from:zWadr:prefix:s -> c3:c4
c3 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c3:c4
c4 :: c3:c4
A__ZWADR :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> c5:c6:c7:c8:c9
c5 :: c5:c6:c7:c8:c9
c6 :: c5:c6:c7:c8:c9
c7 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7:c8:c9
mark :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c8 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7:c8:c9
c9 :: c5:c6:c7:c8:c9
A__PREFIX :: nil:cons:app:from:zWadr:prefix:s -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
app :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c12 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c13 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
from :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c14 :: c3:c4 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
zWadr :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c15 :: c5:c6:c7:c8:c9 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c16 :: c5:c6:c7:c8:c9 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
prefix :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c17 :: c10:c11 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c18 :: c12:c13:c14:c15:c16:c17:c18:c19:c20
c19 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
s :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c20 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
a__app :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
hole_c:c1:c21_21 :: c:c1:c2
hole_nil:cons:app:from:zWadr:prefix:s2_21 :: nil:cons:app:from:zWadr:prefix:s
hole_c12:c13:c14:c15:c16:c17:c18:c19:c203_21 :: c12:c13:c14:c15:c16:c17:c18:c19:c20
hole_c3:c44_21 :: c3:c4
hole_c5:c6:c7:c8:c95_21 :: c5:c6:c7:c8:c9
hole_c10:c116_21 :: c10:c11
gen_nil:cons:app:from:zWadr:prefix:s7_21 :: Nat -> nil:cons:app:from:zWadr:prefix:s
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21 :: Nat -> c12:c13:c14:c15:c16:c17:c18:c19:c20


Lemmas:
mark(gen_nil:cons:app:from:zWadr:prefix:s7_21(n36756_21)) -> gen_nil:cons:app:from:zWadr:prefix:s7_21(n36756_21), rt in Omega(0)


Generator Equations:
gen_nil:cons:app:from:zWadr:prefix:s7_21(0) <=> nil
gen_nil:cons:app:from:zWadr:prefix:s7_21(+(x, 1)) <=> cons(gen_nil:cons:app:from:zWadr:prefix:s7_21(x))
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(0) <=> c18
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(+(x, 1)) <=> c12(c(c18), gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(x))


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

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

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
Innermost TRS:
Rules:
A__APP(nil, z0) -> c(MARK(z0))
A__APP(cons(z0), z2) -> c1(MARK(z0))
A__APP(z0, z1) -> c2
A__FROM(z0) -> c3(MARK(z0))
A__FROM(z0) -> c4
A__ZWADR(nil, z0) -> c5
A__ZWADR(z0, nil) -> c6
A__ZWADR(cons(z0), cons(z2)) -> c7(A__APP(mark(z2), cons(mark(z0))), MARK(z2))
A__ZWADR(cons(z0), cons(z2)) -> c8(A__APP(mark(z2), cons(mark(z0))), MARK(z0))
A__ZWADR(z0, z1) -> c9
A__PREFIX(z0) -> c10
A__PREFIX(z0) -> c11
MARK(app(z0, z1)) -> c12(A__APP(mark(z0), mark(z1)), MARK(z0))
MARK(app(z0, z1)) -> c13(A__APP(mark(z0), mark(z1)), MARK(z1))
MARK(from(z0)) -> c14(A__FROM(mark(z0)), MARK(z0))
MARK(zWadr(z0, z1)) -> c15(A__ZWADR(mark(z0), mark(z1)), MARK(z0))
MARK(zWadr(z0, z1)) -> c16(A__ZWADR(mark(z0), mark(z1)), MARK(z1))
MARK(prefix(z0)) -> c17(A__PREFIX(mark(z0)), MARK(z0))
MARK(nil) -> c18
MARK(cons(z0)) -> c19(MARK(z0))
MARK(s(z0)) -> c20(MARK(z0))
a__app(nil, z0) -> mark(z0)
a__app(cons(z0), z2) -> cons(mark(z0))
a__app(z0, z1) -> app(z0, z1)
a__from(z0) -> cons(mark(z0))
a__from(z0) -> from(z0)
a__zWadr(nil, z0) -> nil
a__zWadr(z0, nil) -> nil
a__zWadr(cons(z0), cons(z2)) -> cons(a__app(mark(z2), cons(mark(z0))))
a__zWadr(z0, z1) -> zWadr(z0, z1)
a__prefix(z0) -> cons(nil)
a__prefix(z0) -> prefix(z0)
mark(app(z0, z1)) -> a__app(mark(z0), mark(z1))
mark(from(z0)) -> a__from(mark(z0))
mark(zWadr(z0, z1)) -> a__zWadr(mark(z0), mark(z1))
mark(prefix(z0)) -> a__prefix(mark(z0))
mark(nil) -> nil
mark(cons(z0)) -> cons(mark(z0))
mark(s(z0)) -> s(mark(z0))

Types:
A__APP :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> c:c1:c2
nil :: nil:cons:app:from:zWadr:prefix:s
c :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
MARK :: nil:cons:app:from:zWadr:prefix:s -> c12:c13:c14:c15:c16:c17:c18:c19:c20
cons :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c1 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c:c1:c2
c2 :: c:c1:c2
A__FROM :: nil:cons:app:from:zWadr:prefix:s -> c3:c4
c3 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c3:c4
c4 :: c3:c4
A__ZWADR :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> c5:c6:c7:c8:c9
c5 :: c5:c6:c7:c8:c9
c6 :: c5:c6:c7:c8:c9
c7 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7:c8:c9
mark :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c8 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c5:c6:c7:c8:c9
c9 :: c5:c6:c7:c8:c9
A__PREFIX :: nil:cons:app:from:zWadr:prefix:s -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
app :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c12 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c13 :: c:c1:c2 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
from :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c14 :: c3:c4 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
zWadr :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c15 :: c5:c6:c7:c8:c9 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c16 :: c5:c6:c7:c8:c9 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
prefix :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c17 :: c10:c11 -> c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
c18 :: c12:c13:c14:c15:c16:c17:c18:c19:c20
c19 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
s :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
c20 :: c12:c13:c14:c15:c16:c17:c18:c19:c20 -> c12:c13:c14:c15:c16:c17:c18:c19:c20
a__app :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s -> nil:cons:app:from:zWadr:prefix:s
hole_c:c1:c21_21 :: c:c1:c2
hole_nil:cons:app:from:zWadr:prefix:s2_21 :: nil:cons:app:from:zWadr:prefix:s
hole_c12:c13:c14:c15:c16:c17:c18:c19:c203_21 :: c12:c13:c14:c15:c16:c17:c18:c19:c20
hole_c3:c44_21 :: c3:c4
hole_c5:c6:c7:c8:c95_21 :: c5:c6:c7:c8:c9
hole_c10:c116_21 :: c10:c11
gen_nil:cons:app:from:zWadr:prefix:s7_21 :: Nat -> nil:cons:app:from:zWadr:prefix:s
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21 :: Nat -> c12:c13:c14:c15:c16:c17:c18:c19:c20


Lemmas:
mark(gen_nil:cons:app:from:zWadr:prefix:s7_21(n36756_21)) -> gen_nil:cons:app:from:zWadr:prefix:s7_21(n36756_21), rt in Omega(0)
MARK(gen_nil:cons:app:from:zWadr:prefix:s7_21(+(1, n40049_21))) -> *9_21, rt in Omega(n40049_21)


Generator Equations:
gen_nil:cons:app:from:zWadr:prefix:s7_21(0) <=> nil
gen_nil:cons:app:from:zWadr:prefix:s7_21(+(x, 1)) <=> cons(gen_nil:cons:app:from:zWadr:prefix:s7_21(x))
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(0) <=> c18
gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(+(x, 1)) <=> c12(c(c18), gen_c12:c13:c14:c15:c16:c17:c18:c19:c208_21(x))


The following defined symbols remain to be analysed:
A__APP, A__FROM

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