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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 431 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), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 300 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), 9456 ms]
        (18) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(inf(x_1)) -> inf(encArg(x_1))
   encArg(nil) -> nil
   encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(length(x_1)) -> length(encArg(x_1))
   encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
   encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
   encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_a__inf(x_1) -> a__inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_a__length(x_1) -> a__length(encArg(x_1))
   encode_length(x_1) -> length(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))


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

(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__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(inf(x_1)) -> inf(encArg(x_1))
   encArg(nil) -> nil
   encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(length(x_1)) -> length(encArg(x_1))
   encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
   encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
   encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_a__inf(x_1) -> a__inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_a__length(x_1) -> a__length(encArg(x_1))
   encode_length(x_1) -> length(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))

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

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of 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:

   a__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(inf(x_1)) -> inf(encArg(x_1))
   encArg(nil) -> nil
   encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(length(x_1)) -> length(encArg(x_1))
   encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
   encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
   encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_a__inf(x_1) -> a__inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_a__length(x_1) -> a__length(encArg(x_1))
   encode_length(x_1) -> length(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))

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:

   a__eq(0', 0') -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0', X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0'
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0') -> 0'
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

   encArg(0') -> 0'
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(inf(x_1)) -> inf(encArg(x_1))
   encArg(nil) -> nil
   encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(length(x_1)) -> length(encArg(x_1))
   encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
   encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
   encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_a__inf(x_1) -> a__inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_a__length(x_1) -> a__length(encArg(x_1))
   encode_length(x_1) -> length(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
a__eq(0', 0') -> true
a__eq(s(X), s(Y)) -> a__eq(X, Y)
a__eq(X, Y) -> false
a__inf(X) -> cons(X, inf(s(X)))
a__take(0', X) -> nil
a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
a__length(nil) -> 0'
a__length(cons(X, L)) -> s(length(L))
mark(eq(X1, X2)) -> a__eq(X1, X2)
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(length(X)) -> a__length(mark(X))
mark(0') -> 0'
mark(true) -> true
mark(s(X)) -> s(X)
mark(false) -> false
mark(cons(X1, X2)) -> cons(X1, X2)
mark(nil) -> nil
a__eq(X1, X2) -> eq(X1, X2)
a__inf(X) -> inf(X)
a__take(X1, X2) -> take(X1, X2)
a__length(X) -> length(X)
encArg(0') -> 0'
encArg(true) -> true
encArg(s(x_1)) -> s(encArg(x_1))
encArg(false) -> false
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(inf(x_1)) -> inf(encArg(x_1))
encArg(nil) -> nil
encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
encArg(length(x_1)) -> length(encArg(x_1))
encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_a__inf(x_1) -> a__inf(encArg(x_1))
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
encode_inf(x_1) -> inf(encArg(x_1))
encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
encode_a__length(x_1) -> a__length(encArg(x_1))
encode_length(x_1) -> length(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
0' :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
true :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
s :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
false :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
nil :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encArg :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_0 :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_true :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_s :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_false :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_cons :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_nil :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
hole_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark1_3 :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3 :: Nat -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__eq, mark, encArg

They will be analysed ascendingly in the following order:
a__eq < mark
a__eq < encArg
mark < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
a__eq(0', 0') -> true
a__eq(s(X), s(Y)) -> a__eq(X, Y)
a__eq(X, Y) -> false
a__inf(X) -> cons(X, inf(s(X)))
a__take(0', X) -> nil
a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
a__length(nil) -> 0'
a__length(cons(X, L)) -> s(length(L))
mark(eq(X1, X2)) -> a__eq(X1, X2)
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(length(X)) -> a__length(mark(X))
mark(0') -> 0'
mark(true) -> true
mark(s(X)) -> s(X)
mark(false) -> false
mark(cons(X1, X2)) -> cons(X1, X2)
mark(nil) -> nil
a__eq(X1, X2) -> eq(X1, X2)
a__inf(X) -> inf(X)
a__take(X1, X2) -> take(X1, X2)
a__length(X) -> length(X)
encArg(0') -> 0'
encArg(true) -> true
encArg(s(x_1)) -> s(encArg(x_1))
encArg(false) -> false
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(inf(x_1)) -> inf(encArg(x_1))
encArg(nil) -> nil
encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
encArg(length(x_1)) -> length(encArg(x_1))
encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_a__inf(x_1) -> a__inf(encArg(x_1))
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
encode_inf(x_1) -> inf(encArg(x_1))
encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
encode_a__length(x_1) -> a__length(encArg(x_1))
encode_length(x_1) -> length(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
0' :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
true :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
s :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
false :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
nil :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encArg :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_0 :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_true :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_s :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_false :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_cons :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_nil :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
hole_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark1_3 :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3 :: Nat -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark


Generator Equations:
gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(0) <=> 0'
gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(+(x, 1)) <=> s(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(x))


The following defined symbols remain to be analysed:
a__eq, mark, encArg

They will be analysed ascendingly in the following order:
a__eq < mark
a__eq < encArg
mark < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(n4_3), gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(n4_3)) -> true, rt in Omega(1 + n4_3)

Induction Base:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(0), gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(0)) ->_R^Omega(1)
true

Induction Step:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(+(n4_3, 1)), gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(+(n4_3, 1))) ->_R^Omega(1)
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(n4_3), gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(n4_3)) ->_IH
true

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:
a__eq(0', 0') -> true
a__eq(s(X), s(Y)) -> a__eq(X, Y)
a__eq(X, Y) -> false
a__inf(X) -> cons(X, inf(s(X)))
a__take(0', X) -> nil
a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
a__length(nil) -> 0'
a__length(cons(X, L)) -> s(length(L))
mark(eq(X1, X2)) -> a__eq(X1, X2)
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(length(X)) -> a__length(mark(X))
mark(0') -> 0'
mark(true) -> true
mark(s(X)) -> s(X)
mark(false) -> false
mark(cons(X1, X2)) -> cons(X1, X2)
mark(nil) -> nil
a__eq(X1, X2) -> eq(X1, X2)
a__inf(X) -> inf(X)
a__take(X1, X2) -> take(X1, X2)
a__length(X) -> length(X)
encArg(0') -> 0'
encArg(true) -> true
encArg(s(x_1)) -> s(encArg(x_1))
encArg(false) -> false
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(inf(x_1)) -> inf(encArg(x_1))
encArg(nil) -> nil
encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
encArg(length(x_1)) -> length(encArg(x_1))
encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_a__inf(x_1) -> a__inf(encArg(x_1))
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
encode_inf(x_1) -> inf(encArg(x_1))
encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
encode_a__length(x_1) -> a__length(encArg(x_1))
encode_length(x_1) -> length(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
0' :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
true :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
s :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
false :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
nil :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encArg :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_0 :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_true :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_s :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_false :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_cons :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_nil :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
hole_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark1_3 :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3 :: Nat -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark


Generator Equations:
gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(0) <=> 0'
gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(+(x, 1)) <=> s(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(x))


The following defined symbols remain to be analysed:
a__eq, mark, encArg

They will be analysed ascendingly in the following order:
a__eq < mark
a__eq < encArg
mark < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
a__eq(0', 0') -> true
a__eq(s(X), s(Y)) -> a__eq(X, Y)
a__eq(X, Y) -> false
a__inf(X) -> cons(X, inf(s(X)))
a__take(0', X) -> nil
a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
a__length(nil) -> 0'
a__length(cons(X, L)) -> s(length(L))
mark(eq(X1, X2)) -> a__eq(X1, X2)
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(length(X)) -> a__length(mark(X))
mark(0') -> 0'
mark(true) -> true
mark(s(X)) -> s(X)
mark(false) -> false
mark(cons(X1, X2)) -> cons(X1, X2)
mark(nil) -> nil
a__eq(X1, X2) -> eq(X1, X2)
a__inf(X) -> inf(X)
a__take(X1, X2) -> take(X1, X2)
a__length(X) -> length(X)
encArg(0') -> 0'
encArg(true) -> true
encArg(s(x_1)) -> s(encArg(x_1))
encArg(false) -> false
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(inf(x_1)) -> inf(encArg(x_1))
encArg(nil) -> nil
encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
encArg(length(x_1)) -> length(encArg(x_1))
encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_a__inf(x_1) -> a__inf(encArg(x_1))
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
encode_inf(x_1) -> inf(encArg(x_1))
encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
encode_a__length(x_1) -> a__length(encArg(x_1))
encode_length(x_1) -> length(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
0' :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
true :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
s :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
false :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
nil :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encArg :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
cons_mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_0 :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_true :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_s :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_false :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_cons :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_inf :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_nil :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_take :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_a__length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_length :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_mark :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
encode_eq :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
hole_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark1_3 :: 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark
gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3 :: Nat -> 0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark


Lemmas:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(n4_3), gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(n4_3)) -> true, rt in Omega(1 + n4_3)


Generator Equations:
gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(0) <=> 0'
gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(+(x, 1)) <=> s(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(x))


The following defined symbols remain to be analysed:
mark, encArg

They will be analysed ascendingly in the following order:
mark < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(n3763_3)) -> gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(n3763_3), rt in Omega(0)

Induction Base:
encArg(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(+(n3763_3, 1))) ->_R^Omega(0)
s(encArg(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(n3763_3))) ->_IH
s(gen_0':true:s:false:inf:cons:nil:take:length:eq:cons_a__eq:cons_a__inf:cons_a__take:cons_a__length:cons_mark2_3(c3764_3))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(18)
BOUNDS(1, INF)