/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 303 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), 471 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), 187 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 132 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 9293 ms]
        (22) 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:

   active(eq(0, 0)) -> mark(true)
   active(eq(s(X), s(Y))) -> mark(eq(X, Y))
   active(eq(X, Y)) -> mark(false)
   active(inf(X)) -> mark(cons(X, inf(s(X))))
   active(take(0, X)) -> mark(nil)
   active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
   active(length(nil)) -> mark(0)
   active(length(cons(X, L))) -> mark(s(length(L)))
   active(inf(X)) -> inf(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(length(X)) -> length(active(X))
   inf(mark(X)) -> mark(inf(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   length(mark(X)) -> mark(length(X))
   proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(s(X)) -> s(proper(X))
   proper(false) -> ok(false)
   proper(inf(X)) -> inf(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(length(X)) -> length(proper(X))
   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
   s(ok(X)) -> ok(s(X))
   inf(ok(X)) -> ok(inf(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   length(ok(X)) -> ok(length(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(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(mark(x_1)) -> mark(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_inf(x_1)) -> inf(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_length(x_1) -> length(encArg(x_1))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))


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

(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:

   active(eq(0, 0)) -> mark(true)
   active(eq(s(X), s(Y))) -> mark(eq(X, Y))
   active(eq(X, Y)) -> mark(false)
   active(inf(X)) -> mark(cons(X, inf(s(X))))
   active(take(0, X)) -> mark(nil)
   active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
   active(length(nil)) -> mark(0)
   active(length(cons(X, L))) -> mark(s(length(L)))
   active(inf(X)) -> inf(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(length(X)) -> length(active(X))
   inf(mark(X)) -> mark(inf(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   length(mark(X)) -> mark(length(X))
   proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(s(X)) -> s(proper(X))
   proper(false) -> ok(false)
   proper(inf(X)) -> inf(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(length(X)) -> length(proper(X))
   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
   s(ok(X)) -> ok(s(X))
   inf(ok(X)) -> ok(inf(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   length(ok(X)) -> ok(length(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(0) -> 0
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_inf(x_1)) -> inf(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_length(x_1) -> length(encArg(x_1))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

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:

   active(eq(0, 0)) -> mark(true)
   active(eq(s(X), s(Y))) -> mark(eq(X, Y))
   active(eq(X, Y)) -> mark(false)
   active(inf(X)) -> mark(cons(X, inf(s(X))))
   active(take(0, X)) -> mark(nil)
   active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
   active(length(nil)) -> mark(0)
   active(length(cons(X, L))) -> mark(s(length(L)))
   active(inf(X)) -> inf(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(length(X)) -> length(active(X))
   inf(mark(X)) -> mark(inf(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   length(mark(X)) -> mark(length(X))
   proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(s(X)) -> s(proper(X))
   proper(false) -> ok(false)
   proper(inf(X)) -> inf(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(length(X)) -> length(proper(X))
   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
   s(ok(X)) -> ok(s(X))
   inf(ok(X)) -> ok(inf(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   length(ok(X)) -> ok(length(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(0) -> 0
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_inf(x_1)) -> inf(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_length(x_1) -> length(encArg(x_1))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

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:

   active(eq(0', 0')) -> mark(true)
   active(eq(s(X), s(Y))) -> mark(eq(X, Y))
   active(eq(X, Y)) -> mark(false)
   active(inf(X)) -> mark(cons(X, inf(s(X))))
   active(take(0', X)) -> mark(nil)
   active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
   active(length(nil)) -> mark(0')
   active(length(cons(X, L))) -> mark(s(length(L)))
   active(inf(X)) -> inf(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(length(X)) -> length(active(X))
   inf(mark(X)) -> mark(inf(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   length(mark(X)) -> mark(length(X))
   proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
   proper(0') -> ok(0')
   proper(true) -> ok(true)
   proper(s(X)) -> s(proper(X))
   proper(false) -> ok(false)
   proper(inf(X)) -> inf(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(length(X)) -> length(proper(X))
   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
   s(ok(X)) -> ok(s(X))
   inf(ok(X)) -> ok(inf(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   length(ok(X)) -> ok(length(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(0') -> 0'
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_inf(x_1)) -> inf(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_length(x_1) -> length(encArg(x_1))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
active(eq(0', 0')) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0', X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0')
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
length(mark(X)) -> mark(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
inf(ok(X)) -> ok(inf(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
encArg(0') -> 0'
encArg(mark(x_1)) -> mark(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(nil) -> nil
encArg(ok(x_1)) -> ok(encArg(x_1))
encArg(cons_active(x_1)) -> active(encArg(x_1))
encArg(cons_inf(x_1)) -> inf(encArg(x_1))
encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
encArg(cons_length(x_1)) -> length(encArg(x_1))
encArg(cons_proper(x_1)) -> proper(encArg(x_1))
encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_top(x_1)) -> top(encArg(x_1))
encode_active(x_1) -> active(encArg(x_1))
encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_inf(x_1) -> inf(encArg(x_1))
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_length(x_1) -> length(encArg(x_1))
encode_proper(x_1) -> proper(encArg(x_1))
encode_ok(x_1) -> ok(encArg(x_1))
encode_top(x_1) -> top(encArg(x_1))

Types:
active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
0' :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encArg :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_0 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
hole_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top1_3 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3 :: Nat -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, eq, cons, inf, s, take, length, proper, top, encArg

They will be analysed ascendingly in the following order:
eq < active
cons < active
inf < active
s < active
take < active
length < active
active < top
active < encArg
eq < proper
eq < encArg
cons < proper
cons < encArg
inf < proper
inf < encArg
s < proper
s < encArg
take < proper
take < encArg
length < proper
length < encArg
proper < top
proper < encArg
top < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
active(eq(0', 0')) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0', X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0')
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
length(mark(X)) -> mark(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
inf(ok(X)) -> ok(inf(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
encArg(0') -> 0'
encArg(mark(x_1)) -> mark(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(nil) -> nil
encArg(ok(x_1)) -> ok(encArg(x_1))
encArg(cons_active(x_1)) -> active(encArg(x_1))
encArg(cons_inf(x_1)) -> inf(encArg(x_1))
encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
encArg(cons_length(x_1)) -> length(encArg(x_1))
encArg(cons_proper(x_1)) -> proper(encArg(x_1))
encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_top(x_1)) -> top(encArg(x_1))
encode_active(x_1) -> active(encArg(x_1))
encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_inf(x_1) -> inf(encArg(x_1))
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_length(x_1) -> length(encArg(x_1))
encode_proper(x_1) -> proper(encArg(x_1))
encode_ok(x_1) -> ok(encArg(x_1))
encode_top(x_1) -> top(encArg(x_1))

Types:
active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
0' :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encArg :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_0 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
hole_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top1_3 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3 :: Nat -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top


Generator Equations:
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(0) <=> 0'
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(x))


The following defined symbols remain to be analysed:
eq, active, cons, inf, s, take, length, proper, top, encArg

They will be analysed ascendingly in the following order:
eq < active
cons < active
inf < active
s < active
take < active
length < active
active < top
active < encArg
eq < proper
eq < encArg
cons < proper
cons < encArg
inf < proper
inf < encArg
s < proper
s < encArg
take < proper
take < encArg
length < proper
length < encArg
proper < top
proper < encArg
top < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inf(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n14_3))) -> *3_3, rt in Omega(n14_3)

Induction Base:
inf(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, 0)))

Induction Step:
inf(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, +(n14_3, 1)))) ->_R^Omega(1)
mark(inf(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n14_3)))) ->_IH
mark(*3_3)

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:
active(eq(0', 0')) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0', X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0')
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
length(mark(X)) -> mark(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
inf(ok(X)) -> ok(inf(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
encArg(0') -> 0'
encArg(mark(x_1)) -> mark(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(nil) -> nil
encArg(ok(x_1)) -> ok(encArg(x_1))
encArg(cons_active(x_1)) -> active(encArg(x_1))
encArg(cons_inf(x_1)) -> inf(encArg(x_1))
encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
encArg(cons_length(x_1)) -> length(encArg(x_1))
encArg(cons_proper(x_1)) -> proper(encArg(x_1))
encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_top(x_1)) -> top(encArg(x_1))
encode_active(x_1) -> active(encArg(x_1))
encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_inf(x_1) -> inf(encArg(x_1))
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_length(x_1) -> length(encArg(x_1))
encode_proper(x_1) -> proper(encArg(x_1))
encode_ok(x_1) -> ok(encArg(x_1))
encode_top(x_1) -> top(encArg(x_1))

Types:
active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
0' :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encArg :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_0 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
hole_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top1_3 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3 :: Nat -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top


Generator Equations:
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(0) <=> 0'
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(x))


The following defined symbols remain to be analysed:
inf, active, s, take, length, proper, top, encArg

They will be analysed ascendingly in the following order:
inf < active
s < active
take < active
length < active
active < top
active < encArg
inf < proper
inf < encArg
s < proper
s < encArg
take < proper
take < encArg
length < proper
length < encArg
proper < top
proper < encArg
top < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
active(eq(0', 0')) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0', X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0')
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
length(mark(X)) -> mark(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
inf(ok(X)) -> ok(inf(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
encArg(0') -> 0'
encArg(mark(x_1)) -> mark(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(nil) -> nil
encArg(ok(x_1)) -> ok(encArg(x_1))
encArg(cons_active(x_1)) -> active(encArg(x_1))
encArg(cons_inf(x_1)) -> inf(encArg(x_1))
encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
encArg(cons_length(x_1)) -> length(encArg(x_1))
encArg(cons_proper(x_1)) -> proper(encArg(x_1))
encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_top(x_1)) -> top(encArg(x_1))
encode_active(x_1) -> active(encArg(x_1))
encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_inf(x_1) -> inf(encArg(x_1))
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_length(x_1) -> length(encArg(x_1))
encode_proper(x_1) -> proper(encArg(x_1))
encode_ok(x_1) -> ok(encArg(x_1))
encode_top(x_1) -> top(encArg(x_1))

Types:
active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
0' :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encArg :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_0 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
hole_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top1_3 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3 :: Nat -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top


Lemmas:
inf(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n14_3))) -> *3_3, rt in Omega(n14_3)


Generator Equations:
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(0) <=> 0'
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(x))


The following defined symbols remain to be analysed:
s, active, take, length, proper, top, encArg

They will be analysed ascendingly in the following order:
s < active
take < active
length < active
active < top
active < encArg
s < proper
s < encArg
take < proper
take < encArg
length < proper
length < encArg
proper < top
proper < encArg
top < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
take(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n674_3)), gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(b)) -> *3_3, rt in Omega(n674_3)

Induction Base:
take(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, 0)), gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(b))

Induction Step:
take(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, +(n674_3, 1))), gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(b)) ->_R^Omega(1)
mark(take(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n674_3)), gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(b))) ->_IH
mark(*3_3)

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

(18)
Obligation:
Innermost TRS:
Rules:
active(eq(0', 0')) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0', X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0')
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
length(mark(X)) -> mark(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
inf(ok(X)) -> ok(inf(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
encArg(0') -> 0'
encArg(mark(x_1)) -> mark(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(nil) -> nil
encArg(ok(x_1)) -> ok(encArg(x_1))
encArg(cons_active(x_1)) -> active(encArg(x_1))
encArg(cons_inf(x_1)) -> inf(encArg(x_1))
encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
encArg(cons_length(x_1)) -> length(encArg(x_1))
encArg(cons_proper(x_1)) -> proper(encArg(x_1))
encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_top(x_1)) -> top(encArg(x_1))
encode_active(x_1) -> active(encArg(x_1))
encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_inf(x_1) -> inf(encArg(x_1))
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_length(x_1) -> length(encArg(x_1))
encode_proper(x_1) -> proper(encArg(x_1))
encode_ok(x_1) -> ok(encArg(x_1))
encode_top(x_1) -> top(encArg(x_1))

Types:
active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
0' :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encArg :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_0 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
hole_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top1_3 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3 :: Nat -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top


Lemmas:
inf(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n14_3))) -> *3_3, rt in Omega(n14_3)
take(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n674_3)), gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(b)) -> *3_3, rt in Omega(n674_3)


Generator Equations:
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(0) <=> 0'
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(x))


The following defined symbols remain to be analysed:
length, active, proper, top, encArg

They will be analysed ascendingly in the following order:
length < active
active < top
active < encArg
length < proper
length < encArg
proper < top
proper < encArg
top < encArg

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n2666_3))) -> *3_3, rt in Omega(n2666_3)

Induction Base:
length(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, 0)))

Induction Step:
length(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, +(n2666_3, 1)))) ->_R^Omega(1)
mark(length(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n2666_3)))) ->_IH
mark(*3_3)

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

(20)
Obligation:
Innermost TRS:
Rules:
active(eq(0', 0')) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0', X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0')
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
length(mark(X)) -> mark(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
inf(ok(X)) -> ok(inf(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
encArg(0') -> 0'
encArg(mark(x_1)) -> mark(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(nil) -> nil
encArg(ok(x_1)) -> ok(encArg(x_1))
encArg(cons_active(x_1)) -> active(encArg(x_1))
encArg(cons_inf(x_1)) -> inf(encArg(x_1))
encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
encArg(cons_length(x_1)) -> length(encArg(x_1))
encArg(cons_proper(x_1)) -> proper(encArg(x_1))
encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_top(x_1)) -> top(encArg(x_1))
encode_active(x_1) -> active(encArg(x_1))
encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_mark(x_1) -> mark(encArg(x_1))
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_inf(x_1) -> inf(encArg(x_1))
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_length(x_1) -> length(encArg(x_1))
encode_proper(x_1) -> proper(encArg(x_1))
encode_ok(x_1) -> ok(encArg(x_1))
encode_top(x_1) -> top(encArg(x_1))

Types:
active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
0' :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encArg :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
cons_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_active :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_eq :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_0 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_mark :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_true :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_s :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_false :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_inf :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_cons :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_take :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_nil :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_length :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_proper :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_ok :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
encode_top :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
hole_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top1_3 :: 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3 :: Nat -> 0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top


Lemmas:
inf(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n14_3))) -> *3_3, rt in Omega(n14_3)
take(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n674_3)), gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(b)) -> *3_3, rt in Omega(n674_3)
length(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(1, n2666_3))) -> *3_3, rt in Omega(n2666_3)


Generator Equations:
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(0) <=> 0'
gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(x))


The following defined symbols remain to be analysed:
active, proper, top, encArg

They will be analysed ascendingly in the following order:
active < top
active < encArg
proper < top
proper < encArg
top < encArg

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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(n20775_3)) -> gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(n20775_3), rt in Omega(0)

Induction Base:
encArg(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(+(n20775_3, 1))) ->_R^Omega(0)
mark(encArg(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(n20775_3))) ->_IH
mark(gen_0':true:mark:false:nil:ok:cons_active:cons_inf:cons_take:cons_length:cons_proper:cons_eq:cons_s:cons_cons:cons_top2_3(c20776_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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(22)
BOUNDS(1, INF)