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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The 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), 337 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 6 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


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

(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(app(nil, YS)) -> mark(YS)
   active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(zWadr(nil, YS)) -> mark(nil)
   active(zWadr(XS, nil)) -> mark(nil)
   active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
   active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
   active(app(X1, X2)) -> app(active(X1), X2)
   active(app(X1, X2)) -> app(X1, active(X2))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
   active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
   active(prefix(X)) -> prefix(active(X))
   app(mark(X1), X2) -> mark(app(X1, X2))
   app(X1, mark(X2)) -> mark(app(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
   zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
   prefix(mark(X)) -> mark(prefix(X))
   proper(app(X1, X2)) -> app(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
   proper(prefix(X)) -> prefix(proper(X))
   app(ok(X1), ok(X2)) -> ok(app(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
   prefix(ok(X)) -> ok(prefix(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(nil) -> nil
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2))
   encArg(cons_prefix(x_1)) -> prefix(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2))
   encode_prefix(x_1) -> prefix(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(app(nil, YS)) -> mark(YS)
   active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(zWadr(nil, YS)) -> mark(nil)
   active(zWadr(XS, nil)) -> mark(nil)
   active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
   active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
   active(app(X1, X2)) -> app(active(X1), X2)
   active(app(X1, X2)) -> app(X1, active(X2))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
   active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
   active(prefix(X)) -> prefix(active(X))
   app(mark(X1), X2) -> mark(app(X1, X2))
   app(X1, mark(X2)) -> mark(app(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
   zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
   prefix(mark(X)) -> mark(prefix(X))
   proper(app(X1, X2)) -> app(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
   proper(prefix(X)) -> prefix(proper(X))
   app(ok(X1), ok(X2)) -> ok(app(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
   prefix(ok(X)) -> ok(prefix(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(nil) -> nil
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2))
   encArg(cons_prefix(x_1)) -> prefix(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2))
   encode_prefix(x_1) -> prefix(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(app(nil, YS)) -> mark(YS)
   active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(zWadr(nil, YS)) -> mark(nil)
   active(zWadr(XS, nil)) -> mark(nil)
   active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
   active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
   active(app(X1, X2)) -> app(active(X1), X2)
   active(app(X1, X2)) -> app(X1, active(X2))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
   active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
   active(prefix(X)) -> prefix(active(X))
   app(mark(X1), X2) -> mark(app(X1, X2))
   app(X1, mark(X2)) -> mark(app(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
   zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
   prefix(mark(X)) -> mark(prefix(X))
   proper(app(X1, X2)) -> app(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
   proper(prefix(X)) -> prefix(proper(X))
   app(ok(X1), ok(X2)) -> ok(app(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
   prefix(ok(X)) -> ok(prefix(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(nil) -> nil
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2))
   encArg(cons_prefix(x_1)) -> prefix(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2))
   encode_prefix(x_1) -> prefix(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) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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(app(nil, YS)) -> mark(YS)
   active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(zWadr(nil, YS)) -> mark(nil)
   active(zWadr(XS, nil)) -> mark(nil)
   active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
   active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
   active(app(X1, X2)) -> app(active(X1), X2)
   active(app(X1, X2)) -> app(X1, active(X2))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
   active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
   active(prefix(X)) -> prefix(active(X))
   app(mark(X1), X2) -> mark(app(X1, X2))
   app(X1, mark(X2)) -> mark(app(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
   zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
   prefix(mark(X)) -> mark(prefix(X))
   proper(app(X1, X2)) -> app(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
   proper(prefix(X)) -> prefix(proper(X))
   app(ok(X1), ok(X2)) -> ok(app(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
   prefix(ok(X)) -> ok(prefix(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(nil) -> nil
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2))
   encArg(cons_prefix(x_1)) -> prefix(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2))
   encode_prefix(x_1) -> prefix(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) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

zWadr(mark(X1), X2) ->^+ mark(zWadr(X1, X2))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X1 / mark(X1)].

The result substitution is [ ].




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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following 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(app(nil, YS)) -> mark(YS)
   active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(zWadr(nil, YS)) -> mark(nil)
   active(zWadr(XS, nil)) -> mark(nil)
   active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
   active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
   active(app(X1, X2)) -> app(active(X1), X2)
   active(app(X1, X2)) -> app(X1, active(X2))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
   active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
   active(prefix(X)) -> prefix(active(X))
   app(mark(X1), X2) -> mark(app(X1, X2))
   app(X1, mark(X2)) -> mark(app(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
   zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
   prefix(mark(X)) -> mark(prefix(X))
   proper(app(X1, X2)) -> app(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
   proper(prefix(X)) -> prefix(proper(X))
   app(ok(X1), ok(X2)) -> ok(app(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
   prefix(ok(X)) -> ok(prefix(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(nil) -> nil
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2))
   encArg(cons_prefix(x_1)) -> prefix(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2))
   encode_prefix(x_1) -> prefix(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
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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(app(nil, YS)) -> mark(YS)
   active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(zWadr(nil, YS)) -> mark(nil)
   active(zWadr(XS, nil)) -> mark(nil)
   active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
   active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
   active(app(X1, X2)) -> app(active(X1), X2)
   active(app(X1, X2)) -> app(X1, active(X2))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
   active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
   active(prefix(X)) -> prefix(active(X))
   app(mark(X1), X2) -> mark(app(X1, X2))
   app(X1, mark(X2)) -> mark(app(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
   zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
   prefix(mark(X)) -> mark(prefix(X))
   proper(app(X1, X2)) -> app(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
   proper(prefix(X)) -> prefix(proper(X))
   app(ok(X1), ok(X2)) -> ok(app(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
   prefix(ok(X)) -> ok(prefix(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(nil) -> nil
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2))
   encArg(cons_prefix(x_1)) -> prefix(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2))
   encode_prefix(x_1) -> prefix(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