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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 49 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 399 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 124 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 100 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 52 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 90 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 48 ms]
            (28) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


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: FULL
----------------------------------------

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
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))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
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))

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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   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(nil) -> ok(nil)
   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: FULL
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   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(nil) -> ok(nil)
   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: FULL
----------------------------------------

(5) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7, 8]
transitions: 
mark0(0) -> 0
nil0() -> 0
ok0(0) -> 0
active0(0) -> 0
app0(0, 0) -> 1
cons0(0, 0) -> 2
from0(0) -> 3
s0(0) -> 4
zWadr0(0, 0) -> 5
prefix0(0) -> 6
proper0(0) -> 7
top0(0) -> 8
app1(0, 0) -> 9
mark1(9) -> 1
cons1(0, 0) -> 10
mark1(10) -> 2
from1(0) -> 11
mark1(11) -> 3
s1(0) -> 12
mark1(12) -> 4
zWadr1(0, 0) -> 13
mark1(13) -> 5
prefix1(0) -> 14
mark1(14) -> 6
nil1() -> 15
ok1(15) -> 7
app1(0, 0) -> 16
ok1(16) -> 1
cons1(0, 0) -> 17
ok1(17) -> 2
from1(0) -> 18
ok1(18) -> 3
s1(0) -> 19
ok1(19) -> 4
zWadr1(0, 0) -> 20
ok1(20) -> 5
prefix1(0) -> 21
ok1(21) -> 6
proper1(0) -> 22
top1(22) -> 8
active1(0) -> 23
top1(23) -> 8
mark1(9) -> 9
mark1(9) -> 16
mark1(10) -> 10
mark1(10) -> 17
mark1(11) -> 11
mark1(11) -> 18
mark1(12) -> 12
mark1(12) -> 19
mark1(13) -> 13
mark1(13) -> 20
mark1(14) -> 14
mark1(14) -> 21
ok1(15) -> 22
ok1(16) -> 9
ok1(16) -> 16
ok1(17) -> 10
ok1(17) -> 17
ok1(18) -> 11
ok1(18) -> 18
ok1(19) -> 12
ok1(19) -> 19
ok1(20) -> 13
ok1(20) -> 20
ok1(21) -> 14
ok1(21) -> 21
active2(15) -> 24
top2(24) -> 8

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

(6)
BOUNDS(1, n^1)

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

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

(8)
Obligation:
The Runtime Complexity (full) of the given CpxTRS 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: FULL
----------------------------------------

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

(10)
Obligation:
TRS:
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))

Types:
active :: nil:mark:ok -> nil:mark:ok
app :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok -> nil:mark:ok
cons :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
from :: nil:mark:ok -> nil:mark:ok
s :: nil:mark:ok -> nil:mark:ok
zWadr :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
prefix :: nil:mark:ok -> nil:mark:ok
proper :: nil:mark:ok -> nil:mark:ok
ok :: nil:mark:ok -> nil:mark:ok
top :: nil:mark:ok -> top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat -> nil:mark:ok

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, cons, app, from, s, zWadr, prefix, proper, top

They will be analysed ascendingly in the following order:
cons < active
app < active
from < active
s < active
zWadr < active
prefix < active
active < top
cons < proper
app < proper
from < proper
s < proper
zWadr < proper
prefix < proper
proper < top

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

(12)
Obligation:
TRS:
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))

Types:
active :: nil:mark:ok -> nil:mark:ok
app :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok -> nil:mark:ok
cons :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
from :: nil:mark:ok -> nil:mark:ok
s :: nil:mark:ok -> nil:mark:ok
zWadr :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
prefix :: nil:mark:ok -> nil:mark:ok
proper :: nil:mark:ok -> nil:mark:ok
ok :: nil:mark:ok -> nil:mark:ok
top :: nil:mark:ok -> top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat -> nil:mark:ok


Generator Equations:
gen_nil:mark:ok3_0(0) <=> nil
gen_nil:mark:ok3_0(+(x, 1)) <=> mark(gen_nil:mark:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, app, from, s, zWadr, prefix, proper, top

They will be analysed ascendingly in the following order:
cons < active
app < active
from < active
s < active
zWadr < active
prefix < active
active < top
cons < proper
app < proper
from < proper
s < proper
zWadr < proper
prefix < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)

Induction Base:
cons(gen_nil:mark:ok3_0(+(1, 0)), gen_nil:mark:ok3_0(b))

Induction Step:
cons(gen_nil:mark:ok3_0(+(1, +(n5_0, 1))), gen_nil:mark:ok3_0(b)) ->_R^Omega(1)
mark(cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b))) ->_IH
mark(*4_0)

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

(14)
Complex Obligation (BEST)

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

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

TRS:
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))

Types:
active :: nil:mark:ok -> nil:mark:ok
app :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok -> nil:mark:ok
cons :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
from :: nil:mark:ok -> nil:mark:ok
s :: nil:mark:ok -> nil:mark:ok
zWadr :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
prefix :: nil:mark:ok -> nil:mark:ok
proper :: nil:mark:ok -> nil:mark:ok
ok :: nil:mark:ok -> nil:mark:ok
top :: nil:mark:ok -> top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat -> nil:mark:ok


Generator Equations:
gen_nil:mark:ok3_0(0) <=> nil
gen_nil:mark:ok3_0(+(x, 1)) <=> mark(gen_nil:mark:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, app, from, s, zWadr, prefix, proper, top

They will be analysed ascendingly in the following order:
cons < active
app < active
from < active
s < active
zWadr < active
prefix < active
active < top
cons < proper
app < proper
from < proper
s < proper
zWadr < proper
prefix < proper
proper < top

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
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))

Types:
active :: nil:mark:ok -> nil:mark:ok
app :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok -> nil:mark:ok
cons :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
from :: nil:mark:ok -> nil:mark:ok
s :: nil:mark:ok -> nil:mark:ok
zWadr :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
prefix :: nil:mark:ok -> nil:mark:ok
proper :: nil:mark:ok -> nil:mark:ok
ok :: nil:mark:ok -> nil:mark:ok
top :: nil:mark:ok -> top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat -> nil:mark:ok


Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_nil:mark:ok3_0(0) <=> nil
gen_nil:mark:ok3_0(+(x, 1)) <=> mark(gen_nil:mark:ok3_0(x))


The following defined symbols remain to be analysed:
app, active, from, s, zWadr, prefix, proper, top

They will be analysed ascendingly in the following order:
app < active
from < active
s < active
zWadr < active
prefix < active
active < top
app < proper
from < proper
s < proper
zWadr < proper
prefix < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:mark:ok3_0(+(1, n972_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n972_0)

Induction Base:
app(gen_nil:mark:ok3_0(+(1, 0)), gen_nil:mark:ok3_0(b))

Induction Step:
app(gen_nil:mark:ok3_0(+(1, +(n972_0, 1))), gen_nil:mark:ok3_0(b)) ->_R^Omega(1)
mark(app(gen_nil:mark:ok3_0(+(1, n972_0)), gen_nil:mark:ok3_0(b))) ->_IH
mark(*4_0)

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

(20)
Obligation:
TRS:
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))

Types:
active :: nil:mark:ok -> nil:mark:ok
app :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok -> nil:mark:ok
cons :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
from :: nil:mark:ok -> nil:mark:ok
s :: nil:mark:ok -> nil:mark:ok
zWadr :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
prefix :: nil:mark:ok -> nil:mark:ok
proper :: nil:mark:ok -> nil:mark:ok
ok :: nil:mark:ok -> nil:mark:ok
top :: nil:mark:ok -> top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat -> nil:mark:ok


Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
app(gen_nil:mark:ok3_0(+(1, n972_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n972_0)


Generator Equations:
gen_nil:mark:ok3_0(0) <=> nil
gen_nil:mark:ok3_0(+(x, 1)) <=> mark(gen_nil:mark:ok3_0(x))


The following defined symbols remain to be analysed:
from, active, s, zWadr, prefix, proper, top

They will be analysed ascendingly in the following order:
from < active
s < active
zWadr < active
prefix < active
active < top
from < proper
s < proper
zWadr < proper
prefix < proper
proper < top

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
from(gen_nil:mark:ok3_0(+(1, n2444_0))) -> *4_0, rt in Omega(n2444_0)

Induction Base:
from(gen_nil:mark:ok3_0(+(1, 0)))

Induction Step:
from(gen_nil:mark:ok3_0(+(1, +(n2444_0, 1)))) ->_R^Omega(1)
mark(from(gen_nil:mark:ok3_0(+(1, n2444_0)))) ->_IH
mark(*4_0)

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

(22)
Obligation:
TRS:
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))

Types:
active :: nil:mark:ok -> nil:mark:ok
app :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok -> nil:mark:ok
cons :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
from :: nil:mark:ok -> nil:mark:ok
s :: nil:mark:ok -> nil:mark:ok
zWadr :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
prefix :: nil:mark:ok -> nil:mark:ok
proper :: nil:mark:ok -> nil:mark:ok
ok :: nil:mark:ok -> nil:mark:ok
top :: nil:mark:ok -> top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat -> nil:mark:ok


Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
app(gen_nil:mark:ok3_0(+(1, n972_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n972_0)
from(gen_nil:mark:ok3_0(+(1, n2444_0))) -> *4_0, rt in Omega(n2444_0)


Generator Equations:
gen_nil:mark:ok3_0(0) <=> nil
gen_nil:mark:ok3_0(+(x, 1)) <=> mark(gen_nil:mark:ok3_0(x))


The following defined symbols remain to be analysed:
s, active, zWadr, prefix, proper, top

They will be analysed ascendingly in the following order:
s < active
zWadr < active
prefix < active
active < top
s < proper
zWadr < proper
prefix < proper
proper < top

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_nil:mark:ok3_0(+(1, n3125_0))) -> *4_0, rt in Omega(n3125_0)

Induction Base:
s(gen_nil:mark:ok3_0(+(1, 0)))

Induction Step:
s(gen_nil:mark:ok3_0(+(1, +(n3125_0, 1)))) ->_R^Omega(1)
mark(s(gen_nil:mark:ok3_0(+(1, n3125_0)))) ->_IH
mark(*4_0)

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

(24)
Obligation:
TRS:
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))

Types:
active :: nil:mark:ok -> nil:mark:ok
app :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok -> nil:mark:ok
cons :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
from :: nil:mark:ok -> nil:mark:ok
s :: nil:mark:ok -> nil:mark:ok
zWadr :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
prefix :: nil:mark:ok -> nil:mark:ok
proper :: nil:mark:ok -> nil:mark:ok
ok :: nil:mark:ok -> nil:mark:ok
top :: nil:mark:ok -> top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat -> nil:mark:ok


Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
app(gen_nil:mark:ok3_0(+(1, n972_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n972_0)
from(gen_nil:mark:ok3_0(+(1, n2444_0))) -> *4_0, rt in Omega(n2444_0)
s(gen_nil:mark:ok3_0(+(1, n3125_0))) -> *4_0, rt in Omega(n3125_0)


Generator Equations:
gen_nil:mark:ok3_0(0) <=> nil
gen_nil:mark:ok3_0(+(x, 1)) <=> mark(gen_nil:mark:ok3_0(x))


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

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

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
zWadr(gen_nil:mark:ok3_0(+(1, n3907_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n3907_0)

Induction Base:
zWadr(gen_nil:mark:ok3_0(+(1, 0)), gen_nil:mark:ok3_0(b))

Induction Step:
zWadr(gen_nil:mark:ok3_0(+(1, +(n3907_0, 1))), gen_nil:mark:ok3_0(b)) ->_R^Omega(1)
mark(zWadr(gen_nil:mark:ok3_0(+(1, n3907_0)), gen_nil:mark:ok3_0(b))) ->_IH
mark(*4_0)

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

(26)
Obligation:
TRS:
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))

Types:
active :: nil:mark:ok -> nil:mark:ok
app :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok -> nil:mark:ok
cons :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
from :: nil:mark:ok -> nil:mark:ok
s :: nil:mark:ok -> nil:mark:ok
zWadr :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
prefix :: nil:mark:ok -> nil:mark:ok
proper :: nil:mark:ok -> nil:mark:ok
ok :: nil:mark:ok -> nil:mark:ok
top :: nil:mark:ok -> top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat -> nil:mark:ok


Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
app(gen_nil:mark:ok3_0(+(1, n972_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n972_0)
from(gen_nil:mark:ok3_0(+(1, n2444_0))) -> *4_0, rt in Omega(n2444_0)
s(gen_nil:mark:ok3_0(+(1, n3125_0))) -> *4_0, rt in Omega(n3125_0)
zWadr(gen_nil:mark:ok3_0(+(1, n3907_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n3907_0)


Generator Equations:
gen_nil:mark:ok3_0(0) <=> nil
gen_nil:mark:ok3_0(+(x, 1)) <=> mark(gen_nil:mark:ok3_0(x))


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

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

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
prefix(gen_nil:mark:ok3_0(+(1, n6101_0))) -> *4_0, rt in Omega(n6101_0)

Induction Base:
prefix(gen_nil:mark:ok3_0(+(1, 0)))

Induction Step:
prefix(gen_nil:mark:ok3_0(+(1, +(n6101_0, 1)))) ->_R^Omega(1)
mark(prefix(gen_nil:mark:ok3_0(+(1, n6101_0)))) ->_IH
mark(*4_0)

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

(28)
Obligation:
TRS:
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))

Types:
active :: nil:mark:ok -> nil:mark:ok
app :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok -> nil:mark:ok
cons :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
from :: nil:mark:ok -> nil:mark:ok
s :: nil:mark:ok -> nil:mark:ok
zWadr :: nil:mark:ok -> nil:mark:ok -> nil:mark:ok
prefix :: nil:mark:ok -> nil:mark:ok
proper :: nil:mark:ok -> nil:mark:ok
ok :: nil:mark:ok -> nil:mark:ok
top :: nil:mark:ok -> top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat -> nil:mark:ok


Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
app(gen_nil:mark:ok3_0(+(1, n972_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n972_0)
from(gen_nil:mark:ok3_0(+(1, n2444_0))) -> *4_0, rt in Omega(n2444_0)
s(gen_nil:mark:ok3_0(+(1, n3125_0))) -> *4_0, rt in Omega(n3125_0)
zWadr(gen_nil:mark:ok3_0(+(1, n3907_0)), gen_nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n3907_0)
prefix(gen_nil:mark:ok3_0(+(1, n6101_0))) -> *4_0, rt in Omega(n6101_0)


Generator Equations:
gen_nil:mark:ok3_0(0) <=> nil
gen_nil:mark:ok3_0(+(x, 1)) <=> mark(gen_nil:mark:ok3_0(x))


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

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