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


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 90 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), 458 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), 88 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 35 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 110 ms]
            (24) 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(dbl(0)) -> mark(0)
   active(dbl(s(X))) -> mark(s(s(dbl(X))))
   active(dbls(nil)) -> mark(nil)
   active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
   active(sel(0, cons(X, Y))) -> mark(X)
   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
   active(indx(nil, X)) -> mark(nil)
   active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(dbl(X)) -> dbl(active(X))
   active(dbls(X)) -> dbls(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   active(indx(X1, X2)) -> indx(active(X1), X2)
   dbl(mark(X)) -> mark(dbl(X))
   dbls(mark(X)) -> mark(dbls(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   indx(mark(X1), X2) -> mark(indx(X1, X2))
   proper(dbl(X)) -> dbl(proper(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(dbls(X)) -> dbls(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   dbl(ok(X)) -> ok(dbl(X))
   s(ok(X)) -> ok(s(X))
   dbls(ok(X)) -> ok(dbls(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
   from(ok(X)) -> ok(from(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(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
proper(dbl(X)) -> dbl(proper(X))
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(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:

   dbl(mark(X)) -> mark(dbl(X))
   dbls(mark(X)) -> mark(dbls(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   indx(mark(X1), X2) -> mark(indx(X1, X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   dbl(ok(X)) -> ok(dbl(X))
   s(ok(X)) -> ok(s(X))
   dbls(ok(X)) -> ok(dbls(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
   from(ok(X)) -> ok(from(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:

   dbl(mark(X)) -> mark(dbl(X))
   dbls(mark(X)) -> mark(dbls(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   indx(mark(X1), X2) -> mark(indx(X1, X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   dbl(ok(X)) -> ok(dbl(X))
   s(ok(X)) -> ok(s(X))
   dbls(ok(X)) -> ok(dbls(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
   from(ok(X)) -> ok(from(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, 9]
transitions: 
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
nil0() -> 0
active0(0) -> 0
dbl0(0) -> 1
dbls0(0) -> 2
sel0(0, 0) -> 3
indx0(0, 0) -> 4
proper0(0) -> 5
s0(0) -> 6
cons0(0, 0) -> 7
from0(0) -> 8
top0(0) -> 9
dbl1(0) -> 10
mark1(10) -> 1
dbls1(0) -> 11
mark1(11) -> 2
sel1(0, 0) -> 12
mark1(12) -> 3
indx1(0, 0) -> 13
mark1(13) -> 4
01() -> 14
ok1(14) -> 5
nil1() -> 15
ok1(15) -> 5
dbl1(0) -> 16
ok1(16) -> 1
s1(0) -> 17
ok1(17) -> 6
dbls1(0) -> 18
ok1(18) -> 2
cons1(0, 0) -> 19
ok1(19) -> 7
sel1(0, 0) -> 20
ok1(20) -> 3
indx1(0, 0) -> 21
ok1(21) -> 4
from1(0) -> 22
ok1(22) -> 8
proper1(0) -> 23
top1(23) -> 9
active1(0) -> 24
top1(24) -> 9
mark1(10) -> 10
mark1(10) -> 16
mark1(11) -> 11
mark1(11) -> 18
mark1(12) -> 12
mark1(12) -> 20
mark1(13) -> 13
mark1(13) -> 21
ok1(14) -> 23
ok1(15) -> 23
ok1(16) -> 10
ok1(16) -> 16
ok1(17) -> 17
ok1(18) -> 11
ok1(18) -> 18
ok1(19) -> 19
ok1(20) -> 12
ok1(20) -> 20
ok1(21) -> 13
ok1(21) -> 21
ok1(22) -> 22
active2(14) -> 25
top2(25) -> 9
active2(15) -> 25

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

(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(dbl(0')) -> mark(0')
   active(dbl(s(X))) -> mark(s(s(dbl(X))))
   active(dbls(nil)) -> mark(nil)
   active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
   active(sel(0', cons(X, Y))) -> mark(X)
   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
   active(indx(nil, X)) -> mark(nil)
   active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(dbl(X)) -> dbl(active(X))
   active(dbls(X)) -> dbls(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   active(indx(X1, X2)) -> indx(active(X1), X2)
   dbl(mark(X)) -> mark(dbl(X))
   dbls(mark(X)) -> mark(dbls(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   indx(mark(X1), X2) -> mark(indx(X1, X2))
   proper(dbl(X)) -> dbl(proper(X))
   proper(0') -> ok(0')
   proper(s(X)) -> s(proper(X))
   proper(dbls(X)) -> dbls(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   dbl(ok(X)) -> ok(dbl(X))
   s(ok(X)) -> ok(s(X))
   dbls(ok(X)) -> ok(dbls(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
   from(ok(X)) -> ok(from(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(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0', cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
dbl(ok(X)) -> ok(dbl(X))
s(ok(X)) -> ok(s(X))
dbls(ok(X)) -> ok(dbls(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:nil:ok -> 0':mark:nil:ok
dbl :: 0':mark:nil:ok -> 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok -> 0':mark:nil:ok
s :: 0':mark:nil:ok -> 0':mark:nil:ok
dbls :: 0':mark:nil:ok -> 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
sel :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
indx :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
from :: 0':mark:nil:ok -> 0':mark:nil:ok
proper :: 0':mark:nil:ok -> 0':mark:nil:ok
ok :: 0':mark:nil:ok -> 0':mark:nil:ok
top :: 0':mark:nil:ok -> top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat -> 0':mark:nil:ok

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, s, dbl, cons, dbls, sel, indx, from, proper, top

They will be analysed ascendingly in the following order:
s < active
dbl < active
cons < active
dbls < active
sel < active
indx < active
from < active
active < top
s < proper
dbl < proper
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
proper < top

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

(12)
Obligation:
TRS:
Rules:
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0', cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
dbl(ok(X)) -> ok(dbl(X))
s(ok(X)) -> ok(s(X))
dbls(ok(X)) -> ok(dbls(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:nil:ok -> 0':mark:nil:ok
dbl :: 0':mark:nil:ok -> 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok -> 0':mark:nil:ok
s :: 0':mark:nil:ok -> 0':mark:nil:ok
dbls :: 0':mark:nil:ok -> 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
sel :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
indx :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
from :: 0':mark:nil:ok -> 0':mark:nil:ok
proper :: 0':mark:nil:ok -> 0':mark:nil:ok
ok :: 0':mark:nil:ok -> 0':mark:nil:ok
top :: 0':mark:nil:ok -> top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat -> 0':mark:nil:ok


Generator Equations:
gen_0':mark:nil:ok3_0(0) <=> 0'
gen_0':mark:nil:ok3_0(+(x, 1)) <=> mark(gen_0':mark:nil:ok3_0(x))


The following defined symbols remain to be analysed:
s, active, dbl, cons, dbls, sel, indx, from, proper, top

They will be analysed ascendingly in the following order:
s < active
dbl < active
cons < active
dbls < active
sel < active
indx < active
from < active
active < top
s < proper
dbl < proper
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) -> *4_0, rt in Omega(n9_0)

Induction Base:
dbl(gen_0':mark:nil:ok3_0(+(1, 0)))

Induction Step:
dbl(gen_0':mark:nil:ok3_0(+(1, +(n9_0, 1)))) ->_R^Omega(1)
mark(dbl(gen_0':mark:nil:ok3_0(+(1, n9_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).
----------------------------------------

(14)
Complex Obligation (BEST)

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

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

TRS:
Rules:
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0', cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
dbl(ok(X)) -> ok(dbl(X))
s(ok(X)) -> ok(s(X))
dbls(ok(X)) -> ok(dbls(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:nil:ok -> 0':mark:nil:ok
dbl :: 0':mark:nil:ok -> 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok -> 0':mark:nil:ok
s :: 0':mark:nil:ok -> 0':mark:nil:ok
dbls :: 0':mark:nil:ok -> 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
sel :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
indx :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
from :: 0':mark:nil:ok -> 0':mark:nil:ok
proper :: 0':mark:nil:ok -> 0':mark:nil:ok
ok :: 0':mark:nil:ok -> 0':mark:nil:ok
top :: 0':mark:nil:ok -> top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat -> 0':mark:nil:ok


Generator Equations:
gen_0':mark:nil:ok3_0(0) <=> 0'
gen_0':mark:nil:ok3_0(+(x, 1)) <=> mark(gen_0':mark:nil:ok3_0(x))


The following defined symbols remain to be analysed:
dbl, active, cons, dbls, sel, indx, from, proper, top

They will be analysed ascendingly in the following order:
dbl < active
cons < active
dbls < active
sel < active
indx < active
from < active
active < top
dbl < proper
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
proper < top

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0', cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
dbl(ok(X)) -> ok(dbl(X))
s(ok(X)) -> ok(s(X))
dbls(ok(X)) -> ok(dbls(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:nil:ok -> 0':mark:nil:ok
dbl :: 0':mark:nil:ok -> 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok -> 0':mark:nil:ok
s :: 0':mark:nil:ok -> 0':mark:nil:ok
dbls :: 0':mark:nil:ok -> 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
sel :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
indx :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
from :: 0':mark:nil:ok -> 0':mark:nil:ok
proper :: 0':mark:nil:ok -> 0':mark:nil:ok
ok :: 0':mark:nil:ok -> 0':mark:nil:ok
top :: 0':mark:nil:ok -> top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat -> 0':mark:nil:ok


Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) -> *4_0, rt in Omega(n9_0)


Generator Equations:
gen_0':mark:nil:ok3_0(0) <=> 0'
gen_0':mark:nil:ok3_0(+(x, 1)) <=> mark(gen_0':mark:nil:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, dbls, sel, indx, from, proper, top

They will be analysed ascendingly in the following order:
cons < active
dbls < active
sel < active
indx < active
from < active
active < top
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
dbls(gen_0':mark:nil:ok3_0(+(1, n397_0))) -> *4_0, rt in Omega(n397_0)

Induction Base:
dbls(gen_0':mark:nil:ok3_0(+(1, 0)))

Induction Step:
dbls(gen_0':mark:nil:ok3_0(+(1, +(n397_0, 1)))) ->_R^Omega(1)
mark(dbls(gen_0':mark:nil:ok3_0(+(1, n397_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).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0', cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
dbl(ok(X)) -> ok(dbl(X))
s(ok(X)) -> ok(s(X))
dbls(ok(X)) -> ok(dbls(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:nil:ok -> 0':mark:nil:ok
dbl :: 0':mark:nil:ok -> 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok -> 0':mark:nil:ok
s :: 0':mark:nil:ok -> 0':mark:nil:ok
dbls :: 0':mark:nil:ok -> 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
sel :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
indx :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
from :: 0':mark:nil:ok -> 0':mark:nil:ok
proper :: 0':mark:nil:ok -> 0':mark:nil:ok
ok :: 0':mark:nil:ok -> 0':mark:nil:ok
top :: 0':mark:nil:ok -> top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat -> 0':mark:nil:ok


Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) -> *4_0, rt in Omega(n9_0)
dbls(gen_0':mark:nil:ok3_0(+(1, n397_0))) -> *4_0, rt in Omega(n397_0)


Generator Equations:
gen_0':mark:nil:ok3_0(0) <=> 0'
gen_0':mark:nil:ok3_0(+(x, 1)) <=> mark(gen_0':mark:nil:ok3_0(x))


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sel(gen_0':mark:nil:ok3_0(+(1, n876_0)), gen_0':mark:nil:ok3_0(b)) -> *4_0, rt in Omega(n876_0)

Induction Base:
sel(gen_0':mark:nil:ok3_0(+(1, 0)), gen_0':mark:nil:ok3_0(b))

Induction Step:
sel(gen_0':mark:nil:ok3_0(+(1, +(n876_0, 1))), gen_0':mark:nil:ok3_0(b)) ->_R^Omega(1)
mark(sel(gen_0':mark:nil:ok3_0(+(1, n876_0)), gen_0':mark:nil: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).
----------------------------------------

(22)
Obligation:
TRS:
Rules:
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0', cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
dbl(ok(X)) -> ok(dbl(X))
s(ok(X)) -> ok(s(X))
dbls(ok(X)) -> ok(dbls(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:nil:ok -> 0':mark:nil:ok
dbl :: 0':mark:nil:ok -> 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok -> 0':mark:nil:ok
s :: 0':mark:nil:ok -> 0':mark:nil:ok
dbls :: 0':mark:nil:ok -> 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
sel :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
indx :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
from :: 0':mark:nil:ok -> 0':mark:nil:ok
proper :: 0':mark:nil:ok -> 0':mark:nil:ok
ok :: 0':mark:nil:ok -> 0':mark:nil:ok
top :: 0':mark:nil:ok -> top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat -> 0':mark:nil:ok


Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) -> *4_0, rt in Omega(n9_0)
dbls(gen_0':mark:nil:ok3_0(+(1, n397_0))) -> *4_0, rt in Omega(n397_0)
sel(gen_0':mark:nil:ok3_0(+(1, n876_0)), gen_0':mark:nil:ok3_0(b)) -> *4_0, rt in Omega(n876_0)


Generator Equations:
gen_0':mark:nil:ok3_0(0) <=> 0'
gen_0':mark:nil:ok3_0(+(x, 1)) <=> mark(gen_0':mark:nil:ok3_0(x))


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

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
indx(gen_0':mark:nil:ok3_0(+(1, n2444_0)), gen_0':mark:nil:ok3_0(b)) -> *4_0, rt in Omega(n2444_0)

Induction Base:
indx(gen_0':mark:nil:ok3_0(+(1, 0)), gen_0':mark:nil:ok3_0(b))

Induction Step:
indx(gen_0':mark:nil:ok3_0(+(1, +(n2444_0, 1))), gen_0':mark:nil:ok3_0(b)) ->_R^Omega(1)
mark(indx(gen_0':mark:nil:ok3_0(+(1, n2444_0)), gen_0':mark:nil: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).
----------------------------------------

(24)
Obligation:
TRS:
Rules:
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0', cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
dbl(ok(X)) -> ok(dbl(X))
s(ok(X)) -> ok(s(X))
dbls(ok(X)) -> ok(dbls(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:nil:ok -> 0':mark:nil:ok
dbl :: 0':mark:nil:ok -> 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok -> 0':mark:nil:ok
s :: 0':mark:nil:ok -> 0':mark:nil:ok
dbls :: 0':mark:nil:ok -> 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
sel :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
indx :: 0':mark:nil:ok -> 0':mark:nil:ok -> 0':mark:nil:ok
from :: 0':mark:nil:ok -> 0':mark:nil:ok
proper :: 0':mark:nil:ok -> 0':mark:nil:ok
ok :: 0':mark:nil:ok -> 0':mark:nil:ok
top :: 0':mark:nil:ok -> top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat -> 0':mark:nil:ok


Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) -> *4_0, rt in Omega(n9_0)
dbls(gen_0':mark:nil:ok3_0(+(1, n397_0))) -> *4_0, rt in Omega(n397_0)
sel(gen_0':mark:nil:ok3_0(+(1, n876_0)), gen_0':mark:nil:ok3_0(b)) -> *4_0, rt in Omega(n876_0)
indx(gen_0':mark:nil:ok3_0(+(1, n2444_0)), gen_0':mark:nil:ok3_0(b)) -> *4_0, rt in Omega(n2444_0)


Generator Equations:
gen_0':mark:nil:ok3_0(0) <=> 0'
gen_0':mark:nil:ok3_0(+(x, 1)) <=> mark(gen_0':mark:nil:ok3_0(x))


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

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