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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 10 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 28 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), 432 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), 85 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 105 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 41 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(f(0)) -> mark(cons(0, f(s(0))))
   active(f(s(0))) -> mark(f(p(s(0))))
   active(p(s(X))) -> mark(X)
   active(f(X)) -> f(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(p(X)) -> p(active(X))
   f(mark(X)) -> mark(f(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   p(mark(X)) -> mark(p(X))
   proper(f(X)) -> f(proper(X))
   proper(0) -> ok(0)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(p(X)) -> p(proper(X))
   f(ok(X)) -> ok(f(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   p(ok(X)) -> ok(p(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(f(0)) -> mark(cons(0, f(s(0))))
active(f(s(0))) -> mark(f(p(s(0))))
active(p(s(X))) -> mark(X)
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(p(X)) -> p(active(X))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(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:

   f(mark(X)) -> mark(f(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   p(mark(X)) -> mark(p(X))
   proper(0) -> ok(0)
   f(ok(X)) -> ok(f(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   p(ok(X)) -> ok(p(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:

   f(mark(X)) -> mark(f(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   p(mark(X)) -> mark(p(X))
   proper(0) -> ok(0)
   f(ok(X)) -> ok(f(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   p(ok(X)) -> ok(p(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]
transitions: 
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
active0(0) -> 0
f0(0) -> 1
cons0(0, 0) -> 2
s0(0) -> 3
p0(0) -> 4
proper0(0) -> 5
top0(0) -> 6
f1(0) -> 7
mark1(7) -> 1
cons1(0, 0) -> 8
mark1(8) -> 2
s1(0) -> 9
mark1(9) -> 3
p1(0) -> 10
mark1(10) -> 4
01() -> 11
ok1(11) -> 5
f1(0) -> 12
ok1(12) -> 1
cons1(0, 0) -> 13
ok1(13) -> 2
s1(0) -> 14
ok1(14) -> 3
p1(0) -> 15
ok1(15) -> 4
proper1(0) -> 16
top1(16) -> 6
active1(0) -> 17
top1(17) -> 6
mark1(7) -> 7
mark1(7) -> 12
mark1(8) -> 8
mark1(8) -> 13
mark1(9) -> 9
mark1(9) -> 14
mark1(10) -> 10
mark1(10) -> 15
ok1(11) -> 16
ok1(12) -> 7
ok1(12) -> 12
ok1(13) -> 8
ok1(13) -> 13
ok1(14) -> 9
ok1(14) -> 14
ok1(15) -> 10
ok1(15) -> 15
active2(11) -> 18
top2(18) -> 6

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

(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(f(0')) -> mark(cons(0', f(s(0'))))
   active(f(s(0'))) -> mark(f(p(s(0'))))
   active(p(s(X))) -> mark(X)
   active(f(X)) -> f(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(p(X)) -> p(active(X))
   f(mark(X)) -> mark(f(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   p(mark(X)) -> mark(p(X))
   proper(f(X)) -> f(proper(X))
   proper(0') -> ok(0')
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(p(X)) -> p(proper(X))
   f(ok(X)) -> ok(f(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   p(ok(X)) -> ok(p(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(f(0')) -> mark(cons(0', f(s(0'))))
active(f(s(0'))) -> mark(f(p(s(0'))))
active(p(s(X))) -> mark(X)
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(p(X)) -> p(active(X))
f(mark(X)) -> mark(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
p(mark(X)) -> mark(p(X))
proper(f(X)) -> f(proper(X))
proper(0') -> ok(0')
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(proper(X))
f(ok(X)) -> ok(f(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
p(ok(X)) -> ok(p(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:ok -> 0':mark:ok
f :: 0':mark:ok -> 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok -> 0':mark:ok
cons :: 0':mark:ok -> 0':mark:ok -> 0':mark:ok
s :: 0':mark:ok -> 0':mark:ok
p :: 0':mark:ok -> 0':mark:ok
proper :: 0':mark:ok -> 0':mark:ok
ok :: 0':mark:ok -> 0':mark:ok
top :: 0':mark:ok -> top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat -> 0':mark:ok

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, cons, f, s, p, proper, top

They will be analysed ascendingly in the following order:
cons < active
f < active
s < active
p < active
active < top
cons < proper
f < proper
s < proper
p < proper
proper < top

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

(12)
Obligation:
TRS:
Rules:
active(f(0')) -> mark(cons(0', f(s(0'))))
active(f(s(0'))) -> mark(f(p(s(0'))))
active(p(s(X))) -> mark(X)
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(p(X)) -> p(active(X))
f(mark(X)) -> mark(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
p(mark(X)) -> mark(p(X))
proper(f(X)) -> f(proper(X))
proper(0') -> ok(0')
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(proper(X))
f(ok(X)) -> ok(f(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
p(ok(X)) -> ok(p(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:ok -> 0':mark:ok
f :: 0':mark:ok -> 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok -> 0':mark:ok
cons :: 0':mark:ok -> 0':mark:ok -> 0':mark:ok
s :: 0':mark:ok -> 0':mark:ok
p :: 0':mark:ok -> 0':mark:ok
proper :: 0':mark:ok -> 0':mark:ok
ok :: 0':mark:ok -> 0':mark:ok
top :: 0':mark:ok -> top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat -> 0':mark:ok


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


The following defined symbols remain to be analysed:
cons, active, f, s, p, proper, top

They will be analysed ascendingly in the following order:
cons < active
f < active
s < active
p < active
active < top
cons < proper
f < proper
s < proper
p < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
cons(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)

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

Induction Step:
cons(gen_0':mark:ok3_0(+(1, +(n5_0, 1))), gen_0':mark:ok3_0(b)) ->_R^Omega(1)
mark(cons(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':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(f(0')) -> mark(cons(0', f(s(0'))))
active(f(s(0'))) -> mark(f(p(s(0'))))
active(p(s(X))) -> mark(X)
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(p(X)) -> p(active(X))
f(mark(X)) -> mark(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
p(mark(X)) -> mark(p(X))
proper(f(X)) -> f(proper(X))
proper(0') -> ok(0')
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(proper(X))
f(ok(X)) -> ok(f(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
p(ok(X)) -> ok(p(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:ok -> 0':mark:ok
f :: 0':mark:ok -> 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok -> 0':mark:ok
cons :: 0':mark:ok -> 0':mark:ok -> 0':mark:ok
s :: 0':mark:ok -> 0':mark:ok
p :: 0':mark:ok -> 0':mark:ok
proper :: 0':mark:ok -> 0':mark:ok
ok :: 0':mark:ok -> 0':mark:ok
top :: 0':mark:ok -> top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat -> 0':mark:ok


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


The following defined symbols remain to be analysed:
cons, active, f, s, p, proper, top

They will be analysed ascendingly in the following order:
cons < active
f < active
s < active
p < active
active < top
cons < proper
f < proper
s < proper
p < proper
proper < top

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(f(0')) -> mark(cons(0', f(s(0'))))
active(f(s(0'))) -> mark(f(p(s(0'))))
active(p(s(X))) -> mark(X)
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(p(X)) -> p(active(X))
f(mark(X)) -> mark(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
p(mark(X)) -> mark(p(X))
proper(f(X)) -> f(proper(X))
proper(0') -> ok(0')
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(proper(X))
f(ok(X)) -> ok(f(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
p(ok(X)) -> ok(p(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:ok -> 0':mark:ok
f :: 0':mark:ok -> 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok -> 0':mark:ok
cons :: 0':mark:ok -> 0':mark:ok -> 0':mark:ok
s :: 0':mark:ok -> 0':mark:ok
p :: 0':mark:ok -> 0':mark:ok
proper :: 0':mark:ok -> 0':mark:ok
ok :: 0':mark:ok -> 0':mark:ok
top :: 0':mark:ok -> top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat -> 0':mark:ok


Lemmas:
cons(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)


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


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_0':mark:ok3_0(+(1, n748_0))) -> *4_0, rt in Omega(n748_0)

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

Induction Step:
f(gen_0':mark:ok3_0(+(1, +(n748_0, 1)))) ->_R^Omega(1)
mark(f(gen_0':mark:ok3_0(+(1, n748_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(f(0')) -> mark(cons(0', f(s(0'))))
active(f(s(0'))) -> mark(f(p(s(0'))))
active(p(s(X))) -> mark(X)
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(p(X)) -> p(active(X))
f(mark(X)) -> mark(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
p(mark(X)) -> mark(p(X))
proper(f(X)) -> f(proper(X))
proper(0') -> ok(0')
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(proper(X))
f(ok(X)) -> ok(f(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
p(ok(X)) -> ok(p(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:ok -> 0':mark:ok
f :: 0':mark:ok -> 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok -> 0':mark:ok
cons :: 0':mark:ok -> 0':mark:ok -> 0':mark:ok
s :: 0':mark:ok -> 0':mark:ok
p :: 0':mark:ok -> 0':mark:ok
proper :: 0':mark:ok -> 0':mark:ok
ok :: 0':mark:ok -> 0':mark:ok
top :: 0':mark:ok -> top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat -> 0':mark:ok


Lemmas:
cons(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
f(gen_0':mark:ok3_0(+(1, n748_0))) -> *4_0, rt in Omega(n748_0)


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


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_0':mark:ok3_0(+(1, n1231_0))) -> *4_0, rt in Omega(n1231_0)

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

Induction Step:
s(gen_0':mark:ok3_0(+(1, +(n1231_0, 1)))) ->_R^Omega(1)
mark(s(gen_0':mark:ok3_0(+(1, n1231_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(f(0')) -> mark(cons(0', f(s(0'))))
active(f(s(0'))) -> mark(f(p(s(0'))))
active(p(s(X))) -> mark(X)
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(p(X)) -> p(active(X))
f(mark(X)) -> mark(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
p(mark(X)) -> mark(p(X))
proper(f(X)) -> f(proper(X))
proper(0') -> ok(0')
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(proper(X))
f(ok(X)) -> ok(f(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
p(ok(X)) -> ok(p(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:ok -> 0':mark:ok
f :: 0':mark:ok -> 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok -> 0':mark:ok
cons :: 0':mark:ok -> 0':mark:ok -> 0':mark:ok
s :: 0':mark:ok -> 0':mark:ok
p :: 0':mark:ok -> 0':mark:ok
proper :: 0':mark:ok -> 0':mark:ok
ok :: 0':mark:ok -> 0':mark:ok
top :: 0':mark:ok -> top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat -> 0':mark:ok


Lemmas:
cons(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
f(gen_0':mark:ok3_0(+(1, n748_0))) -> *4_0, rt in Omega(n748_0)
s(gen_0':mark:ok3_0(+(1, n1231_0))) -> *4_0, rt in Omega(n1231_0)


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


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

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_0':mark:ok3_0(+(1, n1815_0))) -> *4_0, rt in Omega(n1815_0)

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

Induction Step:
p(gen_0':mark:ok3_0(+(1, +(n1815_0, 1)))) ->_R^Omega(1)
mark(p(gen_0':mark:ok3_0(+(1, n1815_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(f(0')) -> mark(cons(0', f(s(0'))))
active(f(s(0'))) -> mark(f(p(s(0'))))
active(p(s(X))) -> mark(X)
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(p(X)) -> p(active(X))
f(mark(X)) -> mark(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
p(mark(X)) -> mark(p(X))
proper(f(X)) -> f(proper(X))
proper(0') -> ok(0')
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(proper(X))
f(ok(X)) -> ok(f(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
p(ok(X)) -> ok(p(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:ok -> 0':mark:ok
f :: 0':mark:ok -> 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok -> 0':mark:ok
cons :: 0':mark:ok -> 0':mark:ok -> 0':mark:ok
s :: 0':mark:ok -> 0':mark:ok
p :: 0':mark:ok -> 0':mark:ok
proper :: 0':mark:ok -> 0':mark:ok
ok :: 0':mark:ok -> 0':mark:ok
top :: 0':mark:ok -> top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat -> 0':mark:ok


Lemmas:
cons(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
f(gen_0':mark:ok3_0(+(1, n748_0))) -> *4_0, rt in Omega(n748_0)
s(gen_0':mark:ok3_0(+(1, n1231_0))) -> *4_0, rt in Omega(n1231_0)
p(gen_0':mark:ok3_0(+(1, n1815_0))) -> *4_0, rt in Omega(n1815_0)


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