/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 37 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), 437 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), 94 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 81 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 92 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(first(0, X)) -> mark(nil)
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   active(s(X)) -> s(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(s(X)) -> s(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1), ok(X2)) -> ok(cons(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(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(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:

   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1), ok(X2)) -> ok(cons(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:

   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1), ok(X2)) -> ok(cons(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]
transitions: 
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
nil0() -> 0
active0(0) -> 0
first0(0, 0) -> 1
s0(0) -> 2
cons0(0, 0) -> 3
from0(0) -> 4
proper0(0) -> 5
top0(0) -> 6
first1(0, 0) -> 7
mark1(7) -> 1
s1(0) -> 8
mark1(8) -> 2
cons1(0, 0) -> 9
mark1(9) -> 3
from1(0) -> 10
mark1(10) -> 4
01() -> 11
ok1(11) -> 5
nil1() -> 12
ok1(12) -> 5
first1(0, 0) -> 13
ok1(13) -> 1
s1(0) -> 14
ok1(14) -> 2
cons1(0, 0) -> 15
ok1(15) -> 3
from1(0) -> 16
ok1(16) -> 4
proper1(0) -> 17
top1(17) -> 6
active1(0) -> 18
top1(18) -> 6
mark1(7) -> 7
mark1(7) -> 13
mark1(8) -> 8
mark1(8) -> 14
mark1(9) -> 9
mark1(9) -> 15
mark1(10) -> 10
mark1(10) -> 16
ok1(11) -> 17
ok1(12) -> 17
ok1(13) -> 7
ok1(13) -> 13
ok1(14) -> 8
ok1(14) -> 14
ok1(15) -> 9
ok1(15) -> 15
ok1(16) -> 10
ok1(16) -> 16
active2(11) -> 19
top2(19) -> 6
active2(12) -> 19

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

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

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

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

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

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

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

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

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


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


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

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

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

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

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

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

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


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


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

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

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

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

(17)
BOUNDS(n^1, INF)

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

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

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


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


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


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
first(gen_0':nil:mark:ok3_0(+(1, n790_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n790_0)

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

Induction Step:
first(gen_0':nil:mark:ok3_0(+(1, +(n790_0, 1))), gen_0':nil:mark:ok3_0(b)) ->_R^Omega(1)
mark(first(gen_0':nil:mark:ok3_0(+(1, n790_0)), gen_0':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(first(0', X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
s(mark(X)) -> mark(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
from(mark(X)) -> mark(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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


Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
first(gen_0':nil:mark:ok3_0(+(1, n790_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n790_0)


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


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
from(gen_0':nil:mark:ok3_0(+(1, n2080_0))) -> *4_0, rt in Omega(n2080_0)

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

Induction Step:
from(gen_0':nil:mark:ok3_0(+(1, +(n2080_0, 1)))) ->_R^Omega(1)
mark(from(gen_0':nil:mark:ok3_0(+(1, n2080_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(first(0', X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
s(mark(X)) -> mark(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
from(mark(X)) -> mark(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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


Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
first(gen_0':nil:mark:ok3_0(+(1, n790_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n790_0)
from(gen_0':nil:mark:ok3_0(+(1, n2080_0))) -> *4_0, rt in Omega(n2080_0)


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


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

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_0':nil:mark:ok3_0(+(1, n2722_0))) -> *4_0, rt in Omega(n2722_0)

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

Induction Step:
s(gen_0':nil:mark:ok3_0(+(1, +(n2722_0, 1)))) ->_R^Omega(1)
mark(s(gen_0':nil:mark:ok3_0(+(1, n2722_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(first(0', X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
s(mark(X)) -> mark(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
from(mark(X)) -> mark(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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


Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
first(gen_0':nil:mark:ok3_0(+(1, n790_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n790_0)
from(gen_0':nil:mark:ok3_0(+(1, n2080_0))) -> *4_0, rt in Omega(n2080_0)
s(gen_0':nil:mark:ok3_0(+(1, n2722_0))) -> *4_0, rt in Omega(n2722_0)


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