/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 88 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), 404 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), 101 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 69 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 120 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(and(true, X)) -> mark(X)
   active(and(false, Y)) -> mark(false)
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(add(0, X)) -> mark(X)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   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(and(X1, X2)) -> and(active(X1), X2)
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(add(X1, X2)) -> add(active(X1), X2)
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   add(mark(X1), X2) -> mark(add(X1, X2))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   s(ok(X)) -> ok(s(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   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(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
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(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
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:

   and(mark(X1), X2) -> mark(and(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   add(mark(X1), X2) -> mark(add(X1, X2))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   s(ok(X)) -> ok(s(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   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:

   and(mark(X1), X2) -> mark(and(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   add(mark(X1), X2) -> mark(add(X1, X2))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   s(ok(X)) -> ok(s(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   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, 7, 8, 9]
transitions: 
mark0(0) -> 0
true0() -> 0
ok0(0) -> 0
false0() -> 0
00() -> 0
nil0() -> 0
active0(0) -> 0
and0(0, 0) -> 1
if0(0, 0, 0) -> 2
add0(0, 0) -> 3
first0(0, 0) -> 4
proper0(0) -> 5
s0(0) -> 6
cons0(0, 0) -> 7
from0(0) -> 8
top0(0) -> 9
and1(0, 0) -> 10
mark1(10) -> 1
if1(0, 0, 0) -> 11
mark1(11) -> 2
add1(0, 0) -> 12
mark1(12) -> 3
first1(0, 0) -> 13
mark1(13) -> 4
true1() -> 14
ok1(14) -> 5
false1() -> 15
ok1(15) -> 5
01() -> 16
ok1(16) -> 5
nil1() -> 17
ok1(17) -> 5
and1(0, 0) -> 18
ok1(18) -> 1
if1(0, 0, 0) -> 19
ok1(19) -> 2
add1(0, 0) -> 20
ok1(20) -> 3
s1(0) -> 21
ok1(21) -> 6
first1(0, 0) -> 22
ok1(22) -> 4
cons1(0, 0) -> 23
ok1(23) -> 7
from1(0) -> 24
ok1(24) -> 8
proper1(0) -> 25
top1(25) -> 9
active1(0) -> 26
top1(26) -> 9
mark1(10) -> 10
mark1(10) -> 18
mark1(11) -> 11
mark1(11) -> 19
mark1(12) -> 12
mark1(12) -> 20
mark1(13) -> 13
mark1(13) -> 22
ok1(14) -> 25
ok1(15) -> 25
ok1(16) -> 25
ok1(17) -> 25
ok1(18) -> 10
ok1(18) -> 18
ok1(19) -> 11
ok1(19) -> 19
ok1(20) -> 12
ok1(20) -> 20
ok1(21) -> 21
ok1(22) -> 13
ok1(22) -> 22
ok1(23) -> 23
ok1(24) -> 24
active2(14) -> 27
top2(27) -> 9
active2(15) -> 27
active2(16) -> 27
active2(17) -> 27

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

(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(and(true, X)) -> mark(X)
   active(and(false, Y)) -> mark(false)
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(add(0', X)) -> mark(X)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   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(and(X1, X2)) -> and(active(X1), X2)
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(add(X1, X2)) -> add(active(X1), X2)
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   add(mark(X1), X2) -> mark(add(X1, X2))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(0') -> ok(0')
   proper(s(X)) -> s(proper(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   s(ok(X)) -> ok(s(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   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(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
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(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
s(ok(X)) -> ok(s(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
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 :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok -> top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat -> true:mark:false:0':nil:ok

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

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

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

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

(12)
Obligation:
TRS:
Rules:
active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
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(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
s(ok(X)) -> ok(s(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
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 :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok -> top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat -> true:mark:false:0':nil:ok


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


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9_0)

Induction Base:
add(gen_true:mark:false:0':nil:ok3_0(+(1, 0)), gen_true:mark:false:0':nil:ok3_0(b))

Induction Step:
add(gen_true:mark:false:0':nil:ok3_0(+(1, +(n9_0, 1))), gen_true:mark:false:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':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).
----------------------------------------

(14)
Complex Obligation (BEST)

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

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

TRS:
Rules:
active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
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(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
s(ok(X)) -> ok(s(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
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 :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok -> top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat -> true:mark:false:0':nil:ok


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


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

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

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
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(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
s(ok(X)) -> ok(s(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
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 :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok -> top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat -> true:mark:false:0':nil:ok


Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9_0)


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


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1003_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1003_0)

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

Induction Step:
first(gen_true:mark:false:0':nil:ok3_0(+(1, +(n1003_0, 1))), gen_true:mark:false:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(first(gen_true:mark:false:0':nil:ok3_0(+(1, n1003_0)), gen_true:mark:false:0':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).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
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(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
s(ok(X)) -> ok(s(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
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 :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok -> top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat -> true:mark:false:0':nil:ok


Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9_0)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1003_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1003_0)


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


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2503_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2503_0)

Induction Base:
and(gen_true:mark:false:0':nil:ok3_0(+(1, 0)), gen_true:mark:false:0':nil:ok3_0(b))

Induction Step:
and(gen_true:mark:false:0':nil:ok3_0(+(1, +(n2503_0, 1))), gen_true:mark:false:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(and(gen_true:mark:false:0':nil:ok3_0(+(1, n2503_0)), gen_true:mark:false:0':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(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
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(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
s(ok(X)) -> ok(s(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
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 :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok -> top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat -> true:mark:false:0':nil:ok


Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9_0)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1003_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1003_0)
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2503_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2503_0)


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


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

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
if(gen_true:mark:false:0':nil:ok3_0(+(1, n4096_0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n4096_0)

Induction Base:
if(gen_true:mark:false:0':nil:ok3_0(+(1, 0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c))

Induction Step:
if(gen_true:mark:false:0':nil:ok3_0(+(1, +(n4096_0, 1))), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c)) ->_R^Omega(1)
mark(if(gen_true:mark:false:0':nil:ok3_0(+(1, n4096_0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c))) ->_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(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0', X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
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(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
s(ok(X)) -> ok(s(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
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 :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok -> true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok -> top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat -> true:mark:false:0':nil:ok


Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9_0)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1003_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1003_0)
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2503_0)), gen_true:mark:false:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2503_0)
if(gen_true:mark:false:0':nil:ok3_0(+(1, n4096_0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n4096_0)


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