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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 95 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 399 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 44 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 66 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 71 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 98 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 90 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 91 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 100 ms]
            (32) 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(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
   active(sqr(0)) -> mark(0)
   active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
   active(dbl(0)) -> mark(0)
   active(dbl(s(X))) -> mark(s(s(dbl(X))))
   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(terms(X)) -> terms(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(recip(X)) -> recip(active(X))
   active(sqr(X)) -> sqr(active(X))
   active(s(X)) -> s(active(X))
   active(add(X1, X2)) -> add(active(X1), X2)
   active(add(X1, X2)) -> add(X1, active(X2))
   active(dbl(X)) -> dbl(active(X))
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   terms(mark(X)) -> mark(terms(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   recip(mark(X)) -> mark(recip(X))
   sqr(mark(X)) -> mark(sqr(X))
   s(mark(X)) -> mark(s(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   dbl(mark(X)) -> mark(dbl(X))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(terms(X)) -> terms(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(recip(X)) -> recip(proper(X))
   proper(sqr(X)) -> sqr(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0) -> ok(0)
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(dbl(X)) -> dbl(proper(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   terms(ok(X)) -> ok(terms(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   recip(ok(X)) -> ok(recip(X))
   sqr(ok(X)) -> ok(sqr(X))
   s(ok(X)) -> ok(s(X))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   dbl(ok(X)) -> ok(dbl(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   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(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))

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

(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:

   terms(mark(X)) -> mark(terms(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   recip(mark(X)) -> mark(recip(X))
   sqr(mark(X)) -> mark(sqr(X))
   s(mark(X)) -> mark(s(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   dbl(mark(X)) -> mark(dbl(X))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   terms(ok(X)) -> ok(terms(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   recip(ok(X)) -> ok(recip(X))
   sqr(ok(X)) -> ok(sqr(X))
   s(ok(X)) -> ok(s(X))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   dbl(ok(X)) -> ok(dbl(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   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:

   terms(mark(X)) -> mark(terms(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   recip(mark(X)) -> mark(recip(X))
   sqr(mark(X)) -> mark(sqr(X))
   s(mark(X)) -> mark(s(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   dbl(mark(X)) -> mark(dbl(X))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   terms(ok(X)) -> ok(terms(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   recip(ok(X)) -> ok(recip(X))
   sqr(ok(X)) -> ok(sqr(X))
   s(ok(X)) -> ok(s(X))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   dbl(ok(X)) -> ok(dbl(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   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, 10]
transitions: 
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
nil0() -> 0
active0(0) -> 0
terms0(0) -> 1
cons0(0, 0) -> 2
recip0(0) -> 3
sqr0(0) -> 4
s0(0) -> 5
add0(0, 0) -> 6
dbl0(0) -> 7
first0(0, 0) -> 8
proper0(0) -> 9
top0(0) -> 10
terms1(0) -> 11
mark1(11) -> 1
cons1(0, 0) -> 12
mark1(12) -> 2
recip1(0) -> 13
mark1(13) -> 3
sqr1(0) -> 14
mark1(14) -> 4
s1(0) -> 15
mark1(15) -> 5
add1(0, 0) -> 16
mark1(16) -> 6
dbl1(0) -> 17
mark1(17) -> 7
first1(0, 0) -> 18
mark1(18) -> 8
01() -> 19
ok1(19) -> 9
nil1() -> 20
ok1(20) -> 9
terms1(0) -> 21
ok1(21) -> 1
cons1(0, 0) -> 22
ok1(22) -> 2
recip1(0) -> 23
ok1(23) -> 3
sqr1(0) -> 24
ok1(24) -> 4
s1(0) -> 25
ok1(25) -> 5
add1(0, 0) -> 26
ok1(26) -> 6
dbl1(0) -> 27
ok1(27) -> 7
first1(0, 0) -> 28
ok1(28) -> 8
proper1(0) -> 29
top1(29) -> 10
active1(0) -> 30
top1(30) -> 10
mark1(11) -> 11
mark1(11) -> 21
mark1(12) -> 12
mark1(12) -> 22
mark1(13) -> 13
mark1(13) -> 23
mark1(14) -> 14
mark1(14) -> 24
mark1(15) -> 15
mark1(15) -> 25
mark1(16) -> 16
mark1(16) -> 26
mark1(17) -> 17
mark1(17) -> 27
mark1(18) -> 18
mark1(18) -> 28
ok1(19) -> 29
ok1(20) -> 29
ok1(21) -> 11
ok1(21) -> 21
ok1(22) -> 12
ok1(22) -> 22
ok1(23) -> 13
ok1(23) -> 23
ok1(24) -> 14
ok1(24) -> 24
ok1(25) -> 15
ok1(25) -> 25
ok1(26) -> 16
ok1(26) -> 26
ok1(27) -> 17
ok1(27) -> 27
ok1(28) -> 18
ok1(28) -> 28
active2(19) -> 31
top2(31) -> 10
active2(20) -> 31

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

(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(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
   active(sqr(0')) -> mark(0')
   active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
   active(dbl(0')) -> mark(0')
   active(dbl(s(X))) -> mark(s(s(dbl(X))))
   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(terms(X)) -> terms(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(recip(X)) -> recip(active(X))
   active(sqr(X)) -> sqr(active(X))
   active(s(X)) -> s(active(X))
   active(add(X1, X2)) -> add(active(X1), X2)
   active(add(X1, X2)) -> add(X1, active(X2))
   active(dbl(X)) -> dbl(active(X))
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   terms(mark(X)) -> mark(terms(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   recip(mark(X)) -> mark(recip(X))
   sqr(mark(X)) -> mark(sqr(X))
   s(mark(X)) -> mark(s(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   dbl(mark(X)) -> mark(dbl(X))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(terms(X)) -> terms(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(recip(X)) -> recip(proper(X))
   proper(sqr(X)) -> sqr(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0') -> ok(0')
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(dbl(X)) -> dbl(proper(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   terms(ok(X)) -> ok(terms(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   recip(ok(X)) -> ok(recip(X))
   sqr(ok(X)) -> ok(sqr(X))
   s(ok(X)) -> ok(s(X))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   dbl(ok(X)) -> ok(dbl(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   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(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, cons, recip, sqr, terms, s, add, dbl, first, proper, top

They will be analysed ascendingly in the following order:
cons < active
recip < active
sqr < active
terms < active
s < active
add < active
dbl < active
first < active
active < top
cons < proper
recip < proper
sqr < proper
terms < proper
s < proper
add < proper
dbl < proper
first < proper
proper < top

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

(12)
Obligation:
TRS:
Rules:
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok


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


The following defined symbols remain to be analysed:
cons, active, recip, sqr, terms, s, add, dbl, first, proper, top

They will be analysed ascendingly in the following order:
cons < active
recip < active
sqr < active
terms < active
s < active
add < active
dbl < active
first < active
active < top
cons < proper
recip < proper
sqr < proper
terms < proper
s < proper
add < proper
dbl < proper
first < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)

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

Induction Step:
cons(gen_mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark: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(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok


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


The following defined symbols remain to be analysed:
cons, active, recip, sqr, terms, s, add, dbl, first, proper, top

They will be analysed ascendingly in the following order:
cons < active
recip < active
sqr < active
terms < active
s < active
add < active
dbl < active
first < active
active < top
cons < proper
recip < proper
sqr < proper
terms < proper
s < proper
add < proper
dbl < proper
first < proper
proper < top

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok


Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)


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


The following defined symbols remain to be analysed:
recip, active, sqr, terms, s, add, dbl, first, proper, top

They will be analysed ascendingly in the following order:
recip < active
sqr < active
terms < active
s < active
add < active
dbl < active
first < active
active < top
recip < proper
sqr < proper
terms < proper
s < proper
add < proper
dbl < proper
first < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0)

Induction Base:
recip(gen_mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
recip(gen_mark:0':nil:ok3_0(+(1, +(n1126_0, 1)))) ->_R^Omega(1)
mark(recip(gen_mark:0':nil:ok3_0(+(1, n1126_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(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok


Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0)


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


The following defined symbols remain to be analysed:
sqr, active, terms, s, add, dbl, first, proper, top

They will be analysed ascendingly in the following order:
sqr < active
terms < active
s < active
add < active
dbl < active
first < active
active < top
sqr < proper
terms < proper
s < proper
add < proper
dbl < proper
first < proper
proper < top

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0)

Induction Base:
sqr(gen_mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
sqr(gen_mark:0':nil:ok3_0(+(1, +(n1690_0, 1)))) ->_R^Omega(1)
mark(sqr(gen_mark:0':nil:ok3_0(+(1, n1690_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(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok


Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0)
sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0)


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


The following defined symbols remain to be analysed:
terms, active, s, add, dbl, first, proper, top

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0)

Induction Base:
terms(gen_mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
terms(gen_mark:0':nil:ok3_0(+(1, +(n2355_0, 1)))) ->_R^Omega(1)
mark(terms(gen_mark:0':nil:ok3_0(+(1, n2355_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(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok


Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0)
sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0)
terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0)


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


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

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

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_mark:0':nil:ok3_0(+(1, n3121_0))) -> *4_0, rt in Omega(n3121_0)

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

Induction Step:
s(gen_mark:0':nil:ok3_0(+(1, +(n3121_0, 1)))) ->_R^Omega(1)
mark(s(gen_mark:0':nil:ok3_0(+(1, n3121_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).
----------------------------------------

(26)
Obligation:
TRS:
Rules:
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok


Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0)
sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0)
terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0)
s(gen_mark:0':nil:ok3_0(+(1, n3121_0))) -> *4_0, rt in Omega(n3121_0)


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


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

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

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_mark:0':nil:ok3_0(+(1, n3988_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3988_0)

Induction Base:
add(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))

Induction Step:
add(gen_mark:0':nil:ok3_0(+(1, +(n3988_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(add(gen_mark:0':nil:ok3_0(+(1, n3988_0)), gen_mark: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).
----------------------------------------

(28)
Obligation:
TRS:
Rules:
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok


Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0)
sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0)
terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0)
s(gen_mark:0':nil:ok3_0(+(1, n3121_0))) -> *4_0, rt in Omega(n3121_0)
add(gen_mark:0':nil:ok3_0(+(1, n3988_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3988_0)


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


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

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

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
dbl(gen_mark:0':nil:ok3_0(+(1, n6446_0))) -> *4_0, rt in Omega(n6446_0)

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

Induction Step:
dbl(gen_mark:0':nil:ok3_0(+(1, +(n6446_0, 1)))) ->_R^Omega(1)
mark(dbl(gen_mark:0':nil:ok3_0(+(1, n6446_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).
----------------------------------------

(30)
Obligation:
TRS:
Rules:
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok


Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0)
sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0)
terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0)
s(gen_mark:0':nil:ok3_0(+(1, n3121_0))) -> *4_0, rt in Omega(n3121_0)
add(gen_mark:0':nil:ok3_0(+(1, n3988_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3988_0)
dbl(gen_mark:0':nil:ok3_0(+(1, n6446_0))) -> *4_0, rt in Omega(n6446_0)


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


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

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

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
first(gen_mark:0':nil:ok3_0(+(1, n7564_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n7564_0)

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

Induction Step:
first(gen_mark:0':nil:ok3_0(+(1, +(n7564_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(first(gen_mark:0':nil:ok3_0(+(1, n7564_0)), gen_mark: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).
----------------------------------------

(32)
Obligation:
TRS:
Rules:
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) -> mark(0')
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) -> mark(0')
active(dbl(s(X))) -> mark(s(s(dbl(X))))
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(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
terms(mark(X)) -> mark(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
sqr(mark(X)) -> mark(sqr(X))
s(mark(X)) -> mark(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0') -> ok(0')
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
terms(ok(X)) -> ok(terms(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(ok(X)) -> ok(recip(X))
sqr(ok(X)) -> ok(sqr(X))
s(ok(X)) -> ok(s(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:0':nil:ok -> mark:0':nil:ok
terms :: mark:0':nil:ok -> mark:0':nil:ok
mark :: mark:0':nil:ok -> mark:0':nil:ok
cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
recip :: mark:0':nil:ok -> mark:0':nil:ok
sqr :: mark:0':nil:ok -> mark:0':nil:ok
s :: mark:0':nil:ok -> mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
dbl :: mark:0':nil:ok -> mark:0':nil:ok
first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok -> mark:0':nil:ok
ok :: mark:0':nil:ok -> mark:0':nil:ok
top :: mark:0':nil:ok -> top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok


Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0)
sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0)
terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0)
s(gen_mark:0':nil:ok3_0(+(1, n3121_0))) -> *4_0, rt in Omega(n3121_0)
add(gen_mark:0':nil:ok3_0(+(1, n3988_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3988_0)
dbl(gen_mark:0':nil:ok3_0(+(1, n6446_0))) -> *4_0, rt in Omega(n6446_0)
first(gen_mark:0':nil:ok3_0(+(1, n7564_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n7564_0)


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