WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxRelTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 334 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 178 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 74 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 2082 ms]
        (18) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   APP(nil, z0) -> c
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(nil, z0) -> c3
   ZWADR(z0, nil) -> c4
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))

The (relative) TRS S consists of the following rules:

   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Rewrite Strategy: INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   APP(nil, z0) -> c
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(nil, z0) -> c3
   ZWADR(z0, nil) -> c4
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))

The (relative) TRS S consists of the following rules:

   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
APP, FROM, ZWADR, PREFIX, prefix, app, from, zWadr

They will be analysed ascendingly in the following order:
APP < ZWADR
ZWADR < PREFIX
prefix < PREFIX
zWadr < prefix
app < zWadr

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

(6)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
APP, FROM, ZWADR, PREFIX, prefix, app, from, zWadr

They will be analysed ascendingly in the following order:
APP < ZWADR
ZWADR < PREFIX
prefix < PREFIX
zWadr < prefix
app < zWadr

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b)) -> gen_c:c16_8(n12_8), rt in Omega(1 + n12_8)

Induction Base:
APP(gen_nil:cons:s7_8(0), gen_nil:cons:s7_8(b)) ->_R^Omega(1)
c

Induction Step:
APP(gen_nil:cons:s7_8(+(n12_8, 1)), gen_nil:cons:s7_8(b)) ->_R^Omega(1)
c1(APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b))) ->_IH
c1(gen_c:c16_8(c13_8))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
APP, FROM, ZWADR, PREFIX, prefix, app, from, zWadr

They will be analysed ascendingly in the following order:
APP < ZWADR
ZWADR < PREFIX
prefix < PREFIX
zWadr < prefix
app < zWadr

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Lemmas:
APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b)) -> gen_c:c16_8(n12_8), rt in Omega(1 + n12_8)


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
FROM, ZWADR, PREFIX, prefix, app, from, zWadr

They will be analysed ascendingly in the following order:
ZWADR < PREFIX
prefix < PREFIX
zWadr < prefix
app < zWadr

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ZWADR(gen_nil:cons:s7_8(n632_8), gen_nil:cons:s7_8(n632_8)) -> gen_c3:c4:c5:c69_8(n632_8), rt in Omega(1 + n632_8)

Induction Base:
ZWADR(gen_nil:cons:s7_8(0), gen_nil:cons:s7_8(0)) ->_R^Omega(1)
c3

Induction Step:
ZWADR(gen_nil:cons:s7_8(+(n632_8, 1)), gen_nil:cons:s7_8(+(n632_8, 1))) ->_R^Omega(1)
c6(ZWADR(gen_nil:cons:s7_8(n632_8), gen_nil:cons:s7_8(n632_8))) ->_IH
c6(gen_c3:c4:c5:c69_8(c633_8))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Lemmas:
APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b)) -> gen_c:c16_8(n12_8), rt in Omega(1 + n12_8)
ZWADR(gen_nil:cons:s7_8(n632_8), gen_nil:cons:s7_8(n632_8)) -> gen_c3:c4:c5:c69_8(n632_8), rt in Omega(1 + n632_8)


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
app, PREFIX, prefix, from, zWadr

They will be analysed ascendingly in the following order:
prefix < PREFIX
zWadr < prefix
app < zWadr

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:cons:s7_8(n1521_8), gen_nil:cons:s7_8(b)) -> gen_nil:cons:s7_8(+(n1521_8, b)), rt in Omega(0)

Induction Base:
app(gen_nil:cons:s7_8(0), gen_nil:cons:s7_8(b)) ->_R^Omega(0)
gen_nil:cons:s7_8(b)

Induction Step:
app(gen_nil:cons:s7_8(+(n1521_8, 1)), gen_nil:cons:s7_8(b)) ->_R^Omega(0)
cons(nil, app(gen_nil:cons:s7_8(n1521_8), gen_nil:cons:s7_8(b))) ->_IH
cons(nil, gen_nil:cons:s7_8(+(b, c1522_8)))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Lemmas:
APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b)) -> gen_c:c16_8(n12_8), rt in Omega(1 + n12_8)
ZWADR(gen_nil:cons:s7_8(n632_8), gen_nil:cons:s7_8(n632_8)) -> gen_c3:c4:c5:c69_8(n632_8), rt in Omega(1 + n632_8)
app(gen_nil:cons:s7_8(n1521_8), gen_nil:cons:s7_8(b)) -> gen_nil:cons:s7_8(+(n1521_8, b)), rt in Omega(0)


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
from, PREFIX, prefix, zWadr

They will be analysed ascendingly in the following order:
prefix < PREFIX
zWadr < prefix

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
zWadr(gen_nil:cons:s7_8(n2878_8), gen_nil:cons:s7_8(n2878_8)) -> *11_8, rt in Omega(0)

Induction Base:
zWadr(gen_nil:cons:s7_8(0), gen_nil:cons:s7_8(0))

Induction Step:
zWadr(gen_nil:cons:s7_8(+(n2878_8, 1)), gen_nil:cons:s7_8(+(n2878_8, 1))) ->_R^Omega(0)
cons(app(nil, cons(nil, nil)), zWadr(gen_nil:cons:s7_8(n2878_8), gen_nil:cons:s7_8(n2878_8))) ->_L^Omega(0)
cons(gen_nil:cons:s7_8(+(0, +(0, 1))), zWadr(gen_nil:cons:s7_8(n2878_8), gen_nil:cons:s7_8(n2878_8))) ->_IH
cons(gen_nil:cons:s7_8(1), *11_8)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(18)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Lemmas:
APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b)) -> gen_c:c16_8(n12_8), rt in Omega(1 + n12_8)
ZWADR(gen_nil:cons:s7_8(n632_8), gen_nil:cons:s7_8(n632_8)) -> gen_c3:c4:c5:c69_8(n632_8), rt in Omega(1 + n632_8)
app(gen_nil:cons:s7_8(n1521_8), gen_nil:cons:s7_8(b)) -> gen_nil:cons:s7_8(+(n1521_8, b)), rt in Omega(0)
zWadr(gen_nil:cons:s7_8(n2878_8), gen_nil:cons:s7_8(n2878_8)) -> *11_8, rt in Omega(0)


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
prefix, PREFIX

They will be analysed ascendingly in the following order:
prefix < PREFIX