The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0, cons(X, Z)) → mark(X)
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0, cons(X, Z)) → a__quote(X)
a__first1(0, Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0) → 01
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01) → 0
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01) → 01
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
mark(sel(from(X125125_3), X2)) →⁺ a__sel(cons(mark(mark(X125125_3)), from(s(mark(X125125_3)))), mark(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X125125_3 / sel(from(X125125_3), X2)].
The result substitution is [ ].

The rewrite sequence
mark(sel(from(X125125_3), X2)) →⁺ a__sel(cons(mark(mark(X125125_3)), from(s(mark(X125125_3)))), mark(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0].
The pumping substitution is [X125125_3 / sel(from(X125125_3), X2)].
The result substitution is [ ].

(2) BOUNDS(2^n, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
a__sel, mark, a__from, a__sel1, a__quote, a__first1, a__quote1, a__unquote, a__unquote1, a__fcons

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(8) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
mark, a__sel, a__from, a__sel1, a__quote, a__first1, a__quote1, a__unquote, a__unquote1, a__fcons

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0)) →_R^Ω(1)
0'

Induction Step:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(n4_0, 1))) →_R^Ω(1)
s(mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0))) →_IH
s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(c5_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__sel, a__from, a__sel1, a__quote, a__first1, a__quote1, a__unquote, a__unquote1, a__fcons

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__sel.

(13) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__from, a__sel1, a__quote, a__first1, a__quote1, a__unquote, a__unquote1, a__fcons

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__from.

(15) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__sel1, a__quote, a__first1, a__quote1, a__unquote, a__unquote1, a__fcons

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__sel1.

(17) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__quote, a__first1, a__quote1, a__unquote, a__unquote1, a__fcons

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Induction Base:
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, 0)))

Induction Step:
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, +(n14758796_0, 1)))) →_R^Ω(1)
s1(a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0)))) →_IH
s1(*3_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__first1, a__sel, mark, a__from, a__sel1, a__quote1, a__unquote, a__unquote1, a__fcons

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__first1.

(22) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__quote1, a__sel, mark, a__from, a__sel1, a__unquote, a__unquote1, a__fcons

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__quote1.

(24) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__unquote, a__sel, mark, a__from, a__sel1, a__unquote1, a__fcons

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__unquote.

(26) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__unquote1, a__sel, mark, a__from, a__sel1, a__fcons

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__unquote1.

(28) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__fcons, a__sel, mark, a__from, a__sel1

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__fcons.

(30) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
mark, a__sel, a__from, a__sel1

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(31) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0), rt ∈ Ω(1 + n14760349₀)

Induction Base:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0)) →_R^Ω(1)
0'

Induction Step:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(n14760349_0, 1))) →_R^Ω(1)
s(mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0))) →_IH
s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(c14760350_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(32) Complex Obligation (BEST)

(33) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0), rt ∈ Ω(1 + n14760349₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__sel, a__from, a__sel1

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(34) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__sel.

(35) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0), rt ∈ Ω(1 + n14760349₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__from, a__sel1

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(36) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__from.

(37) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0), rt ∈ Ω(1 + n14760349₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

The following defined symbols remain to be analysed:
a__sel1

They will be analysed ascendingly in the following order:
a__sel = mark
a__sel = a__from
a__sel = a__sel1
a__sel = a__quote
a__sel = a__first1
a__sel = a__quote1
a__sel = a__unquote
a__sel = a__unquote1
a__sel = a__fcons
mark = a__from
mark = a__sel1
mark = a__quote
mark = a__first1
mark = a__quote1
mark = a__unquote
mark = a__unquote1
mark = a__fcons
a__from = a__sel1
a__from = a__quote
a__from = a__first1
a__from = a__quote1
a__from = a__unquote
a__from = a__unquote1
a__from = a__fcons
a__sel1 = a__quote
a__sel1 = a__first1
a__sel1 = a__quote1
a__sel1 = a__unquote
a__sel1 = a__unquote1
a__sel1 = a__fcons
a__quote = a__first1
a__quote = a__quote1
a__quote = a__unquote
a__quote = a__unquote1
a__quote = a__fcons
a__first1 = a__quote1
a__first1 = a__unquote
a__first1 = a__unquote1
a__first1 = a__fcons
a__quote1 = a__unquote
a__quote1 = a__unquote1
a__quote1 = a__fcons
a__unquote = a__unquote1
a__unquote = a__fcons
a__unquote1 = a__fcons

(38) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__sel1.

(39) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0), rt ∈ Ω(1 + n14760349₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

No more defined symbols left to analyse.

(40) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0), rt ∈ Ω(1 + n14760349₀)

(41) BOUNDS(n^1, INF)

(42) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0), rt ∈ Ω(1 + n14760349₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

No more defined symbols left to analyse.

(43) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n14760349_0), rt ∈ Ω(1 + n14760349₀)

(44) BOUNDS(n^1, INF)

(45) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)
a__quote(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(1, n14758796_0))) → *3_0, rt ∈ Ω(n14758796₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

No more defined symbols left to analyse.

(46) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)

(47) BOUNDS(n^1, INF)

(48) Obligation:

TRS:
Rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0', cons(X, Z)) → mark(X)
a__first(0', Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0', cons(X, Z)) → a__quote(X)
a__first1(0', Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0') → 01'
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01') → 0'
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01') → 01'
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)

Types:
a__sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
mark :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
0' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
from :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
nil1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
cons1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
01' :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
s1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
a__fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
sel1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
first1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
quote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
unquote1 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
fcons :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
hole_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons1_0 :: s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0 :: Nat → s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons

Lemmas:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(0) ⇔ 0'
gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(+(x, 1)) ⇔ s(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(x))

No more defined symbols left to analyse.

(49) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0)) → gen_s:cons:0':nil:first:from:nil1:cons1:01':s1:sel:sel1:quote:first1:quote1:unquote:unquote1:fcons2_0(n4_0), rt ∈ Ω(1 + n4₀)

(50) BOUNDS(n^1, INF)