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

(0) Obligation:

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

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Types:
a__terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
cons :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
recip :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
mark :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
s :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
0' :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
nil :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
hole_recip:s:terms:cons:0':nil:first:sqr:add:dbl1_0 :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0 :: Nat → recip:s:terms:cons:0':nil:first:sqr:add:dbl

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
a__terms, a__sqr, mark, a__add, a__dbl

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__add
a__terms = a__dbl
a__sqr = mark
a__sqr = a__add
a__sqr = a__dbl
mark = a__add
mark = a__dbl
a__add = a__dbl

(6) Obligation:

Innermost TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Types:
a__terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
cons :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
recip :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
mark :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
s :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
0' :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
nil :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
hole_recip:s:terms:cons:0':nil:first:sqr:add:dbl1_0 :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0 :: Nat → recip:s:terms:cons:0':nil:first:sqr:add:dbl

Generator Equations:
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(0) ⇔ 0'
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(+(x, 1)) ⇔ cons(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(x), 0')

The following defined symbols remain to be analysed:
a__sqr, a__terms, mark, a__add, a__dbl

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__add
a__terms = a__dbl
a__sqr = mark
a__sqr = a__add
a__sqr = a__dbl
mark = a__add
mark = a__dbl
a__add = a__dbl

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(8) Obligation:

Innermost TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Types:
a__terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
cons :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
recip :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
mark :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
s :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
0' :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
nil :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
hole_recip:s:terms:cons:0':nil:first:sqr:add:dbl1_0 :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0 :: Nat → recip:s:terms:cons:0':nil:first:sqr:add:dbl

Generator Equations:
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(0) ⇔ 0'
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(+(x, 1)) ⇔ cons(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(x), 0')

The following defined symbols remain to be analysed:
a__add, a__terms, mark, a__dbl

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__add
a__terms = a__dbl
a__sqr = mark
a__sqr = a__add
a__sqr = a__dbl
mark = a__add
mark = a__dbl
a__add = a__dbl

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

Innermost TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Types:
a__terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
cons :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
recip :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
mark :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
s :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
0' :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
nil :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
hole_recip:s:terms:cons:0':nil:first:sqr:add:dbl1_0 :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0 :: Nat → recip:s:terms:cons:0':nil:first:sqr:add:dbl

Generator Equations:
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(0) ⇔ 0'
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(+(x, 1)) ⇔ cons(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(x), 0')

The following defined symbols remain to be analysed:
mark, a__terms, a__dbl

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__add
a__terms = a__dbl
a__sqr = mark
a__sqr = a__add
a__sqr = a__dbl
mark = a__add
mark = a__dbl
a__add = a__dbl

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0)) → gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0), rt ∈ Ω(1 + n45911₀)

Induction Base:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(0)) →_R^Ω(1)
0'

Induction Step:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(+(n45911_0, 1))) →_R^Ω(1)
cons(mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0)), 0') →_IH
cons(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(c45912_0), 0')

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Types:
a__terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
cons :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
recip :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
mark :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
s :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
0' :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
nil :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
hole_recip:s:terms:cons:0':nil:first:sqr:add:dbl1_0 :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0 :: Nat → recip:s:terms:cons:0':nil:first:sqr:add:dbl

Lemmas:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0)) → gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0), rt ∈ Ω(1 + n45911₀)

Generator Equations:
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(0) ⇔ 0'
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(+(x, 1)) ⇔ cons(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(x), 0')

The following defined symbols remain to be analysed:
a__terms, a__sqr, a__add, a__dbl

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__add
a__terms = a__dbl
a__sqr = mark
a__sqr = a__add
a__sqr = a__dbl
mark = a__add
mark = a__dbl
a__add = a__dbl

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

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

(15) Obligation:

Innermost TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Types:
a__terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
cons :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
recip :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
mark :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
s :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
0' :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
nil :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
hole_recip:s:terms:cons:0':nil:first:sqr:add:dbl1_0 :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0 :: Nat → recip:s:terms:cons:0':nil:first:sqr:add:dbl

Lemmas:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0)) → gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0), rt ∈ Ω(1 + n45911₀)

Generator Equations:
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(0) ⇔ 0'
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(+(x, 1)) ⇔ cons(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(x), 0')

The following defined symbols remain to be analysed:
a__dbl, a__sqr, a__add

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__add
a__terms = a__dbl
a__sqr = mark
a__sqr = a__add
a__sqr = a__dbl
mark = a__add
mark = a__dbl
a__add = a__dbl

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

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

(17) Obligation:

Innermost TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Types:
a__terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
cons :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
recip :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
mark :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
s :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
0' :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
nil :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
hole_recip:s:terms:cons:0':nil:first:sqr:add:dbl1_0 :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0 :: Nat → recip:s:terms:cons:0':nil:first:sqr:add:dbl

Lemmas:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0)) → gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0), rt ∈ Ω(1 + n45911₀)

Generator Equations:
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(0) ⇔ 0'
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(+(x, 1)) ⇔ cons(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(x), 0')

The following defined symbols remain to be analysed:
a__sqr, a__add

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__add
a__terms = a__dbl
a__sqr = mark
a__sqr = a__add
a__sqr = a__dbl
mark = a__add
mark = a__dbl
a__add = a__dbl

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

Innermost TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Types:
a__terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
cons :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
recip :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
mark :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
s :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
0' :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
nil :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
hole_recip:s:terms:cons:0':nil:first:sqr:add:dbl1_0 :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0 :: Nat → recip:s:terms:cons:0':nil:first:sqr:add:dbl

Lemmas:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0)) → gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0), rt ∈ Ω(1 + n45911₀)

Generator Equations:
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(0) ⇔ 0'
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(+(x, 1)) ⇔ cons(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(x), 0')

The following defined symbols remain to be analysed:
a__add

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__add
a__terms = a__dbl
a__sqr = mark
a__sqr = a__add
a__sqr = a__dbl
mark = a__add
mark = a__dbl
a__add = a__dbl

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(21) Obligation:

Innermost TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Types:
a__terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
cons :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
recip :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
mark :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
s :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
0' :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
nil :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
hole_recip:s:terms:cons:0':nil:first:sqr:add:dbl1_0 :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0 :: Nat → recip:s:terms:cons:0':nil:first:sqr:add:dbl

Lemmas:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0)) → gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0), rt ∈ Ω(1 + n45911₀)

Generator Equations:
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(0) ⇔ 0'
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(+(x, 1)) ⇔ cons(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(x), 0')

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0)) → gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0), rt ∈ Ω(1 + n45911₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

Innermost TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0', X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

Types:
a__terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
cons :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
recip :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
mark :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
terms :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
s :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
0' :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
a__first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
nil :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
first :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
sqr :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
add :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
dbl :: recip:s:terms:cons:0':nil:first:sqr:add:dbl → recip:s:terms:cons:0':nil:first:sqr:add:dbl
hole_recip:s:terms:cons:0':nil:first:sqr:add:dbl1_0 :: recip:s:terms:cons:0':nil:first:sqr:add:dbl
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0 :: Nat → recip:s:terms:cons:0':nil:first:sqr:add:dbl

Lemmas:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0)) → gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0), rt ∈ Ω(1 + n45911₀)

Generator Equations:
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(0) ⇔ 0'
gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(+(x, 1)) ⇔ cons(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(x), 0')

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0)) → gen_recip:s:terms:cons:0':nil:first:sqr:add:dbl2_0(n45911_0), rt ∈ Ω(1 + n45911₀)

(26) BOUNDS(n^1, INF)