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__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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

The rewrite sequence
mark(terms(X)) →⁺ cons(recip(a__sqr(mark(mark(X)))), terms(s(mark(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / terms(X)].
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__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: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
cons/1

(6) Obligation:

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

a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
cons :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
recip :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
mark :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
0' :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
s :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
nil :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
terms :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
hole_recip:cons:0':s:nil:terms:sqr:add:dbl:first1_0 :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0 :: Nat → recip:cons:0':s:nil:terms:sqr:add:dbl:first

(9) 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

(10) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
cons :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
recip :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
mark :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
0' :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
s :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
nil :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
terms :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
hole_recip:cons:0':s:nil:terms:sqr:add:dbl:first1_0 :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0 :: Nat → recip:cons:0':s:nil:terms:sqr:add:dbl:first

Generator Equations:
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(0) ⇔ 0'
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(+(x, 1)) ⇔ cons(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(x))

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

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
cons :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
recip :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
mark :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
0' :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
s :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
nil :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
terms :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
hole_recip:cons:0':s:nil:terms:sqr:add:dbl:first1_0 :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0 :: Nat → recip:cons:0':s:nil:terms:sqr:add:dbl:first

Generator Equations:
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(0) ⇔ 0'
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(+(x, 1)) ⇔ cons(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(x))

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

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
cons :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
recip :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
mark :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
0' :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
s :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
nil :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
terms :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
hole_recip:cons:0':s:nil:terms:sqr:add:dbl:first1_0 :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0 :: Nat → recip:cons:0':s:nil:terms:sqr:add:dbl:first

Generator Equations:
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(0) ⇔ 0'
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(+(x, 1)) ⇔ cons(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(x))

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

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0)) → gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0), rt ∈ Ω(1 + n42031₀)

Induction Base:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(0)) →_R^Ω(1)
0'

Induction Step:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(+(n42031_0, 1))) →_R^Ω(1)
cons(mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0))) →_IH
cons(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(c42032_0))

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
cons :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
recip :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
mark :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
0' :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
s :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
nil :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
terms :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
hole_recip:cons:0':s:nil:terms:sqr:add:dbl:first1_0 :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0 :: Nat → recip:cons:0':s:nil:terms:sqr:add:dbl:first

Lemmas:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0)) → gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0), rt ∈ Ω(1 + n42031₀)

Generator Equations:
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(0) ⇔ 0'
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(+(x, 1)) ⇔ cons(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(x))

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

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

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

(19) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
cons :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
recip :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
mark :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
0' :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
s :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
nil :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
terms :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
hole_recip:cons:0':s:nil:terms:sqr:add:dbl:first1_0 :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0 :: Nat → recip:cons:0':s:nil:terms:sqr:add:dbl:first

Lemmas:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0)) → gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0), rt ∈ Ω(1 + n42031₀)

Generator Equations:
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(0) ⇔ 0'
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(+(x, 1)) ⇔ cons(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(x))

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

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

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

(21) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
cons :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
recip :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
mark :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
0' :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
s :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
nil :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
terms :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
hole_recip:cons:0':s:nil:terms:sqr:add:dbl:first1_0 :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0 :: Nat → recip:cons:0':s:nil:terms:sqr:add:dbl:first

Lemmas:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0)) → gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0), rt ∈ Ω(1 + n42031₀)

Generator Equations:
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(0) ⇔ 0'
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(+(x, 1)) ⇔ cons(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(x))

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

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(23) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
cons :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
recip :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
mark :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
0' :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
s :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
nil :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
terms :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
hole_recip:cons:0':s:nil:terms:sqr:add:dbl:first1_0 :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0 :: Nat → recip:cons:0':s:nil:terms:sqr:add:dbl:first

Lemmas:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0)) → gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0), rt ∈ Ω(1 + n42031₀)

Generator Equations:
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(0) ⇔ 0'
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(+(x, 1)) ⇔ cons(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(x))

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

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(25) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
cons :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
recip :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
mark :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
0' :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
s :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
nil :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
terms :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
hole_recip:cons:0':s:nil:terms:sqr:add:dbl:first1_0 :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0 :: Nat → recip:cons:0':s:nil:terms:sqr:add:dbl:first

Lemmas:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0)) → gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0), rt ∈ Ω(1 + n42031₀)

Generator Equations:
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(0) ⇔ 0'
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(+(x, 1)) ⇔ cons(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0)) → gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0), rt ∈ Ω(1 + n42031₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(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)) → cons(mark(Y))
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)) → cons(mark(X1))
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:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
cons :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
recip :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
mark :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
0' :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
s :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
a__first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
nil :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
terms :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
sqr :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
add :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
dbl :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
first :: recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first → recip:cons:0':s:nil:terms:sqr:add:dbl:first
hole_recip:cons:0':s:nil:terms:sqr:add:dbl:first1_0 :: recip:cons:0':s:nil:terms:sqr:add:dbl:first
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0 :: Nat → recip:cons:0':s:nil:terms:sqr:add:dbl:first

Lemmas:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0)) → gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0), rt ∈ Ω(1 + n42031₀)

Generator Equations:
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(0) ⇔ 0'
gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(+(x, 1)) ⇔ cons(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0)) → gen_recip:cons:0':s:nil:terms:sqr:add:dbl:first2_0(n42031_0), rt ∈ Ω(1 + n42031₀)

(30) BOUNDS(n^1, INF)