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

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
half(0) → 0
half(s(0)) → 0
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

terms'(N) → cons'(recip'(sqr'(N)), n__terms'(n__s'(N)))
sqr'(0') → 0'
sqr'(s'(X)) → s'(add'(sqr'(X), dbl'(X)))
dbl'(0') → 0'
dbl'(s'(X)) → s'(s'(dbl'(X)))
add'(0', X) → X
add'(s'(X), Y) → s'(add'(X, Y))
first'(0', X) → nil'
first'(s'(X), cons'(Y, Z)) → cons'(Y, n__first'(X, activate'(Z)))
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(X))) → s'(half'(X))
half'(dbl'(X)) → X
terms'(X) → n__terms'(X)
s'(X) → n__s'(X)
first'(X1, X2) → n__first'(X1, X2)
activate'(n__terms'(X)) → terms'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(n__first'(X1, X2)) → first'(activate'(X1), activate'(X2))
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
terms'(N) → cons'(recip'(sqr'(N)), n__terms'(n__s'(N)))
sqr'(0') → 0'
sqr'(s'(X)) → s'(add'(sqr'(X), dbl'(X)))
dbl'(0') → 0'
dbl'(s'(X)) → s'(s'(dbl'(X)))
add'(0', X) → X
add'(s'(X), Y) → s'(add'(X, Y))
first'(0', X) → nil'
first'(s'(X), cons'(Y, Z)) → cons'(Y, n__first'(X, activate'(Z)))
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(X))) → s'(half'(X))
half'(dbl'(X)) → X
terms'(X) → n__terms'(X)
s'(X) → n__s'(X)
first'(X1, X2) → n__first'(X1, X2)
activate'(n__terms'(X)) → terms'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(n__first'(X1, X2)) → first'(activate'(X1), activate'(X2))
activate'(X) → X

Types:
terms' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
cons' :: recip' → n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
recip' :: n__s':n__terms':cons':0':nil':n__first' → recip'
sqr' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
n__terms' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
n__s' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
0' :: n__s':n__terms':cons':0':nil':n__first'
s' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
add' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
dbl' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
first' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
nil' :: n__s':n__terms':cons':0':nil':n__first'
n__first' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
activate' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
half' :: n__s':n__terms':cons':0':nil':n__first' → n__s':n__terms':cons':0':nil':n__first'
_hole_n__s':n__terms':cons':0':nil':n__first'1 :: n__s':n__terms':cons':0':nil':n__first'
_hole_recip'2 :: recip'
_gen_n__s':n__terms':cons':0':nil':n__first'3 :: Nat → n__s':n__terms':cons':0':nil':n__first'

Heuristically decided to analyse the following defined symbols:
sqr', add', dbl', activate', half'

They will be analysed ascendingly in the following order:
add' < sqr'
dbl' < sqr'

Generator Equations:
_gen_n__s':n__terms':cons':0':nil':n__first'3(0) ⇔ 0'
_gen_n__s':n__terms':cons':0':nil':n__first'3(+(x, 1)) ⇔ cons'(recip'(0'), _gen_n__s':n__terms':cons':0':nil':n__first'3(x))

The following defined symbols remain to be analysed:
add', sqr', dbl', activate', half'

They will be analysed ascendingly in the following order:
add' < sqr'
dbl' < sqr'

Could not prove a rewrite lemma for the defined symbol add'.

The following defined symbols remain to be analysed:
dbl', sqr', activate', half'

They will be analysed ascendingly in the following order:
dbl' < sqr'

Could not prove a rewrite lemma for the defined symbol dbl'.

The following defined symbols remain to be analysed:
sqr', activate', half'

Could not prove a rewrite lemma for the defined symbol sqr'.

The following defined symbols remain to be analysed:
activate', half'

Could not prove a rewrite lemma for the defined symbol activate'.

The following defined symbols remain to be analysed:
half'

Could not prove a rewrite lemma for the defined symbol half'.

No more defined symbols left to analyse.