The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).

(0) Obligation:

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

terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0) → 0
half(s(0)) → 0
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
sqr(s(X)) →⁺ s(add(sqr(X), dbl(X)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X / s(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, 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:

terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: 0':s → cons:nil
cons :: recip → cons:nil
recip :: 0':s → recip
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
dbl :: 0':s → 0':s
first :: 0':s → cons:nil → cons:nil
nil :: cons:nil
half :: 0':s → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat → 0':s

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

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

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

(8) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: 0':s → cons:nil
cons :: recip → cons:nil
recip :: 0':s → recip
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
dbl :: 0':s → 0':s
first :: 0':s → cons:nil → cons:nil
nil :: cons:nil
half :: 0':s → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

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

Proved the following rewrite lemma:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Induction Base:
add(gen_0':s4_0(0), gen_0':s4_0(b)) →_R^Ω(1)
gen_0':s4_0(b)

Induction Step:
add(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(b)) →_R^Ω(1)
s(add(gen_0':s4_0(n6_0), gen_0':s4_0(b))) →_IH
s(gen_0':s4_0(+(b, c7_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:
terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: 0':s → cons:nil
cons :: recip → cons:nil
recip :: 0':s → recip
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
dbl :: 0':s → 0':s
first :: 0':s → cons:nil → cons:nil
nil :: cons:nil
half :: 0':s → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat → 0':s

Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
dbl(gen_0':s4_0(n567_0)) → gen_0':s4_0(*(2, n567_0)), rt ∈ Ω(1 + n567₀)

Induction Base:
dbl(gen_0':s4_0(0)) →_R^Ω(1)
0'

Induction Step:
dbl(gen_0':s4_0(+(n567_0, 1))) →_R^Ω(1)
s(s(dbl(gen_0':s4_0(n567_0)))) →_IH
s(s(gen_0':s4_0(*(2, c568_0))))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: 0':s → cons:nil
cons :: recip → cons:nil
recip :: 0':s → recip
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
dbl :: 0':s → 0':s
first :: 0':s → cons:nil → cons:nil
nil :: cons:nil
half :: 0':s → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat → 0':s

Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
dbl(gen_0':s4_0(n567_0)) → gen_0':s4_0(*(2, n567_0)), rt ∈ Ω(1 + n567₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

Proved the following rewrite lemma:
sqr(gen_0':s4_0(n827_0)) → gen_0':s4_0(*(n827_0, n827_0)), rt ∈ Ω(1 + n827₀ + n827₀² + n827₀³)

Induction Base:
sqr(gen_0':s4_0(0)) →_R^Ω(1)
0'

Induction Step:
sqr(gen_0':s4_0(+(n827_0, 1))) →_R^Ω(1)
s(add(sqr(gen_0':s4_0(n827_0)), dbl(gen_0':s4_0(n827_0)))) →_IH
s(add(gen_0':s4_0(*(c828_0, c828_0)), dbl(gen_0':s4_0(n827_0)))) →_L^{Ω(1 + n827₀)}
s(add(gen_0':s4_0(*(n827_0, n827_0)), gen_0':s4_0(*(2, n827_0)))) →_L^{Ω(1 + n827₀2)}
s(gen_0':s4_0(+(*(n827_0, n827_0), *(2, n827_0))))

We have rt ∈ Ω(n³) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n³).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: 0':s → cons:nil
cons :: recip → cons:nil
recip :: 0':s → recip
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
dbl :: 0':s → 0':s
first :: 0':s → cons:nil → cons:nil
nil :: cons:nil
half :: 0':s → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat → 0':s

Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
dbl(gen_0':s4_0(n567_0)) → gen_0':s4_0(*(2, n567_0)), rt ∈ Ω(1 + n567₀)
sqr(gen_0':s4_0(n827_0)) → gen_0':s4_0(*(n827_0, n827_0)), rt ∈ Ω(1 + n827₀ + n827₀² + n827₀³)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
half

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

Proved the following rewrite lemma:
half(gen_0':s4_0(*(2, n1160_0))) → gen_0':s4_0(n1160_0), rt ∈ Ω(1 + n1160₀)

Induction Base:
half(gen_0':s4_0(*(2, 0))) →_R^Ω(1)
0'

Induction Step:
half(gen_0':s4_0(*(2, +(n1160_0, 1)))) →_R^Ω(1)
s(half(gen_0':s4_0(*(2, n1160_0)))) →_IH
s(gen_0':s4_0(c1161_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:
terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: 0':s → cons:nil
cons :: recip → cons:nil
recip :: 0':s → recip
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
dbl :: 0':s → 0':s
first :: 0':s → cons:nil → cons:nil
nil :: cons:nil
half :: 0':s → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat → 0':s

Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
dbl(gen_0':s4_0(n567_0)) → gen_0':s4_0(*(2, n567_0)), rt ∈ Ω(1 + n567₀)
sqr(gen_0':s4_0(n827_0)) → gen_0':s4_0(*(n827_0, n827_0)), rt ∈ Ω(1 + n827₀ + n827₀² + n827₀³)
half(gen_0':s4_0(*(2, n1160_0))) → gen_0':s4_0(n1160_0), rt ∈ Ω(1 + n1160₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
sqr(gen_0':s4_0(n827_0)) → gen_0':s4_0(*(n827_0, n827_0)), rt ∈ Ω(1 + n827₀ + n827₀² + n827₀³)

(22) BOUNDS(n^3, INF)

(23) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: 0':s → cons:nil
cons :: recip → cons:nil
recip :: 0':s → recip
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
dbl :: 0':s → 0':s
first :: 0':s → cons:nil → cons:nil
nil :: cons:nil
half :: 0':s → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat → 0':s

Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
dbl(gen_0':s4_0(n567_0)) → gen_0':s4_0(*(2, n567_0)), rt ∈ Ω(1 + n567₀)
sqr(gen_0':s4_0(n827_0)) → gen_0':s4_0(*(n827_0, n827_0)), rt ∈ Ω(1 + n827₀ + n827₀² + n827₀³)
half(gen_0':s4_0(*(2, n1160_0))) → gen_0':s4_0(n1160_0), rt ∈ Ω(1 + n1160₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
sqr(gen_0':s4_0(n827_0)) → gen_0':s4_0(*(n827_0, n827_0)), rt ∈ Ω(1 + n827₀ + n827₀² + n827₀³)

(25) BOUNDS(n^3, INF)

(26) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: 0':s → cons:nil
cons :: recip → cons:nil
recip :: 0':s → recip
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
dbl :: 0':s → 0':s
first :: 0':s → cons:nil → cons:nil
nil :: cons:nil
half :: 0':s → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat → 0':s

Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
dbl(gen_0':s4_0(n567_0)) → gen_0':s4_0(*(2, n567_0)), rt ∈ Ω(1 + n567₀)
sqr(gen_0':s4_0(n827_0)) → gen_0':s4_0(*(n827_0, n827_0)), rt ∈ Ω(1 + n827₀ + n827₀² + n827₀³)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
sqr(gen_0':s4_0(n827_0)) → gen_0':s4_0(*(n827_0, n827_0)), rt ∈ Ω(1 + n827₀ + n827₀² + n827₀³)

(28) BOUNDS(n^3, INF)

(29) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: 0':s → cons:nil
cons :: recip → cons:nil
recip :: 0':s → recip
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
dbl :: 0':s → 0':s
first :: 0':s → cons:nil → cons:nil
nil :: cons:nil
half :: 0':s → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat → 0':s

Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
dbl(gen_0':s4_0(n567_0)) → gen_0':s4_0(*(2, n567_0)), rt ∈ Ω(1 + n567₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(31) BOUNDS(n^1, INF)

(32) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(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)) → cons(Y)
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: 0':s → cons:nil
cons :: recip → cons:nil
recip :: 0':s → recip
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
dbl :: 0':s → 0':s
first :: 0':s → cons:nil → cons:nil
nil :: cons:nil
half :: 0':s → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat → 0':s

Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(34) BOUNDS(n^1, INF)