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

(0) Obligation:

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

sqr(0) → 0
sqr(s(x)) → +(sqr(x), s(double(x)))
double(0) → 0
double(s(x)) → s(s(double(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
sqr(s(x)) → s(+(sqr(x), double(x)))

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:

sqr(0') → 0'
sqr(s(x)) → +'(sqr(x), s(double(x)))
double(0') → 0'
double(s(x)) → s(s(double(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
sqr(s(x)) → s(+'(sqr(x), double(x)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
sqr(0') → 0'
sqr(s(x)) → +'(sqr(x), s(double(x)))
double(0') → 0'
double(s(x)) → s(s(double(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
sqr(s(x)) → s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
sqr, +', double

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

(6) Obligation:

Innermost TRS:
Rules:
sqr(0') → 0'
sqr(s(x)) → +'(sqr(x), s(double(x)))
double(0') → 0'
double(s(x)) → s(s(double(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
sqr(s(x)) → s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
+', sqr, double

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

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Induction Base:
+'(gen_0':s2_0(a), gen_0':s2_0(0)) →_R^Ω(1)
gen_0':s2_0(a)

Induction Step:
+'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) →_R^Ω(1)
s(+'(gen_0':s2_0(a), gen_0':s2_0(n4_0))) →_IH
s(gen_0':s2_0(+(a, c5_0)))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
sqr(0') → 0'
sqr(s(x)) → +'(sqr(x), s(double(x)))
double(0') → 0'
double(s(x)) → s(s(double(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
sqr(s(x)) → s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

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

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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
double(gen_0':s2_0(n527_0)) → gen_0':s2_0(*(2, n527_0)), rt ∈ Ω(1 + n527₀)

Induction Base:
double(gen_0':s2_0(0)) →_R^Ω(1)
0'

Induction Step:
double(gen_0':s2_0(+(n527_0, 1))) →_R^Ω(1)
s(s(double(gen_0':s2_0(n527_0)))) →_IH
s(s(gen_0':s2_0(*(2, c528_0))))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
sqr(0') → 0'
sqr(s(x)) → +'(sqr(x), s(double(x)))
double(0') → 0'
double(s(x)) → s(s(double(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
sqr(s(x)) → s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
double(gen_0':s2_0(n527_0)) → gen_0':s2_0(*(2, n527_0)), rt ∈ Ω(1 + n527₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
sqr

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sqr(gen_0':s2_0(n791_0)) → gen_0':s2_0(*(n791_0, n791_0)), rt ∈ Ω(1 + n791₀ + n791₀²)

Induction Base:
sqr(gen_0':s2_0(0)) →_R^Ω(1)
0'

Induction Step:
sqr(gen_0':s2_0(+(n791_0, 1))) →_R^Ω(1)
+'(sqr(gen_0':s2_0(n791_0)), s(double(gen_0':s2_0(n791_0)))) →_IH
+'(gen_0':s2_0(*(c792_0, c792_0)), s(double(gen_0':s2_0(n791_0)))) →_L^{Ω(1 + n791₀)}
+'(gen_0':s2_0(*(n791_0, n791_0)), s(gen_0':s2_0(*(2, n791_0)))) →_L^{Ω(2 + 2·n791₀)}
gen_0':s2_0(+(+(*(2, n791_0), 1), *(n791_0, n791_0)))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
sqr(0') → 0'
sqr(s(x)) → +'(sqr(x), s(double(x)))
double(0') → 0'
double(s(x)) → s(s(double(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
sqr(s(x)) → s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
double(gen_0':s2_0(n527_0)) → gen_0':s2_0(*(2, n527_0)), rt ∈ Ω(1 + n527₀)
sqr(gen_0':s2_0(n791_0)) → gen_0':s2_0(*(n791_0, n791_0)), rt ∈ Ω(1 + n791₀ + n791₀²)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
sqr(gen_0':s2_0(n791_0)) → gen_0':s2_0(*(n791_0, n791_0)), rt ∈ Ω(1 + n791₀ + n791₀²)

(17) BOUNDS(n^2, INF)

(18) Obligation:

Innermost TRS:
Rules:
sqr(0') → 0'
sqr(s(x)) → +'(sqr(x), s(double(x)))
double(0') → 0'
double(s(x)) → s(s(double(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
sqr(s(x)) → s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
double(gen_0':s2_0(n527_0)) → gen_0':s2_0(*(2, n527_0)), rt ∈ Ω(1 + n527₀)
sqr(gen_0':s2_0(n791_0)) → gen_0':s2_0(*(n791_0, n791_0)), rt ∈ Ω(1 + n791₀ + n791₀²)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
sqr(gen_0':s2_0(n791_0)) → gen_0':s2_0(*(n791_0, n791_0)), rt ∈ Ω(1 + n791₀ + n791₀²)

(20) BOUNDS(n^2, INF)

(21) Obligation:

Innermost TRS:
Rules:
sqr(0') → 0'
sqr(s(x)) → +'(sqr(x), s(double(x)))
double(0') → 0'
double(s(x)) → s(s(double(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
sqr(s(x)) → s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
double(gen_0':s2_0(n527_0)) → gen_0':s2_0(*(2, n527_0)), rt ∈ Ω(1 + n527₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

Innermost TRS:
Rules:
sqr(0') → 0'
sqr(s(x)) → +'(sqr(x), s(double(x)))
double(0') → 0'
double(s(x)) → s(s(double(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
sqr(s(x)) → s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(26) BOUNDS(n^1, INF)