### (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: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

### (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:

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: FULL

Infered types.

### (6) Obligation:

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

### (7) 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

### (8) Obligation:

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

### (9) 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)) →RΩ(1)
gen_0':s2_0(a)

Induction Step:
+'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) →RΩ(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 ircR ∈ Ω(n).

### (11) Obligation:

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

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

Proved the following rewrite lemma:
double(gen_0':s2_0(n457_0)) → gen_0':s2_0(*(2, n457_0)), rt ∈ Ω(1 + n4570)

Induction Base:
double(gen_0':s2_0(0)) →RΩ(1)
0'

Induction Step:
double(gen_0':s2_0(+(n457_0, 1))) →RΩ(1)
s(s(double(gen_0':s2_0(n457_0)))) →IH
s(s(gen_0':s2_0(*(2, c458_0))))

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

### (14) Obligation:

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(n457_0)) → gen_0':s2_0(*(2, n457_0)), rt ∈ Ω(1 + n4570)

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

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

Proved the following rewrite lemma:
sqr(gen_0':s2_0(n693_0)) → gen_0':s2_0(*(n693_0, n693_0)), rt ∈ Ω(1 + n6930 + n69302)

Induction Base:
sqr(gen_0':s2_0(0)) →RΩ(1)
0'

Induction Step:
sqr(gen_0':s2_0(+(n693_0, 1))) →RΩ(1)
+'(sqr(gen_0':s2_0(n693_0)), s(double(gen_0':s2_0(n693_0)))) →IH
+'(gen_0':s2_0(*(c694_0, c694_0)), s(double(gen_0':s2_0(n693_0)))) →LΩ(1 + n6930)
+'(gen_0':s2_0(*(n693_0, n693_0)), s(gen_0':s2_0(*(2, n693_0)))) →LΩ(2 + 2·n6930)
gen_0':s2_0(+(+(*(2, n693_0), 1), *(n693_0, n693_0)))

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

### (17) Obligation:

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(n457_0)) → gen_0':s2_0(*(2, n457_0)), rt ∈ Ω(1 + n4570)
sqr(gen_0':s2_0(n693_0)) → gen_0':s2_0(*(n693_0, n693_0)), rt ∈ Ω(1 + n6930 + n69302)

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.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
sqr(gen_0':s2_0(n693_0)) → gen_0':s2_0(*(n693_0, n693_0)), rt ∈ Ω(1 + n6930 + n69302)

### (20) Obligation:

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(n457_0)) → gen_0':s2_0(*(2, n457_0)), rt ∈ Ω(1 + n4570)
sqr(gen_0':s2_0(n693_0)) → gen_0':s2_0(*(n693_0, n693_0)), rt ∈ Ω(1 + n6930 + n69302)

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.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
sqr(gen_0':s2_0(n693_0)) → gen_0':s2_0(*(n693_0, n693_0)), rt ∈ Ω(1 + n6930 + n69302)

### (23) Obligation:

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(n457_0)) → gen_0':s2_0(*(2, n457_0)), rt ∈ Ω(1 + n4570)

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.

### (24) 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) Obligation:

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.

### (27) 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)