### (0) Obligation:

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

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
log, le, double, square, plus

They will be analysed ascendingly in the following order:
le < log
double < log
square < log
double < square
plus < square

### (8) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
le, log, double, square, plus

They will be analysed ascendingly in the following order:
le < log
double < log
square < log
double < square
plus < square

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

Proved the following rewrite lemma:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Induction Base:
le(gen_s:0'3_0(0), gen_s:0'3_0(0)) →RΩ(1)
true

Induction Step:
le(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) →RΩ(1)
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) →IH
true

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

### (11) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
double, log, square, plus

They will be analysed ascendingly in the following order:
double < log
square < log
double < square
plus < square

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

Proved the following rewrite lemma:
double(gen_s:0'3_0(n288_0)) → gen_s:0'3_0(*(2, n288_0)), rt ∈ Ω(1 + n2880)

Induction Base:
double(gen_s:0'3_0(0)) →RΩ(1)
0'

Induction Step:
double(gen_s:0'3_0(+(n288_0, 1))) →RΩ(1)
s(s(double(gen_s:0'3_0(n288_0)))) →IH
s(s(gen_s:0'3_0(*(2, c289_0))))

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

### (14) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
double(gen_s:0'3_0(n288_0)) → gen_s:0'3_0(*(2, n288_0)), rt ∈ Ω(1 + n2880)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
plus, log, square

They will be analysed ascendingly in the following order:
square < log
plus < square

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

Proved the following rewrite lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n544_0)) → gen_s:0'3_0(+(n544_0, a)), rt ∈ Ω(1 + n5440)

Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) →RΩ(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n544_0, 1))) →RΩ(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n544_0))) →IH
s(gen_s:0'3_0(+(a, c545_0)))

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

### (17) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
double(gen_s:0'3_0(n288_0)) → gen_s:0'3_0(*(2, n288_0)), rt ∈ Ω(1 + n2880)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n544_0)) → gen_s:0'3_0(+(n544_0, a)), rt ∈ Ω(1 + n5440)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
square, log

They will be analysed ascendingly in the following order:
square < log

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

Proved the following rewrite lemma:
square(gen_s:0'3_0(n1123_0)) → gen_s:0'3_0(*(n1123_0, n1123_0)), rt ∈ Ω(1 + n11230 + n112302)

Induction Base:
square(gen_s:0'3_0(0)) →RΩ(1)
0'

Induction Step:
square(gen_s:0'3_0(+(n1123_0, 1))) →RΩ(1)
s(plus(square(gen_s:0'3_0(n1123_0)), double(gen_s:0'3_0(n1123_0)))) →IH
s(plus(gen_s:0'3_0(*(c1124_0, c1124_0)), double(gen_s:0'3_0(n1123_0)))) →LΩ(1 + n11230)
s(plus(gen_s:0'3_0(*(n1123_0, n1123_0)), gen_s:0'3_0(*(2, n1123_0)))) →LΩ(1 + 2·n11230)
s(gen_s:0'3_0(+(*(2, n1123_0), *(n1123_0, n1123_0))))

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

### (20) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
double(gen_s:0'3_0(n288_0)) → gen_s:0'3_0(*(2, n288_0)), rt ∈ Ω(1 + n2880)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n544_0)) → gen_s:0'3_0(+(n544_0, a)), rt ∈ Ω(1 + n5440)
square(gen_s:0'3_0(n1123_0)) → gen_s:0'3_0(*(n1123_0, n1123_0)), rt ∈ Ω(1 + n11230 + n112302)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
log

### (21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (22) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
double(gen_s:0'3_0(n288_0)) → gen_s:0'3_0(*(2, n288_0)), rt ∈ Ω(1 + n2880)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n544_0)) → gen_s:0'3_0(+(n544_0, a)), rt ∈ Ω(1 + n5440)
square(gen_s:0'3_0(n1123_0)) → gen_s:0'3_0(*(n1123_0, n1123_0)), rt ∈ Ω(1 + n11230 + n112302)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
square(gen_s:0'3_0(n1123_0)) → gen_s:0'3_0(*(n1123_0, n1123_0)), rt ∈ Ω(1 + n11230 + n112302)

### (25) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
double(gen_s:0'3_0(n288_0)) → gen_s:0'3_0(*(2, n288_0)), rt ∈ Ω(1 + n2880)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n544_0)) → gen_s:0'3_0(+(n544_0, a)), rt ∈ Ω(1 + n5440)
square(gen_s:0'3_0(n1123_0)) → gen_s:0'3_0(*(n1123_0, n1123_0)), rt ∈ Ω(1 + n11230 + n112302)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
square(gen_s:0'3_0(n1123_0)) → gen_s:0'3_0(*(n1123_0, n1123_0)), rt ∈ Ω(1 + n11230 + n112302)

### (28) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
double(gen_s:0'3_0(n288_0)) → gen_s:0'3_0(*(2, n288_0)), rt ∈ Ω(1 + n2880)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n544_0)) → gen_s:0'3_0(+(n544_0, a)), rt ∈ Ω(1 + n5440)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

### (31) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
double(gen_s:0'3_0(n288_0)) → gen_s:0'3_0(*(2, n288_0)), rt ∈ Ω(1 + n2880)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

### (34) Obligation:

TRS:
Rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0')
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0', v) → true
le(s(u), 0') → false
le(s(u), s(v)) → le(u, v)
double(0') → 0'
double(s(x)) → s(s(double(x)))
square(0') → 0'
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
log :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
le :: s:0' → s:0' → true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' → s:0'
square :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)