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:

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: 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:

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: INNERMOST

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

Infered types.

(4) Obligation:

Innermost 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'

(5) 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

(6) Obligation:

Innermost 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

(7) 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)) →_R^Ω(1)
true

Induction Step:
le(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) →_R^Ω(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 irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost 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

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

Proved the following rewrite lemma:
double(gen_s:0'3_0(n342_0)) → gen_s:0'3_0(*(2, n342_0)), rt ∈ Ω(1 + n342₀)

Induction Base:
double(gen_s:0'3_0(0)) →_R^Ω(1)
0'

Induction Step:
double(gen_s:0'3_0(+(n342_0, 1))) →_R^Ω(1)
s(s(double(gen_s:0'3_0(n342_0)))) →_IH
s(s(gen_s:0'3_0(*(2, c343_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:
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(n342_0)) → gen_s:0'3_0(*(2, n342_0)), rt ∈ Ω(1 + n342₀)

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

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

Proved the following rewrite lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n634_0)) → gen_s:0'3_0(+(n634_0, a)), rt ∈ Ω(1 + n634₀)

Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) →_R^Ω(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n634_0, 1))) →_R^Ω(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n634_0))) →_IH
s(gen_s:0'3_0(+(a, c635_0)))

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost 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(n342_0)) → gen_s:0'3_0(*(2, n342_0)), rt ∈ Ω(1 + n342₀)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n634_0)) → gen_s:0'3_0(+(n634_0, a)), rt ∈ Ω(1 + n634₀)

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

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
square(gen_s:0'3_0(n1303_0)) → gen_s:0'3_0(*(n1303_0, n1303_0)), rt ∈ Ω(1 + n1303₀ + n1303₀²)

Induction Base:
square(gen_s:0'3_0(0)) →_R^Ω(1)
0'

Induction Step:
square(gen_s:0'3_0(+(n1303_0, 1))) →_R^Ω(1)
s(plus(square(gen_s:0'3_0(n1303_0)), double(gen_s:0'3_0(n1303_0)))) →_IH
s(plus(gen_s:0'3_0(*(c1304_0, c1304_0)), double(gen_s:0'3_0(n1303_0)))) →_L^{Ω(1 + n1303₀)}
s(plus(gen_s:0'3_0(*(n1303_0, n1303_0)), gen_s:0'3_0(*(2, n1303_0)))) →_L^{Ω(1 + 2·n1303₀)}
s(gen_s:0'3_0(+(*(2, n1303_0), *(n1303_0, n1303_0))))

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

(17) Complex Obligation (BEST)

(18) Obligation:

Innermost 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(n342_0)) → gen_s:0'3_0(*(2, n342_0)), rt ∈ Ω(1 + n342₀)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n634_0)) → gen_s:0'3_0(+(n634_0, a)), rt ∈ Ω(1 + n634₀)
square(gen_s:0'3_0(n1303_0)) → gen_s:0'3_0(*(n1303_0, n1303_0)), rt ∈ Ω(1 + n1303₀ + n1303₀²)

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

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

Innermost 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(n342_0)) → gen_s:0'3_0(*(2, n342_0)), rt ∈ Ω(1 + n342₀)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n634_0)) → gen_s:0'3_0(+(n634_0, a)), rt ∈ Ω(1 + n634₀)
square(gen_s:0'3_0(n1303_0)) → gen_s:0'3_0(*(n1303_0, n1303_0)), rt ∈ Ω(1 + n1303₀ + n1303₀²)

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.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
square(gen_s:0'3_0(n1303_0)) → gen_s:0'3_0(*(n1303_0, n1303_0)), rt ∈ Ω(1 + n1303₀ + n1303₀²)

(22) BOUNDS(n^2, INF)

(23) Obligation:

Innermost 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(n342_0)) → gen_s:0'3_0(*(2, n342_0)), rt ∈ Ω(1 + n342₀)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n634_0)) → gen_s:0'3_0(+(n634_0, a)), rt ∈ Ω(1 + n634₀)
square(gen_s:0'3_0(n1303_0)) → gen_s:0'3_0(*(n1303_0, n1303_0)), rt ∈ Ω(1 + n1303₀ + n1303₀²)

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.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
square(gen_s:0'3_0(n1303_0)) → gen_s:0'3_0(*(n1303_0, n1303_0)), rt ∈ Ω(1 + n1303₀ + n1303₀²)

(25) BOUNDS(n^2, INF)

(26) Obligation:

Innermost 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(n342_0)) → gen_s:0'3_0(*(2, n342_0)), rt ∈ Ω(1 + n342₀)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n634_0)) → gen_s:0'3_0(+(n634_0, a)), rt ∈ Ω(1 + n634₀)

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.

(27) 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)

(28) BOUNDS(n^1, INF)

(29) Obligation:

Innermost 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(n342_0)) → gen_s:0'3_0(*(2, n342_0)), rt ∈ Ω(1 + n342₀)

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.

(30) 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) BOUNDS(n^1, INF)

(32) Obligation:

Innermost 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.

(33) 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) BOUNDS(n^1, INF)