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:

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0, x, y, s(0))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

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:

plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s → 0':s
help :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
plus, times, exp, ge, towerIter

They will be analysed ascendingly in the following order:
plus < times
times < exp
exp < towerIter
ge < towerIter

(6) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s → 0':s
help :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
plus, times, exp, ge, towerIter

They will be analysed ascendingly in the following order:
plus < times
times < exp
exp < towerIter
ge < towerIter

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

Proved the following rewrite lemma:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Induction Base:
plus(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
plus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
s(plus(gen_0':s3_0(n5_0), gen_0':s3_0(b))) →_IH
s(gen_0':s3_0(+(b, c6_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:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s → 0':s
help :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
times, exp, ge, towerIter

They will be analysed ascendingly in the following order:
times < exp
exp < towerIter
ge < towerIter

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

Proved the following rewrite lemma:
times(gen_0':s3_0(n666_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n666_0, b)), rt ∈ Ω(1 + b·n666₀ + n666₀)

Induction Base:
times(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_0':s3_0(+(n666_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
plus(gen_0':s3_0(b), times(gen_0':s3_0(n666_0), gen_0':s3_0(b))) →_IH
plus(gen_0':s3_0(b), gen_0':s3_0(*(c667_0, b))) →_L^{Ω(1 + b)}
gen_0':s3_0(+(b, *(n666_0, b)))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s → 0':s
help :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
times(gen_0':s3_0(n666_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n666_0, b)), rt ∈ Ω(1 + b·n666₀ + n666₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
exp, ge, towerIter

They will be analysed ascendingly in the following order:
exp < towerIter
ge < towerIter

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s → 0':s
help :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
times(gen_0':s3_0(n666_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n666_0, b)), rt ∈ Ω(1 + b·n666₀ + n666₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
ge, towerIter

They will be analysed ascendingly in the following order:
ge < towerIter

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

Proved the following rewrite lemma:
ge(gen_0':s3_0(n5087_0), gen_0':s3_0(n5087_0)) → true, rt ∈ Ω(1 + n5087₀)

Induction Base:
ge(gen_0':s3_0(0), gen_0':s3_0(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_0':s3_0(+(n5087_0, 1)), gen_0':s3_0(+(n5087_0, 1))) →_R^Ω(1)
ge(gen_0':s3_0(n5087_0), gen_0':s3_0(n5087_0)) →_IH
true

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

(16) Complex Obligation (BEST)

(17) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s → 0':s
help :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
times(gen_0':s3_0(n666_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n666_0, b)), rt ∈ Ω(1 + b·n666₀ + n666₀)
ge(gen_0':s3_0(n5087_0), gen_0':s3_0(n5087_0)) → true, rt ∈ Ω(1 + n5087₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
towerIter

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s → 0':s
help :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
times(gen_0':s3_0(n666_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n666_0, b)), rt ∈ Ω(1 + b·n666₀ + n666₀)
ge(gen_0':s3_0(n5087_0), gen_0':s3_0(n5087_0)) → true, rt ∈ Ω(1 + n5087₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s3_0(n666_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n666_0, b)), rt ∈ Ω(1 + b·n666₀ + n666₀)

(21) BOUNDS(n^2, INF)

(22) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s → 0':s
help :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
times(gen_0':s3_0(n666_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n666_0, b)), rt ∈ Ω(1 + b·n666₀ + n666₀)
ge(gen_0':s3_0(n5087_0), gen_0':s3_0(n5087_0)) → true, rt ∈ Ω(1 + n5087₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s3_0(n666_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n666_0, b)), rt ∈ Ω(1 + b·n666₀ + n666₀)

(24) BOUNDS(n^2, INF)

(25) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s → 0':s
help :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
times(gen_0':s3_0(n666_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n666_0, b)), rt ∈ Ω(1 + b·n666₀ + n666₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s3_0(n666_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n666_0, b)), rt ∈ Ω(1 + b·n666₀ + n666₀)

(27) BOUNDS(n^2, INF)

(28) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0', x, y, s(0'))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s → 0':s
help :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(30) BOUNDS(n^1, INF)