The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

ge(0, 0) → true
ge(s(x), 0) → ge(x, 0)
ge(0, s(0)) → false
ge(0, s(s(x))) → ge(0, s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0, 0) → 0
minus(0, s(x)) → minus(0, x)
minus(s(x), 0) → s(minus(x, 0))
minus(s(x), s(y)) → minus(x, y)
plus(0, 0) → 0
plus(0, s(x)) → s(plus(0, x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0)), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
ge(s(x), 0) →⁺ ge(x, 0)
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 [ ].

(2) BOUNDS(n^1, INF)

(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:

ge(0', 0') → true
ge(s(x), 0') → ge(x, 0')
ge(0', s(0')) → false
ge(0', s(s(x))) → ge(0', s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0', 0') → 0'
minus(0', s(x)) → minus(0', x)
minus(s(x), 0') → s(minus(x, 0'))
minus(s(x), s(y)) → minus(x, y)
plus(0', 0') → 0'
plus(0', s(x)) → s(plus(0', x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0')), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0'
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
ge(0', 0') → true
ge(s(x), 0') → ge(x, 0')
ge(0', s(0')) → false
ge(0', s(s(x))) → ge(0', s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0', 0') → 0'
minus(0', s(x)) → minus(0', x)
minus(s(x), 0') → s(minus(x, 0'))
minus(s(x), s(y)) → minus(x, y)
plus(0', 0') → 0'
plus(0', s(x)) → s(plus(0', x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0')), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0'
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

Types:
ge :: 0':s:divByZeroError → 0':s:divByZeroError → true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError → 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
plus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
div :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
ify :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat → 0':s:divByZeroError

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

Heuristically decided to analyse the following defined symbols:
ge, minus, plus, div

They will be analysed ascendingly in the following order:
ge < div
minus < div
plus < div

(8) Obligation:

TRS:
Rules:
ge(0', 0') → true
ge(s(x), 0') → ge(x, 0')
ge(0', s(0')) → false
ge(0', s(s(x))) → ge(0', s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0', 0') → 0'
minus(0', s(x)) → minus(0', x)
minus(s(x), 0') → s(minus(x, 0'))
minus(s(x), s(y)) → minus(x, y)
plus(0', 0') → 0'
plus(0', s(x)) → s(plus(0', x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0')), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0'
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

Types:
ge :: 0':s:divByZeroError → 0':s:divByZeroError → true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError → 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
plus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
div :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
ify :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat → 0':s:divByZeroError

Generator Equations:
gen_0':s:divByZeroError3_0(0) ⇔ 0'
gen_0':s:divByZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:divByZeroError3_0(x))

The following defined symbols remain to be analysed:
ge, minus, plus, div

They will be analysed ascendingly in the following order:
ge < div
minus < div
plus < div

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

Proved the following rewrite lemma:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
ge(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_0':s:divByZeroError3_0(+(n5_0, 1)), gen_0':s:divByZeroError3_0(0)) →_R^Ω(1)
ge(gen_0':s:divByZeroError3_0(n5_0), 0') →_IH
true

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
ge(0', 0') → true
ge(s(x), 0') → ge(x, 0')
ge(0', s(0')) → false
ge(0', s(s(x))) → ge(0', s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0', 0') → 0'
minus(0', s(x)) → minus(0', x)
minus(s(x), 0') → s(minus(x, 0'))
minus(s(x), s(y)) → minus(x, y)
plus(0', 0') → 0'
plus(0', s(x)) → s(plus(0', x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0')), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0'
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

Types:
ge :: 0':s:divByZeroError → 0':s:divByZeroError → true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError → 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
plus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
div :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
ify :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat → 0':s:divByZeroError

Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:divByZeroError3_0(0) ⇔ 0'
gen_0':s:divByZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:divByZeroError3_0(x))

The following defined symbols remain to be analysed:
minus, plus, div

They will be analysed ascendingly in the following order:
minus < div
plus < div

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

Proved the following rewrite lemma:
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n7130_0)) → gen_0':s:divByZeroError3_0(0), rt ∈ Ω(1 + n7130₀)

Induction Base:
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(0)) →_R^Ω(1)
0'

Induction Step:
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(+(n7130_0, 1))) →_R^Ω(1)
minus(0', gen_0':s:divByZeroError3_0(n7130_0)) →_IH
gen_0':s:divByZeroError3_0(0)

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
ge(0', 0') → true
ge(s(x), 0') → ge(x, 0')
ge(0', s(0')) → false
ge(0', s(s(x))) → ge(0', s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0', 0') → 0'
minus(0', s(x)) → minus(0', x)
minus(s(x), 0') → s(minus(x, 0'))
minus(s(x), s(y)) → minus(x, y)
plus(0', 0') → 0'
plus(0', s(x)) → s(plus(0', x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0')), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0'
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

Types:
ge :: 0':s:divByZeroError → 0':s:divByZeroError → true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError → 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
plus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
div :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
ify :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat → 0':s:divByZeroError

Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n7130_0)) → gen_0':s:divByZeroError3_0(0), rt ∈ Ω(1 + n7130₀)

Generator Equations:
gen_0':s:divByZeroError3_0(0) ⇔ 0'
gen_0':s:divByZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:divByZeroError3_0(x))

The following defined symbols remain to be analysed:
plus, div

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

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

Proved the following rewrite lemma:
plus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n10596_0)) → gen_0':s:divByZeroError3_0(n10596_0), rt ∈ Ω(1 + n10596₀)

Induction Base:
plus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(0)) →_R^Ω(1)
0'

Induction Step:
plus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(+(n10596_0, 1))) →_R^Ω(1)
s(plus(0', gen_0':s:divByZeroError3_0(n10596_0))) →_IH
s(gen_0':s:divByZeroError3_0(c10597_0))

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
ge(0', 0') → true
ge(s(x), 0') → ge(x, 0')
ge(0', s(0')) → false
ge(0', s(s(x))) → ge(0', s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0', 0') → 0'
minus(0', s(x)) → minus(0', x)
minus(s(x), 0') → s(minus(x, 0'))
minus(s(x), s(y)) → minus(x, y)
plus(0', 0') → 0'
plus(0', s(x)) → s(plus(0', x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0')), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0'
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

Types:
ge :: 0':s:divByZeroError → 0':s:divByZeroError → true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError → 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
plus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
div :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
ify :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat → 0':s:divByZeroError

Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n7130_0)) → gen_0':s:divByZeroError3_0(0), rt ∈ Ω(1 + n7130₀)
plus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n10596_0)) → gen_0':s:divByZeroError3_0(n10596_0), rt ∈ Ω(1 + n10596₀)

Generator Equations:
gen_0':s:divByZeroError3_0(0) ⇔ 0'
gen_0':s:divByZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:divByZeroError3_0(x))

The following defined symbols remain to be analysed:
div

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

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

(19) Obligation:

TRS:
Rules:
ge(0', 0') → true
ge(s(x), 0') → ge(x, 0')
ge(0', s(0')) → false
ge(0', s(s(x))) → ge(0', s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0', 0') → 0'
minus(0', s(x)) → minus(0', x)
minus(s(x), 0') → s(minus(x, 0'))
minus(s(x), s(y)) → minus(x, y)
plus(0', 0') → 0'
plus(0', s(x)) → s(plus(0', x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0')), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0'
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

Types:
ge :: 0':s:divByZeroError → 0':s:divByZeroError → true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError → 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
plus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
div :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
ify :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat → 0':s:divByZeroError

Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n7130_0)) → gen_0':s:divByZeroError3_0(0), rt ∈ Ω(1 + n7130₀)
plus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n10596_0)) → gen_0':s:divByZeroError3_0(n10596_0), rt ∈ Ω(1 + n10596₀)

Generator Equations:
gen_0':s:divByZeroError3_0(0) ⇔ 0'
gen_0':s:divByZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:divByZeroError3_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
ge(0', 0') → true
ge(s(x), 0') → ge(x, 0')
ge(0', s(0')) → false
ge(0', s(s(x))) → ge(0', s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0', 0') → 0'
minus(0', s(x)) → minus(0', x)
minus(s(x), 0') → s(minus(x, 0'))
minus(s(x), s(y)) → minus(x, y)
plus(0', 0') → 0'
plus(0', s(x)) → s(plus(0', x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0')), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0'
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

Types:
ge :: 0':s:divByZeroError → 0':s:divByZeroError → true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError → 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
plus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
div :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
ify :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat → 0':s:divByZeroError

Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n7130_0)) → gen_0':s:divByZeroError3_0(0), rt ∈ Ω(1 + n7130₀)
plus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n10596_0)) → gen_0':s:divByZeroError3_0(n10596_0), rt ∈ Ω(1 + n10596₀)

Generator Equations:
gen_0':s:divByZeroError3_0(0) ⇔ 0'
gen_0':s:divByZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:divByZeroError3_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)

(24) BOUNDS(n^1, INF)

(25) Obligation:

TRS:
Rules:
ge(0', 0') → true
ge(s(x), 0') → ge(x, 0')
ge(0', s(0')) → false
ge(0', s(s(x))) → ge(0', s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0', 0') → 0'
minus(0', s(x)) → minus(0', x)
minus(s(x), 0') → s(minus(x, 0'))
minus(s(x), s(y)) → minus(x, y)
plus(0', 0') → 0'
plus(0', s(x)) → s(plus(0', x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0')), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0'
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

Types:
ge :: 0':s:divByZeroError → 0':s:divByZeroError → true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError → 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
plus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
div :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
ify :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat → 0':s:divByZeroError

Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n7130_0)) → gen_0':s:divByZeroError3_0(0), rt ∈ Ω(1 + n7130₀)

Generator Equations:
gen_0':s:divByZeroError3_0(0) ⇔ 0'
gen_0':s:divByZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:divByZeroError3_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

TRS:
Rules:
ge(0', 0') → true
ge(s(x), 0') → ge(x, 0')
ge(0', s(0')) → false
ge(0', s(s(x))) → ge(0', s(x))
ge(s(x), s(y)) → ge(x, y)
minus(0', 0') → 0'
minus(0', s(x)) → minus(0', x)
minus(s(x), 0') → s(minus(x, 0'))
minus(s(x), s(y)) → minus(x, y)
plus(0', 0') → 0'
plus(0', s(x)) → s(plus(0', x))
plus(s(x), y) → s(plus(x, y))
div(x, y) → ify(ge(y, s(0')), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0'
if(true, x, y) → s(div(minus(x, y), y))
div(plus(x, y), z) → plus(div(x, z), div(y, z))

Types:
ge :: 0':s:divByZeroError → 0':s:divByZeroError → true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError → 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
plus :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
div :: 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
ify :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false → 0':s:divByZeroError → 0':s:divByZeroError → 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat → 0':s:divByZeroError

Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:divByZeroError3_0(0) ⇔ 0'
gen_0':s:divByZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:divByZeroError3_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) → true, rt ∈ Ω(1 + n5₀)

(30) BOUNDS(n^1, INF)