The Runtime Complexity (innermost) 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:

lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0)
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0

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:

lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0'

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0'

Types:
lt :: 0':s → 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
plus :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
lt, plus, help

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

(6) Obligation:

Innermost TRS:
Rules:
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0'

Types:
lt :: 0':s → 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
plus :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
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:
lt, plus, help

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

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

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

Induction Base:
lt(gen_0':s3_0(0), gen_0':s3_0(0)) →_R^Ω(1)
false

Induction Step:
lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) →_IH
false

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:
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0'

Types:
lt :: 0':s → 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
plus :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, 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:
plus, help

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

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

Proved the following rewrite lemma:
plus(gen_0':s3_0(a), gen_0':s3_0(n306_0)) → gen_0':s3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

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

Induction Step:
plus(gen_0':s3_0(a), gen_0':s3_0(+(n306_0, 1))) →_R^Ω(1)
s(plus(gen_0':s3_0(a), gen_0':s3_0(n306_0))) →_IH
s(gen_0':s3_0(+(a, c307_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:
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0'

Types:
lt :: 0':s → 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
plus :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(a), gen_0':s3_0(n306_0)) → gen_0':s3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

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:
help

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

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

(14) Obligation:

Innermost TRS:
Rules:
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0'

Types:
lt :: 0':s → 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
plus :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(a), gen_0':s3_0(n306_0)) → gen_0':s3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

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.

(15) LowerBoundsProof (EQUIVALENT transformation)

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

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0'

Types:
lt :: 0':s → 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
plus :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(a), gen_0':s3_0(n306_0)) → gen_0':s3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

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

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0'

Types:
lt :: 0':s → 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
plus :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, 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.

(21) LowerBoundsProof (EQUIVALENT transformation)

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

(22) BOUNDS(n^1, INF)