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:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(0, s(y)) → 0
minus(s(x), s(y)) → minus(x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0) → 0
mod(x, s(y)) → help(x, s(y), 0)
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

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:

le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(x, 0') → x
minus(0', s(y)) → 0'
minus(s(x), s(y)) → minus(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0') → 0'
mod(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(x, 0') → x
minus(0', s(y)) → 0'
minus(s(x), s(y)) → minus(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0') → 0'
mod(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
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:
le, minus, plus, help

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

(6) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(x, 0') → x
minus(0', s(y)) → 0'
minus(s(x), s(y)) → minus(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0') → 0'
mod(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
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:
le, minus, plus, help

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

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

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

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

Induction Step:
le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
le(gen_0':s3_0(n5_0), gen_0':s3_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:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(x, 0') → x
minus(0', s(y)) → 0'
minus(s(x), s(y)) → minus(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0') → 0'
mod(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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:
minus, plus, help

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

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

Proved the following rewrite lemma:
minus(gen_0':s3_0(n354_0), gen_0':s3_0(n354_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n354₀)

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

Induction Step:
minus(gen_0':s3_0(+(n354_0, 1)), gen_0':s3_0(+(n354_0, 1))) →_R^Ω(1)
minus(gen_0':s3_0(n354_0), gen_0':s3_0(n354_0)) →_IH
gen_0':s3_0(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:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(x, 0') → x
minus(0', s(y)) → 0'
minus(s(x), s(y)) → minus(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0') → 0'
mod(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
minus(gen_0':s3_0(n354_0), gen_0':s3_0(n354_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n354₀)

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

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

Proved the following rewrite lemma:
plus(gen_0':s3_0(a), gen_0':s3_0(n846_0)) → gen_0':s3_0(+(n846_0, a)), rt ∈ Ω(1 + n846₀)

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(+(n846_0, 1))) →_R^Ω(1)
s(plus(gen_0':s3_0(a), gen_0':s3_0(n846_0))) →_IH
s(gen_0':s3_0(+(a, c847_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:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(x, 0') → x
minus(0', s(y)) → 0'
minus(s(x), s(y)) → minus(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0') → 0'
mod(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
minus(gen_0':s3_0(n354_0), gen_0':s3_0(n354_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n354₀)
plus(gen_0':s3_0(a), gen_0':s3_0(n846_0)) → gen_0':s3_0(+(n846_0, a)), rt ∈ Ω(1 + n846₀)

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

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(x, 0') → x
minus(0', s(y)) → 0'
minus(s(x), s(y)) → minus(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0') → 0'
mod(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
minus(gen_0':s3_0(n354_0), gen_0':s3_0(n354_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n354₀)
plus(gen_0':s3_0(a), gen_0':s3_0(n846_0)) → gen_0':s3_0(+(n846_0, a)), rt ∈ Ω(1 + n846₀)

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:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(x, 0') → x
minus(0', s(y)) → 0'
minus(s(x), s(y)) → minus(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0') → 0'
mod(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
minus(gen_0':s3_0(n354_0), gen_0':s3_0(n354_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n354₀)
plus(gen_0':s3_0(a), gen_0':s3_0(n846_0)) → gen_0':s3_0(+(n846_0, a)), rt ∈ Ω(1 + n846₀)

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:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(x, 0') → x
minus(0', s(y)) → 0'
minus(s(x), s(y)) → minus(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0') → 0'
mod(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
minus(gen_0':s3_0(n354_0), gen_0':s3_0(n354_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n354₀)

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.

(24) LowerBoundsProof (EQUIVALENT transformation)

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

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(x, 0') → x
minus(0', s(y)) → 0'
minus(s(x), s(y)) → minus(x, y)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
mod(s(x), 0') → 0'
mod(x, s(y)) → help(x, s(y), 0')
help(x, s(y), c) → if(le(c, x), x, s(y), c)
if(true, x, s(y), c) → help(x, s(y), plus(c, s(y)))
if(false, x, s(y), c) → minus(x, minus(c, s(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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.

(27) LowerBoundsProof (EQUIVALENT transformation)

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

(28) BOUNDS(n^1, INF)