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:

minus(X, 0) → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0, s(Y)) → 0
div(s(X), s(Y)) → s(div(minus(X, Y), 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:

minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

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

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

(6) Obligation:

Innermost TRS:
Rules:
minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

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

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

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

Proved the following rewrite lemma:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
minus(gen_0':s2_0(+(1, 0)), gen_0':s2_0(+(1, 0)))

Induction Step:
minus(gen_0':s2_0(+(1, +(n4_0, 1))), gen_0':s2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
p(minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0)))) →_IH
p(*3_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:
minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
div

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

Proved the following rewrite lemma:
div(gen_0':s2_0(n2205_0), gen_0':s2_0(1)) → gen_0':s2_0(n2205_0), rt ∈ Ω(1 + n2205₀)

Induction Base:
div(gen_0':s2_0(0), gen_0':s2_0(1)) →_R^Ω(1)
0'

Induction Step:
div(gen_0':s2_0(+(n2205_0, 1)), gen_0':s2_0(1)) →_R^Ω(1)
s(div(minus(gen_0':s2_0(n2205_0), gen_0':s2_0(0)), s(gen_0':s2_0(0)))) →_R^Ω(1)
s(div(gen_0':s2_0(n2205_0), s(gen_0':s2_0(0)))) →_IH
s(gen_0':s2_0(c2206_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:
minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)
div(gen_0':s2_0(n2205_0), gen_0':s2_0(1)) → gen_0':s2_0(n2205_0), rt ∈ Ω(1 + n2205₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(14) BOUNDS(n^1, INF)

(15) Obligation:

Innermost TRS:
Rules:
minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)
div(gen_0':s2_0(n2205_0), gen_0':s2_0(1)) → gen_0':s2_0(n2205_0), rt ∈ Ω(1 + n2205₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(20) BOUNDS(n^1, INF)