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:

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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
minus(s(X), s(Y)) →⁺ p(minus(X, Y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / s(X), Y / s(Y)].
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:

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: FULL

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

Infered types.

(6) Obligation:

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

(7) 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

(8) Obligation:

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

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

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:
div

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
div(gen_0':s2_0(n1451_0), gen_0':s2_0(1)) → gen_0':s2_0(n1451_0), rt ∈ Ω(1 + n1451₀)

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

Induction Step:
div(gen_0':s2_0(+(n1451_0, 1)), gen_0':s2_0(1)) →_R^Ω(1)
s(div(minus(gen_0':s2_0(n1451_0), gen_0':s2_0(0)), s(gen_0':s2_0(0)))) →_R^Ω(1)
s(div(gen_0':s2_0(n1451_0), s(gen_0':s2_0(0)))) →_IH
s(gen_0':s2_0(c1452_0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

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:
div(gen_0':s2_0(n1451_0), gen_0':s2_0(1)) → gen_0':s2_0(n1451_0), rt ∈ Ω(1 + n1451₀)

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.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
div(gen_0':s2_0(n1451_0), gen_0':s2_0(1)) → gen_0':s2_0(n1451_0), rt ∈ Ω(1 + n1451₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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:
div(gen_0':s2_0(n1451_0), gen_0':s2_0(1)) → gen_0':s2_0(n1451_0), rt ∈ Ω(1 + n1451₀)

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.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
div(gen_0':s2_0(n1451_0), gen_0':s2_0(1)) → gen_0':s2_0(n1451_0), rt ∈ Ω(1 + n1451₀)

(18) BOUNDS(n^1, INF)