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:

pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(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(x, s(y)) →⁺ pred(minus(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [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:

pred(s(x)) → x
minus(x, 0') → x
minus(x, s(y)) → pred(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
pred(s(x)) → x
minus(x, 0') → x
minus(x, s(y)) → pred(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))

Types:
pred :: s:0' → s:0'
s :: s:0' → s:0'
minus :: s:0' → s:0' → s:0'
0' :: s:0'
quot :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

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

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

(8) Obligation:

TRS:
Rules:
pred(s(x)) → x
minus(x, 0') → x
minus(x, s(y)) → pred(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))

Types:
pred :: s:0' → s:0'
s :: s:0' → s:0'
minus :: s:0' → s:0' → s:0'
0' :: s:0'
quot :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

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

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

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

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

(10) Obligation:

TRS:
Rules:
pred(s(x)) → x
minus(x, 0') → x
minus(x, s(y)) → pred(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))

Types:
pred :: s:0' → s:0'
s :: s:0' → s:0'
minus :: s:0' → s:0' → s:0'
0' :: s:0'
quot :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
quot

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

Proved the following rewrite lemma:
quot(gen_s:0'2_0(n1423_0), gen_s:0'2_0(1)) → gen_s:0'2_0(n1423_0), rt ∈ Ω(1 + n1423₀)

Induction Base:
quot(gen_s:0'2_0(0), gen_s:0'2_0(1)) →_R^Ω(1)
0'

Induction Step:
quot(gen_s:0'2_0(+(n1423_0, 1)), gen_s:0'2_0(1)) →_R^Ω(1)
s(quot(minus(gen_s:0'2_0(n1423_0), gen_s:0'2_0(0)), s(gen_s:0'2_0(0)))) →_R^Ω(1)
s(quot(gen_s:0'2_0(n1423_0), s(gen_s:0'2_0(0)))) →_IH
s(gen_s:0'2_0(c1424_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:
pred(s(x)) → x
minus(x, 0') → x
minus(x, s(y)) → pred(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))

Types:
pred :: s:0' → s:0'
s :: s:0' → s:0'
minus :: s:0' → s:0' → s:0'
0' :: s:0'
quot :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
quot(gen_s:0'2_0(n1423_0), gen_s:0'2_0(1)) → gen_s:0'2_0(n1423_0), rt ∈ Ω(1 + n1423₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
quot(gen_s:0'2_0(n1423_0), gen_s:0'2_0(1)) → gen_s:0'2_0(n1423_0), rt ∈ Ω(1 + n1423₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
pred(s(x)) → x
minus(x, 0') → x
minus(x, s(y)) → pred(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))

Types:
pred :: s:0' → s:0'
s :: s:0' → s:0'
minus :: s:0' → s:0' → s:0'
0' :: s:0'
quot :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
quot(gen_s:0'2_0(n1423_0), gen_s:0'2_0(1)) → gen_s:0'2_0(n1423_0), rt ∈ Ω(1 + n1423₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
quot(gen_s:0'2_0(n1423_0), gen_s:0'2_0(1)) → gen_s:0'2_0(n1423_0), rt ∈ Ω(1 + n1423₀)

(18) BOUNDS(n^1, INF)