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, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))

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, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

Types:
minus :: s:0' → s:0' → s:0'
cond :: s:0' → s:0' → s:0' → s:0'
min :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

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

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

(6) Obligation:

Innermost TRS:
Rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

Types:
minus :: s:0' → s:0' → s:0'
cond :: s:0' → s:0' → s:0' → s:0'
min :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
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:
min, minus

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

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

Proved the following rewrite lemma:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

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

Induction Step:
min(gen_s:0'2_0(+(n4_0, 1)), gen_s:0'2_0(+(n4_0, 1))) →_R^Ω(1)
s(min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0))) →_IH
s(gen_s:0'2_0(c5_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, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

Types:
minus :: s:0' → s:0' → s:0'
cond :: s:0' → s:0' → s:0' → s:0'
min :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

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

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

Types:
minus :: s:0' → s:0' → s:0'
cond :: s:0' → s:0' → s:0' → s:0'
min :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

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.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))

Types:
minus :: s:0' → s:0' → s:0'
cond :: s:0' → s:0' → s:0' → s:0'
min :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

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.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

(16) BOUNDS(n^1, INF)