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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
min(s(u), s(v)) →⁺ s(min(u, v))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [u / s(u), v / s(v)].
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, y) → cond(equal(min(x, y), y), x, y)
cond(true, x, y) → s(minus(x, s(y)))
min(0', v) → 0'
min(u, 0') → 0'
min(s(u), s(v)) → s(min(u, v))
equal(0', 0') → true
equal(s(x), 0') → false
equal(0', s(y)) → false
equal(s(x), s(y)) → equal(x, y)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
minus :: s:0' → s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
equal :: s:0' → s:0' → true:false
min :: s:0' → s:0' → s:0'
true :: true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

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

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

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

(8) Obligation:

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

Types:
minus :: s:0' → s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
equal :: s:0' → s:0' → true:false
min :: s:0' → s:0' → s:0'
true :: true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
equal, minus, min

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

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
equal(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
equal(gen_s:0'3_0(0), gen_s:0'3_0(0)) →_R^Ω(1)
true

Induction Step:
equal(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) →_R^Ω(1)
equal(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) →_IH
true

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

(10) Complex Obligation (BEST)

(11) Obligation:

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

Types:
minus :: s:0' → s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
equal :: s:0' → s:0' → true:false
min :: s:0' → s:0' → s:0'
true :: true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
equal(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

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

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

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
min(gen_s:0'3_0(n504_0), gen_s:0'3_0(n504_0)) → gen_s:0'3_0(n504_0), rt ∈ Ω(1 + n504₀)

Induction Base:
min(gen_s:0'3_0(0), gen_s:0'3_0(0)) →_R^Ω(1)
0'

Induction Step:
min(gen_s:0'3_0(+(n504_0, 1)), gen_s:0'3_0(+(n504_0, 1))) →_R^Ω(1)
s(min(gen_s:0'3_0(n504_0), gen_s:0'3_0(n504_0))) →_IH
s(gen_s:0'3_0(c505_0))

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

(13) Complex Obligation (BEST)

(14) Obligation:

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

Types:
minus :: s:0' → s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
equal :: s:0' → s:0' → true:false
min :: s:0' → s:0' → s:0'
true :: true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
equal(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
min(gen_s:0'3_0(n504_0), gen_s:0'3_0(n504_0)) → gen_s:0'3_0(n504_0), rt ∈ Ω(1 + n504₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
minus

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

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

Types:
minus :: s:0' → s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
equal :: s:0' → s:0' → true:false
min :: s:0' → s:0' → s:0'
true :: true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
equal(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
min(gen_s:0'3_0(n504_0), gen_s:0'3_0(n504_0)) → gen_s:0'3_0(n504_0), rt ∈ Ω(1 + n504₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
equal(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)

(19) Obligation:

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

Types:
minus :: s:0' → s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
equal :: s:0' → s:0' → true:false
min :: s:0' → s:0' → s:0'
true :: true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
equal(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
min(gen_s:0'3_0(n504_0), gen_s:0'3_0(n504_0)) → gen_s:0'3_0(n504_0), rt ∈ Ω(1 + n504₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
equal(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

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

Types:
minus :: s:0' → s:0' → s:0'
cond :: true:false → s:0' → s:0' → s:0'
equal :: s:0' → s:0' → true:false
min :: s:0' → s:0' → s:0'
true :: true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
equal(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
equal(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(24) BOUNDS(n^1, INF)