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(gt(x, y), x, y)
cond(false, x, y) → 0
cond(true, x, y) → s(minus(x, s(y)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(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(gt(x, y), x, y)
cond(false, x, y) → 0'
cond(true, x, y) → s(minus(x, s(y)))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(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(gt(x, y), x, y)
cond(false, x, y) → 0'
cond(true, x, y) → s(minus(x, s(y)))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)

Types:
minus :: 0':s → 0':s → 0':s
cond :: false:true → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → false:true
false :: false:true
0' :: 0':s
true :: false:true
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

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

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

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

(6) Obligation:

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

Types:
minus :: 0':s → 0':s → 0':s
cond :: false:true → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → false:true
false :: false:true
0' :: 0':s
true :: false:true
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

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

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

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

Proved the following rewrite lemma:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Induction Base:
gt(gen_0':s3_0(0), gen_0':s3_0(0)) →_R^Ω(1)
false

Induction Step:
gt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) →_IH
false

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(gt(x, y), x, y)
cond(false, x, y) → 0'
cond(true, x, y) → s(minus(x, s(y)))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)

Types:
minus :: 0':s → 0':s → 0':s
cond :: false:true → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → false:true
false :: false:true
0' :: 0':s
true :: false:true
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_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(gt(x, y), x, y)
cond(false, x, y) → 0'
cond(true, x, y) → s(minus(x, s(y)))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)

Types:
minus :: 0':s → 0':s → 0':s
cond :: false:true → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → false:true
false :: false:true
0' :: 0':s
true :: false:true
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

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

Types:
minus :: 0':s → 0':s → 0':s
cond :: false:true → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → false:true
false :: false:true
0' :: 0':s
true :: false:true
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)