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:

diff(x, y) → cond1(equal(x, y), x, y)
cond1(true, x, y) → 0
cond1(false, x, y) → cond2(gt(x, y), x, y)
cond2(true, x, y) → s(diff(x, s(y)))
cond2(false, x, y) → s(diff(s(x), y))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(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: 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:

diff(x, y) → cond1(equal(x, y), x, y)
cond1(true, x, y) → 0'
cond1(false, x, y) → cond2(gt(x, y), x, y)
cond2(true, x, y) → s(diff(x, s(y)))
cond2(false, x, y) → s(diff(s(x), y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(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: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
diff(x, y) → cond1(equal(x, y), x, y)
cond1(true, x, y) → 0'
cond1(false, x, y) → cond2(gt(x, y), x, y)
cond2(true, x, y) → s(diff(x, s(y)))
cond2(false, x, y) → s(diff(s(x), y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(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:
diff :: 0':s → 0':s → 0':s
cond1 :: true:false → 0':s → 0':s → 0':s
equal :: 0':s → 0':s → true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
diff, equal, gt

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

(6) Obligation:

Innermost TRS:
Rules:
diff(x, y) → cond1(equal(x, y), x, y)
cond1(true, x, y) → 0'
cond1(false, x, y) → cond2(gt(x, y), x, y)
cond2(true, x, y) → s(diff(x, s(y)))
cond2(false, x, y) → s(diff(s(x), y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(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:
diff :: 0':s → 0':s → 0':s
cond1 :: true:false → 0':s → 0':s → 0':s
equal :: 0':s → 0':s → true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
equal, diff, gt

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

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

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

Induction Base:
equal(gen_0':s3_0(0), gen_0':s3_0(0)) →_R^Ω(1)
true

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

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:
diff(x, y) → cond1(equal(x, y), x, y)
cond1(true, x, y) → 0'
cond1(false, x, y) → cond2(gt(x, y), x, y)
cond2(true, x, y) → s(diff(x, s(y)))
cond2(false, x, y) → s(diff(s(x), y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(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:
diff :: 0':s → 0':s → 0':s
cond1 :: true:false → 0':s → 0':s → 0':s
equal :: 0':s → 0':s → true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
equal(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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:
gt, diff

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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
gt(gen_0':s3_0(n552_0), gen_0':s3_0(n552_0)) → false, rt ∈ Ω(1 + n552₀)

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

Induction Step:
gt(gen_0':s3_0(+(n552_0, 1)), gen_0':s3_0(+(n552_0, 1))) →_R^Ω(1)
gt(gen_0':s3_0(n552_0), gen_0':s3_0(n552_0)) →_IH
false

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
diff(x, y) → cond1(equal(x, y), x, y)
cond1(true, x, y) → 0'
cond1(false, x, y) → cond2(gt(x, y), x, y)
cond2(true, x, y) → s(diff(x, s(y)))
cond2(false, x, y) → s(diff(s(x), y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(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:
diff :: 0':s → 0':s → 0':s
cond1 :: true:false → 0':s → 0':s → 0':s
equal :: 0':s → 0':s → true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
equal(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
gt(gen_0':s3_0(n552_0), gen_0':s3_0(n552_0)) → false, rt ∈ Ω(1 + n552₀)

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:
diff

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

Innermost TRS:
Rules:
diff(x, y) → cond1(equal(x, y), x, y)
cond1(true, x, y) → 0'
cond1(false, x, y) → cond2(gt(x, y), x, y)
cond2(true, x, y) → s(diff(x, s(y)))
cond2(false, x, y) → s(diff(s(x), y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(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:
diff :: 0':s → 0':s → 0':s
cond1 :: true:false → 0':s → 0':s → 0':s
equal :: 0':s → 0':s → true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
equal(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
gt(gen_0':s3_0(n552_0), gen_0':s3_0(n552_0)) → false, rt ∈ Ω(1 + n552₀)

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:
equal(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
diff(x, y) → cond1(equal(x, y), x, y)
cond1(true, x, y) → 0'
cond1(false, x, y) → cond2(gt(x, y), x, y)
cond2(true, x, y) → s(diff(x, s(y)))
cond2(false, x, y) → s(diff(s(x), y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(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:
diff :: 0':s → 0':s → 0':s
cond1 :: true:false → 0':s → 0':s → 0':s
equal :: 0':s → 0':s → true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
equal(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
gt(gen_0':s3_0(n552_0), gen_0':s3_0(n552_0)) → false, rt ∈ Ω(1 + n552₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

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

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
diff(x, y) → cond1(equal(x, y), x, y)
cond1(true, x, y) → 0'
cond1(false, x, y) → cond2(gt(x, y), x, y)
cond2(true, x, y) → s(diff(x, s(y)))
cond2(false, x, y) → s(diff(s(x), y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(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:
diff :: 0':s → 0':s → 0':s
cond1 :: true:false → 0':s → 0':s → 0':s
equal :: 0':s → 0':s → true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
equal(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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.

(21) LowerBoundsProof (EQUIVALENT transformation)

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

(22) BOUNDS(n^1, INF)