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:

sub(0, 0) → 0
sub(s(x), 0) → s(x)
sub(0, s(x)) → 0
sub(s(x), s(y)) → sub(x, y)
zero(nil) → zero2(0, nil)
zero(cons(x, xs)) → zero2(sub(x, x), cons(x, xs))
zero2(0, nil) → nil
zero2(0, cons(x, xs)) → cons(sub(x, x), zero(xs))
zero2(s(y), nil) → zero(nil)
zero2(s(y), cons(x, xs)) → zero(cons(x, xs))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
sub(s(x), s(y)) →⁺ sub(x, y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / s(x), 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:

sub(0', 0') → 0'
sub(s(x), 0') → s(x)
sub(0', s(x)) → 0'
sub(s(x), s(y)) → sub(x, y)
zero(nil) → zero2(0', nil)
zero(cons(x, xs)) → zero2(sub(x, x), cons(x, xs))
zero2(0', nil) → nil
zero2(0', cons(x, xs)) → cons(sub(x, x), zero(xs))
zero2(s(y), nil) → zero(nil)
zero2(s(y), cons(x, xs)) → zero(cons(x, xs))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
sub(0', 0') → 0'
sub(s(x), 0') → s(x)
sub(0', s(x)) → 0'
sub(s(x), s(y)) → sub(x, y)
zero(nil) → zero2(0', nil)
zero(cons(x, xs)) → zero2(sub(x, x), cons(x, xs))
zero2(0', nil) → nil
zero2(0', cons(x, xs)) → cons(sub(x, x), zero(xs))
zero2(s(y), nil) → zero(nil)
zero2(s(y), cons(x, xs)) → zero(cons(x, xs))

Types:
sub :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: nil:cons → nil:cons
nil :: nil:cons
zero2 :: 0':s → nil:cons → nil:cons
cons :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
sub, zero

They will be analysed ascendingly in the following order:
sub < zero

(8) Obligation:

TRS:
Rules:
sub(0', 0') → 0'
sub(s(x), 0') → s(x)
sub(0', s(x)) → 0'
sub(s(x), s(y)) → sub(x, y)
zero(nil) → zero2(0', nil)
zero(cons(x, xs)) → zero2(sub(x, x), cons(x, xs))
zero2(0', nil) → nil
zero2(0', cons(x, xs)) → cons(sub(x, x), zero(xs))
zero2(s(y), nil) → zero(nil)
zero2(s(y), cons(x, xs)) → zero(cons(x, xs))

Types:
sub :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: nil:cons → nil:cons
nil :: nil:cons
zero2 :: 0':s → nil:cons → nil:cons
cons :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
sub, zero

They will be analysed ascendingly in the following order:
sub < zero

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

Proved the following rewrite lemma:
sub(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

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

Induction Step:
sub(gen_0':s3_0(+(n6_0, 1)), gen_0':s3_0(+(n6_0, 1))) →_R^Ω(1)
sub(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) →_IH
gen_0':s3_0(0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
sub(0', 0') → 0'
sub(s(x), 0') → s(x)
sub(0', s(x)) → 0'
sub(s(x), s(y)) → sub(x, y)
zero(nil) → zero2(0', nil)
zero(cons(x, xs)) → zero2(sub(x, x), cons(x, xs))
zero2(0', nil) → nil
zero2(0', cons(x, xs)) → cons(sub(x, x), zero(xs))
zero2(s(y), nil) → zero(nil)
zero2(s(y), cons(x, xs)) → zero(cons(x, xs))

Types:
sub :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: nil:cons → nil:cons
nil :: nil:cons
zero2 :: 0':s → nil:cons → nil:cons
cons :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
sub(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
zero

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

Proved the following rewrite lemma:
zero(gen_nil:cons4_0(n750_0)) → gen_nil:cons4_0(n750_0), rt ∈ Ω(1 + n750₀)

Induction Base:
zero(gen_nil:cons4_0(0)) →_R^Ω(1)
zero2(0', nil) →_R^Ω(1)
nil

Induction Step:
zero(gen_nil:cons4_0(+(n750_0, 1))) →_R^Ω(1)
zero2(sub(0', 0'), cons(0', gen_nil:cons4_0(n750_0))) →_L^Ω(1)
zero2(gen_0':s3_0(0), cons(0', gen_nil:cons4_0(n750_0))) →_R^Ω(1)
cons(sub(0', 0'), zero(gen_nil:cons4_0(n750_0))) →_L^Ω(1)
cons(gen_0':s3_0(0), zero(gen_nil:cons4_0(n750_0))) →_IH
cons(gen_0':s3_0(0), gen_nil:cons4_0(c751_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:
sub(0', 0') → 0'
sub(s(x), 0') → s(x)
sub(0', s(x)) → 0'
sub(s(x), s(y)) → sub(x, y)
zero(nil) → zero2(0', nil)
zero(cons(x, xs)) → zero2(sub(x, x), cons(x, xs))
zero2(0', nil) → nil
zero2(0', cons(x, xs)) → cons(sub(x, x), zero(xs))
zero2(s(y), nil) → zero(nil)
zero2(s(y), cons(x, xs)) → zero(cons(x, xs))

Types:
sub :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: nil:cons → nil:cons
nil :: nil:cons
zero2 :: 0':s → nil:cons → nil:cons
cons :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
sub(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)
zero(gen_nil:cons4_0(n750_0)) → gen_nil:cons4_0(n750_0), rt ∈ Ω(1 + n750₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sub(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

TRS:
Rules:
sub(0', 0') → 0'
sub(s(x), 0') → s(x)
sub(0', s(x)) → 0'
sub(s(x), s(y)) → sub(x, y)
zero(nil) → zero2(0', nil)
zero(cons(x, xs)) → zero2(sub(x, x), cons(x, xs))
zero2(0', nil) → nil
zero2(0', cons(x, xs)) → cons(sub(x, x), zero(xs))
zero2(s(y), nil) → zero(nil)
zero2(s(y), cons(x, xs)) → zero(cons(x, xs))

Types:
sub :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: nil:cons → nil:cons
nil :: nil:cons
zero2 :: 0':s → nil:cons → nil:cons
cons :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
sub(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)
zero(gen_nil:cons4_0(n750_0)) → gen_nil:cons4_0(n750_0), rt ∈ Ω(1 + n750₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sub(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

TRS:
Rules:
sub(0', 0') → 0'
sub(s(x), 0') → s(x)
sub(0', s(x)) → 0'
sub(s(x), s(y)) → sub(x, y)
zero(nil) → zero2(0', nil)
zero(cons(x, xs)) → zero2(sub(x, x), cons(x, xs))
zero2(0', nil) → nil
zero2(0', cons(x, xs)) → cons(sub(x, x), zero(xs))
zero2(s(y), nil) → zero(nil)
zero2(s(y), cons(x, xs)) → zero(cons(x, xs))

Types:
sub :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: nil:cons → nil:cons
nil :: nil:cons
zero2 :: 0':s → nil:cons → nil:cons
cons :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
sub(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sub(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

(22) BOUNDS(n^1, INF)