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:

dec(Cons(Nil, Nil)) → Nil
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs))
dec(Cons(Cons(x, xs), Nil)) → dec(Nil)
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs))
isNilNil(Cons(Nil, Nil)) → True
isNilNil(Cons(Nil, Cons(x, xs))) → False
isNilNil(Cons(Cons(x, xs), Nil)) → False
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False
nestdec(Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs)))
number17(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(x) → nestdec(x)

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:

dec(Cons(Nil, Nil)) → Nil
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs))
dec(Cons(Cons(x, xs), Nil)) → dec(Nil)
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs))
isNilNil(Cons(Nil, Nil)) → True
isNilNil(Cons(Nil, Cons(x, xs))) → False
isNilNil(Cons(Cons(x, xs), Nil)) → False
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False
nestdec(Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs)))
number17(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(x) → nestdec(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
dec(Cons(Nil, Nil)) → Nil
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs))
dec(Cons(Cons(x, xs), Nil)) → dec(Nil)
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs))
isNilNil(Cons(Nil, Nil)) → True
isNilNil(Cons(Nil, Cons(x, xs))) → False
isNilNil(Cons(Cons(x, xs), Nil)) → False
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False
nestdec(Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs)))
number17(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(x) → nestdec(x)

Types:
dec :: Nil:Cons → Nil:Cons
Cons :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
isNilNil :: Nil:Cons → True:False
True :: True:False
False :: True:False
nestdec :: Nil:Cons → Nil:Cons
number17 :: a → Nil:Cons
goal :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_True:False2_0 :: True:False
hole_a3_0 :: a
gen_Nil:Cons4_0 :: Nat → Nil:Cons

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

Heuristically decided to analyse the following defined symbols:
dec, nestdec

They will be analysed ascendingly in the following order:
dec < nestdec

(6) Obligation:

Innermost TRS:
Rules:
dec(Cons(Nil, Nil)) → Nil
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs))
dec(Cons(Cons(x, xs), Nil)) → dec(Nil)
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs))
isNilNil(Cons(Nil, Nil)) → True
isNilNil(Cons(Nil, Cons(x, xs))) → False
isNilNil(Cons(Cons(x, xs), Nil)) → False
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False
nestdec(Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs)))
number17(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(x) → nestdec(x)

Types:
dec :: Nil:Cons → Nil:Cons
Cons :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
isNilNil :: Nil:Cons → True:False
True :: True:False
False :: True:False
nestdec :: Nil:Cons → Nil:Cons
number17 :: a → Nil:Cons
goal :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_True:False2_0 :: True:False
hole_a3_0 :: a
gen_Nil:Cons4_0 :: Nat → Nil:Cons

Generator Equations:
gen_Nil:Cons4_0(0) ⇔ Nil
gen_Nil:Cons4_0(+(x, 1)) ⇔ Cons(Nil, gen_Nil:Cons4_0(x))

The following defined symbols remain to be analysed:
dec, nestdec

They will be analysed ascendingly in the following order:
dec < nestdec

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

Proved the following rewrite lemma:
dec(gen_Nil:Cons4_0(+(1, n6_0))) → gen_Nil:Cons4_0(0), rt ∈ Ω(1 + n6₀)

Induction Base:
dec(gen_Nil:Cons4_0(+(1, 0))) →_R^Ω(1)
Nil

Induction Step:
dec(gen_Nil:Cons4_0(+(1, +(n6_0, 1)))) →_R^Ω(1)
dec(Cons(Nil, gen_Nil:Cons4_0(n6_0))) →_IH
gen_Nil:Cons4_0(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:
dec(Cons(Nil, Nil)) → Nil
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs))
dec(Cons(Cons(x, xs), Nil)) → dec(Nil)
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs))
isNilNil(Cons(Nil, Nil)) → True
isNilNil(Cons(Nil, Cons(x, xs))) → False
isNilNil(Cons(Cons(x, xs), Nil)) → False
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False
nestdec(Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs)))
number17(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(x) → nestdec(x)

Types:
dec :: Nil:Cons → Nil:Cons
Cons :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
isNilNil :: Nil:Cons → True:False
True :: True:False
False :: True:False
nestdec :: Nil:Cons → Nil:Cons
number17 :: a → Nil:Cons
goal :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_True:False2_0 :: True:False
hole_a3_0 :: a
gen_Nil:Cons4_0 :: Nat → Nil:Cons

Lemmas:
dec(gen_Nil:Cons4_0(+(1, n6_0))) → gen_Nil:Cons4_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_Nil:Cons4_0(0) ⇔ Nil
gen_Nil:Cons4_0(+(x, 1)) ⇔ Cons(Nil, gen_Nil:Cons4_0(x))

The following defined symbols remain to be analysed:
nestdec

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

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

(11) Obligation:

Innermost TRS:
Rules:
dec(Cons(Nil, Nil)) → Nil
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs))
dec(Cons(Cons(x, xs), Nil)) → dec(Nil)
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs))
isNilNil(Cons(Nil, Nil)) → True
isNilNil(Cons(Nil, Cons(x, xs))) → False
isNilNil(Cons(Cons(x, xs), Nil)) → False
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False
nestdec(Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs)))
number17(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(x) → nestdec(x)

Types:
dec :: Nil:Cons → Nil:Cons
Cons :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
isNilNil :: Nil:Cons → True:False
True :: True:False
False :: True:False
nestdec :: Nil:Cons → Nil:Cons
number17 :: a → Nil:Cons
goal :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_True:False2_0 :: True:False
hole_a3_0 :: a
gen_Nil:Cons4_0 :: Nat → Nil:Cons

Lemmas:
dec(gen_Nil:Cons4_0(+(1, n6_0))) → gen_Nil:Cons4_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_Nil:Cons4_0(0) ⇔ Nil
gen_Nil:Cons4_0(+(x, 1)) ⇔ Cons(Nil, gen_Nil:Cons4_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
dec(gen_Nil:Cons4_0(+(1, n6_0))) → gen_Nil:Cons4_0(0), rt ∈ Ω(1 + n6₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
dec(Cons(Nil, Nil)) → Nil
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs))
dec(Cons(Cons(x, xs), Nil)) → dec(Nil)
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs))
isNilNil(Cons(Nil, Nil)) → True
isNilNil(Cons(Nil, Cons(x, xs))) → False
isNilNil(Cons(Cons(x, xs), Nil)) → False
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False
nestdec(Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs)))
number17(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(x) → nestdec(x)

Types:
dec :: Nil:Cons → Nil:Cons
Cons :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
isNilNil :: Nil:Cons → True:False
True :: True:False
False :: True:False
nestdec :: Nil:Cons → Nil:Cons
number17 :: a → Nil:Cons
goal :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_True:False2_0 :: True:False
hole_a3_0 :: a
gen_Nil:Cons4_0 :: Nat → Nil:Cons

Lemmas:
dec(gen_Nil:Cons4_0(+(1, n6_0))) → gen_Nil:Cons4_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_Nil:Cons4_0(0) ⇔ Nil
gen_Nil:Cons4_0(+(x, 1)) ⇔ Cons(Nil, 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:
dec(gen_Nil:Cons4_0(+(1, n6_0))) → gen_Nil:Cons4_0(0), rt ∈ Ω(1 + n6₀)

(16) BOUNDS(n^1, INF)