The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

0 CpxTRS
↳1 RenamingProof (⇔, 0 ms)
↳2 CpxRelTRS
↳3 SlicingProof (LOWER BOUND(ID), 0 ms)
↳4 CpxRelTRS
↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 typed CpxTrs
↳7 OrderProof (LOWER BOUND(ID), 0 ms)
↳8 typed CpxTrs
↳9 RewriteLemmaProof (LOWER BOUND(ID), 305 ms)
↳10 BEST
    ↳11 typed CpxTrs
        ↳12 NoRewriteLemmaProof (LOWER BOUND(ID), 24 ms)
        ↳13 typed CpxTrs
        ↳14 LowerBoundsProof (⇔, 0 ms)
        ↳15 BOUNDS(n^1, INF)
    ↳16 typed CpxTrs
        ↳17 LowerBoundsProof (⇔, 0 ms)
        ↳18 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) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
number17/0

(4) 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 → 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

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

Infered types.

(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 → 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 :: Nil:Cons
goal :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_True:False2_0 :: True:False
gen_Nil:Cons3_0 :: Nat → Nil:Cons

(7) 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

(8) 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 → 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 :: Nil:Cons
goal :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_True:False2_0 :: True:False
gen_Nil:Cons3_0 :: Nat → Nil:Cons

Generator Equations:
gen_Nil:Cons3_0(0) ⇔ Nil
gen_Nil:Cons3_0(+(x, 1)) ⇔ Cons(Nil, gen_Nil:Cons3_0(x))

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

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

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

Proved the following rewrite lemma:
dec(gen_Nil:Cons3_0(+(1, n5_0))) → gen_Nil:Cons3_0(0), rt ∈ Ω(1 + n50)

Induction Base:
dec(gen_Nil:Cons3_0(+(1, 0))) →RΩ(1)
Nil

Induction Step:
dec(gen_Nil:Cons3_0(+(1, +(n5_0, 1)))) →RΩ(1)
dec(Cons(Nil, gen_Nil:Cons3_0(n5_0))) →IH
gen_Nil:Cons3_0(0)

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

(10) Complex Obligation (BEST)

(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 → 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 :: Nil:Cons
goal :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_True:False2_0 :: True:False
gen_Nil:Cons3_0 :: Nat → Nil:Cons

Lemmas:
dec(gen_Nil:Cons3_0(+(1, n5_0))) → gen_Nil:Cons3_0(0), rt ∈ Ω(1 + n50)

Generator Equations:
gen_Nil:Cons3_0(0) ⇔ Nil
gen_Nil:Cons3_0(+(x, 1)) ⇔ Cons(Nil, gen_Nil:Cons3_0(x))

The following defined symbols remain to be analysed:
nestdec

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) 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 → 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 :: Nil:Cons
goal :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_True:False2_0 :: True:False
gen_Nil:Cons3_0 :: Nat → Nil:Cons

Lemmas:
dec(gen_Nil:Cons3_0(+(1, n5_0))) → gen_Nil:Cons3_0(0), rt ∈ Ω(1 + n50)

Generator Equations:
gen_Nil:Cons3_0(0) ⇔ Nil
gen_Nil:Cons3_0(+(x, 1)) ⇔ Cons(Nil, gen_Nil:Cons3_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
dec(gen_Nil:Cons3_0(+(1, n5_0))) → gen_Nil:Cons3_0(0), rt ∈ Ω(1 + n50)

(15) BOUNDS(n^1, INF)

(16) 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 → 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 :: Nil:Cons
goal :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_True:False2_0 :: True:False
gen_Nil:Cons3_0 :: Nat → Nil:Cons

Lemmas:
dec(gen_Nil:Cons3_0(+(1, n5_0))) → gen_Nil:Cons3_0(0), rt ∈ Ω(1 + n50)

Generator Equations:
gen_Nil:Cons3_0(0) ⇔ Nil
gen_Nil:Cons3_0(+(x, 1)) ⇔ Cons(Nil, gen_Nil:Cons3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
dec(gen_Nil:Cons3_0(+(1, n5_0))) → gen_Nil:Cons3_0(0), rt ∈ Ω(1 + n50)

(18) BOUNDS(n^1, INF)