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

0 CpxTRS
↳1 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxWeightedTrs
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 2 ms)
↳4 CpxTypedWeightedTrs
↳5 CompletionProof (UPPER BOUND(ID), 0 ms)
↳6 CpxTypedWeightedCompleteTrs
↳7 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CpxTypedWeightedCompleteTrs
↳9 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)
↳10 CpxRNTS
↳11 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CpxRNTS
↳13 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CpxRNTS
↳15 IntTrsBoundProof (UPPER BOUND(ID), 172 ms)
↳16 CpxRNTS
↳17 IntTrsBoundProof (UPPER BOUND(ID), 9 ms)
↳18 CpxRNTS
↳19 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳20 CpxRNTS
↳21 IntTrsBoundProof (UPPER BOUND(ID), 425 ms)
↳22 CpxRNTS
↳23 IntTrsBoundProof (UPPER BOUND(ID), 159 ms)
↳24 CpxRNTS
↳25 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳26 CpxRNTS
↳27 IntTrsBoundProof (UPPER BOUND(ID), 127 ms)
↳28 CpxRNTS
↳29 IntTrsBoundProof (UPPER BOUND(ID), 31 ms)
↳30 CpxRNTS
↳31 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳32 CpxRNTS
↳33 IntTrsBoundProof (UPPER BOUND(ID), 494 ms)
↳34 CpxRNTS
↳35 IntTrsBoundProof (UPPER BOUND(ID), 357 ms)
↳36 CpxRNTS
↳37 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳38 CpxRNTS
↳39 IntTrsBoundProof (UPPER BOUND(ID), 164 ms)
↳40 CpxRNTS
↳41 IntTrsBoundProof (UPPER BOUND(ID), 63 ms)
↳42 CpxRNTS
↳43 FinalProof (⇔, 0 ms)
↳44 BOUNDS(1, n^1)

(0) Obligation:

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


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) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(2) Obligation:

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


The TRS R consists of the following rules:

dec(Cons(Nil, Nil)) → Nil [1]
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs)) [1]
dec(Cons(Cons(x, xs), Nil)) → dec(Nil) [1]
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs)) [1]
isNilNil(Cons(Nil, Nil)) → True [1]
isNilNil(Cons(Nil, Cons(x, xs))) → False [1]
isNilNil(Cons(Cons(x, xs), Nil)) → False [1]
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False [1]
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))))))))))))))))) [1]
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs))) [1]
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))))))))))))))))) [1]
goal(x) → nestdec(x) [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

dec(Cons(Nil, Nil)) → Nil [1]
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs)) [1]
dec(Cons(Cons(x, xs), Nil)) → dec(Nil) [1]
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs)) [1]
isNilNil(Cons(Nil, Nil)) → True [1]
isNilNil(Cons(Nil, Cons(x, xs))) → False [1]
isNilNil(Cons(Cons(x, xs), Nil)) → False [1]
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False [1]
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))))))))))))))))) [1]
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs))) [1]
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))))))))))))))))) [1]
goal(x) → nestdec(x) [1]

The TRS has the following type information:
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

Rewrite Strategy: INNERMOST

(5) CompletionProof (UPPER BOUND(ID) transformation)

The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:


isNilNil
nestdec
number17
goal

(c) The following functions are completely defined:

dec

Due to the following rules being added:

dec(v0) → Nil [0]

And the following fresh constants:

const

(6) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

dec(Cons(Nil, Nil)) → Nil [1]
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs)) [1]
dec(Cons(Cons(x, xs), Nil)) → dec(Nil) [1]
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs)) [1]
isNilNil(Cons(Nil, Nil)) → True [1]
isNilNil(Cons(Nil, Cons(x, xs))) → False [1]
isNilNil(Cons(Cons(x, xs), Nil)) → False [1]
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False [1]
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))))))))))))))))) [1]
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs))) [1]
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))))))))))))))))) [1]
goal(x) → nestdec(x) [1]
dec(v0) → Nil [0]

The TRS has the following type information:
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
const :: a

Rewrite Strategy: INNERMOST

(7) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(8) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

dec(Cons(Nil, Nil)) → Nil [1]
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs)) [1]
dec(Cons(Cons(x, xs), Nil)) → dec(Nil) [1]
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs)) [1]
isNilNil(Cons(Nil, Nil)) → True [1]
isNilNil(Cons(Nil, Cons(x, xs))) → False [1]
isNilNil(Cons(Cons(x, xs), Nil)) → False [1]
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False [1]
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))))))))))))))))) [1]
nestdec(Cons(Nil, Nil)) → nestdec(Nil) [2]
nestdec(Cons(Nil, Cons(x'', xs''))) → nestdec(dec(Cons(x'', xs''))) [2]
nestdec(Cons(Cons(x1, xs1), Nil)) → nestdec(dec(Nil)) [2]
nestdec(Cons(Cons(x''', xs'''), Cons(x2, xs2))) → nestdec(dec(Cons(x2, xs2))) [2]
nestdec(Cons(x, xs)) → nestdec(Nil) [1]
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))))))))))))))))) [1]
goal(x) → nestdec(x) [1]
dec(v0) → Nil [0]

The TRS has the following type information:
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
const :: a

Rewrite Strategy: INNERMOST

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

Nil => 0
True => 1
False => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 1 }→ dec(0) :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
goal(z) -{ 1 }→ nestdec(x) :|: x >= 0, z = x
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 2 }→ nestdec(dec(0)) :|: x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 2 }→ nestdec(dec(1 + x'' + xs'')) :|: xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 2 }→ nestdec(dec(1 + x2 + xs2)) :|: z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: n >= 0, z = n

(11) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(12) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 1 }→ dec(0) :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 2 }→ nestdec(dec(0)) :|: x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 2 }→ nestdec(dec(1 + x'' + xs'')) :|: xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 2 }→ nestdec(dec(1 + x2 + xs2)) :|: z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ number17 }
{ dec }
{ isNilNil }
{ nestdec }
{ goal }

(14) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 1 }→ dec(0) :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 2 }→ nestdec(dec(0)) :|: x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 2 }→ nestdec(dec(1 + x'' + xs'')) :|: xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 2 }→ nestdec(dec(1 + x2 + xs2)) :|: z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {number17}, {dec}, {isNilNil}, {nestdec}, {goal}

(15) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: number17
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 17

(16) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 1 }→ dec(0) :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 2 }→ nestdec(dec(0)) :|: x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 2 }→ nestdec(dec(1 + x'' + xs'')) :|: xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 2 }→ nestdec(dec(1 + x2 + xs2)) :|: z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {number17}, {dec}, {isNilNil}, {nestdec}, {goal}
Previous analysis results are:
number17: runtime: ?, size: O(1) [17]

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: number17
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(18) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 1 }→ dec(0) :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 2 }→ nestdec(dec(0)) :|: x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 2 }→ nestdec(dec(1 + x'' + xs'')) :|: xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 2 }→ nestdec(dec(1 + x2 + xs2)) :|: z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {dec}, {isNilNil}, {nestdec}, {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(20) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 1 }→ dec(0) :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 2 }→ nestdec(dec(0)) :|: x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 2 }→ nestdec(dec(1 + x'' + xs'')) :|: xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 2 }→ nestdec(dec(1 + x2 + xs2)) :|: z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {dec}, {isNilNil}, {nestdec}, {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: dec
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(22) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 1 }→ dec(0) :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 2 }→ nestdec(dec(0)) :|: x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 2 }→ nestdec(dec(1 + x'' + xs'')) :|: xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 2 }→ nestdec(dec(1 + x2 + xs2)) :|: z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {dec}, {isNilNil}, {nestdec}, {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: ?, size: O(1) [0]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using PUBS for: dec
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 2 + z

(24) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 1 }→ dec(0) :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 1 }→ dec(1 + x + xs) :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 2 }→ nestdec(dec(0)) :|: x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 2 }→ nestdec(dec(1 + x'' + xs'')) :|: xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 2 }→ nestdec(dec(1 + x2 + xs2)) :|: z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {isNilNil}, {nestdec}, {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: O(n1) [2 + z], size: O(1) [0]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(26) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 4 + x + xs }→ s :|: s >= 0, s <= 0, xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 3 }→ s' :|: s' >= 0, s' <= 0, xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 4 + x + xs }→ s'' :|: s'' >= 0, s'' <= 0, xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 5 + x'' + xs'' }→ nestdec(s1) :|: s1 >= 0, s1 <= 0, xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 4 }→ nestdec(s2) :|: s2 >= 0, s2 <= 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 5 + x2 + xs2 }→ nestdec(s3) :|: s3 >= 0, s3 <= 0, z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {isNilNil}, {nestdec}, {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: O(n1) [2 + z], size: O(1) [0]

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: isNilNil
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(28) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 4 + x + xs }→ s :|: s >= 0, s <= 0, xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 3 }→ s' :|: s' >= 0, s' <= 0, xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 4 + x + xs }→ s'' :|: s'' >= 0, s'' <= 0, xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 5 + x'' + xs'' }→ nestdec(s1) :|: s1 >= 0, s1 <= 0, xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 4 }→ nestdec(s2) :|: s2 >= 0, s2 <= 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 5 + x2 + xs2 }→ nestdec(s3) :|: s3 >= 0, s3 <= 0, z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {isNilNil}, {nestdec}, {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: O(n1) [2 + z], size: O(1) [0]
isNilNil: runtime: ?, size: O(1) [1]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: isNilNil
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(30) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 4 + x + xs }→ s :|: s >= 0, s <= 0, xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 3 }→ s' :|: s' >= 0, s' <= 0, xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 4 + x + xs }→ s'' :|: s'' >= 0, s'' <= 0, xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 5 + x'' + xs'' }→ nestdec(s1) :|: s1 >= 0, s1 <= 0, xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 4 }→ nestdec(s2) :|: s2 >= 0, s2 <= 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 5 + x2 + xs2 }→ nestdec(s3) :|: s3 >= 0, s3 <= 0, z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {nestdec}, {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: O(n1) [2 + z], size: O(1) [0]
isNilNil: runtime: O(1) [1], size: O(1) [1]

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(32) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 4 + x + xs }→ s :|: s >= 0, s <= 0, xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 3 }→ s' :|: s' >= 0, s' <= 0, xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 4 + x + xs }→ s'' :|: s'' >= 0, s'' <= 0, xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 5 + x'' + xs'' }→ nestdec(s1) :|: s1 >= 0, s1 <= 0, xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 4 }→ nestdec(s2) :|: s2 >= 0, s2 <= 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 5 + x2 + xs2 }→ nestdec(s3) :|: s3 >= 0, s3 <= 0, z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {nestdec}, {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: O(n1) [2 + z], size: O(1) [0]
isNilNil: runtime: O(1) [1], size: O(1) [1]

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: nestdec
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 17

(34) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 4 + x + xs }→ s :|: s >= 0, s <= 0, xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 3 }→ s' :|: s' >= 0, s' <= 0, xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 4 + x + xs }→ s'' :|: s'' >= 0, s'' <= 0, xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 5 + x'' + xs'' }→ nestdec(s1) :|: s1 >= 0, s1 <= 0, xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 4 }→ nestdec(s2) :|: s2 >= 0, s2 <= 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 5 + x2 + xs2 }→ nestdec(s3) :|: s3 >= 0, s3 <= 0, z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {nestdec}, {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: O(n1) [2 + z], size: O(1) [0]
isNilNil: runtime: O(1) [1], size: O(1) [1]
nestdec: runtime: ?, size: O(1) [17]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: nestdec
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 5 + z

(36) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 4 + x + xs }→ s :|: s >= 0, s <= 0, xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 3 }→ s' :|: s' >= 0, s' <= 0, xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 4 + x + xs }→ s'' :|: s'' >= 0, s'' <= 0, xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 1 }→ nestdec(z) :|: z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 5 + x'' + xs'' }→ nestdec(s1) :|: s1 >= 0, s1 <= 0, xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 4 }→ nestdec(s2) :|: s2 >= 0, s2 <= 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 5 + x2 + xs2 }→ nestdec(s3) :|: s3 >= 0, s3 <= 0, z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 2 }→ nestdec(0) :|: z = 1 + 0 + 0
nestdec(z) -{ 1 }→ nestdec(0) :|: z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: O(n1) [2 + z], size: O(1) [0]
isNilNil: runtime: O(1) [1], size: O(1) [1]
nestdec: runtime: O(n1) [5 + z], size: O(1) [17]

(37) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(38) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 4 + x + xs }→ s :|: s >= 0, s <= 0, xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 3 }→ s' :|: s' >= 0, s' <= 0, xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 4 + x + xs }→ s'' :|: s'' >= 0, s'' <= 0, xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 6 + z }→ s9 :|: s9 >= 0, s9 <= 17, z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 7 }→ s4 :|: s4 >= 0, s4 <= 17, z = 1 + 0 + 0
nestdec(z) -{ 10 + s1 + x'' + xs'' }→ s5 :|: s5 >= 0, s5 <= 17, s1 >= 0, s1 <= 0, xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 9 + s2 }→ s6 :|: s6 >= 0, s6 <= 17, s2 >= 0, s2 <= 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 10 + s3 + x2 + xs2 }→ s7 :|: s7 >= 0, s7 <= 17, s3 >= 0, s3 <= 0, z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 6 }→ s8 :|: s8 >= 0, s8 <= 17, z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: O(n1) [2 + z], size: O(1) [0]
isNilNil: runtime: O(1) [1], size: O(1) [1]
nestdec: runtime: O(n1) [5 + z], size: O(1) [17]

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 17

(40) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 4 + x + xs }→ s :|: s >= 0, s <= 0, xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 3 }→ s' :|: s' >= 0, s' <= 0, xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 4 + x + xs }→ s'' :|: s'' >= 0, s'' <= 0, xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 6 + z }→ s9 :|: s9 >= 0, s9 <= 17, z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 7 }→ s4 :|: s4 >= 0, s4 <= 17, z = 1 + 0 + 0
nestdec(z) -{ 10 + s1 + x'' + xs'' }→ s5 :|: s5 >= 0, s5 <= 17, s1 >= 0, s1 <= 0, xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 9 + s2 }→ s6 :|: s6 >= 0, s6 <= 17, s2 >= 0, s2 <= 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 10 + s3 + x2 + xs2 }→ s7 :|: s7 >= 0, s7 <= 17, s3 >= 0, s3 <= 0, z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 6 }→ s8 :|: s8 >= 0, s8 <= 17, z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: O(n1) [2 + z], size: O(1) [0]
isNilNil: runtime: O(1) [1], size: O(1) [1]
nestdec: runtime: O(n1) [5 + z], size: O(1) [17]
goal: runtime: ?, size: O(1) [17]

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 6 + z

(42) Obligation:

Complexity RNTS consisting of the following rules:

dec(z) -{ 4 + x + xs }→ s :|: s >= 0, s <= 0, xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
dec(z) -{ 3 }→ s' :|: s' >= 0, s' <= 0, xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
dec(z) -{ 4 + x + xs }→ s'' :|: s'' >= 0, s'' <= 0, xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
dec(z) -{ 1 }→ 0 :|: z = 1 + 0 + 0
dec(z) -{ 0 }→ 0 :|: z >= 0
goal(z) -{ 6 + z }→ s9 :|: s9 >= 0, s9 <= 17, z >= 0
isNilNil(z) -{ 1 }→ 1 :|: z = 1 + 0 + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + 0 + (1 + x + xs)
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, x >= 0, z = 1 + (1 + x + xs) + 0
isNilNil(z) -{ 1 }→ 0 :|: xs >= 0, z = 1 + (1 + x' + xs') + (1 + x + xs), x' >= 0, xs' >= 0, x >= 0
nestdec(z) -{ 7 }→ s4 :|: s4 >= 0, s4 <= 17, z = 1 + 0 + 0
nestdec(z) -{ 10 + s1 + x'' + xs'' }→ s5 :|: s5 >= 0, s5 <= 17, s1 >= 0, s1 <= 0, xs'' >= 0, x'' >= 0, z = 1 + 0 + (1 + x'' + xs'')
nestdec(z) -{ 9 + s2 }→ s6 :|: s6 >= 0, s6 <= 17, s2 >= 0, s2 <= 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, xs1 >= 0
nestdec(z) -{ 10 + s3 + x2 + xs2 }→ s7 :|: s7 >= 0, s7 <= 17, s3 >= 0, s3 <= 0, z = 1 + (1 + x''' + xs''') + (1 + x2 + xs2), xs2 >= 0, xs''' >= 0, x''' >= 0, x2 >= 0
nestdec(z) -{ 6 }→ s8 :|: s8 >= 0, s8 <= 17, z = 1 + x + xs, xs >= 0, x >= 0
nestdec(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0
number17(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed:
Previous analysis results are:
number17: runtime: O(1) [1], size: O(1) [17]
dec: runtime: O(n1) [2 + z], size: O(1) [0]
isNilNil: runtime: O(1) [1], size: O(1) [1]
nestdec: runtime: O(n1) [5 + z], size: O(1) [17]
goal: runtime: O(n1) [6 + z], size: O(1) [17]

(43) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(44) BOUNDS(1, n^1)