Runtime Complexity (innermost) proof of /tmp/tmpMSYb1S/member.xml

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

0 CpxRelTRS

↳1 RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CpxWeightedTrs

↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 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), 4 ms)

↳10 CpxRNTS

↳11 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CpxRNTS

↳13 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 11 ms)

↳14 CpxRNTS

↳15 IntTrsBoundProof (UPPER BOUND(ID), 187 ms)

↳16 CpxRNTS

↳17 IntTrsBoundProof (UPPER BOUND(ID), 10 ms)

↳18 CpxRNTS

↳19 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳20 CpxRNTS

↳21 IntTrsBoundProof (UPPER BOUND(ID), 518 ms)

↳22 CpxRNTS

↳23 IntTrsBoundProof (UPPER BOUND(ID), 23 ms)

↳24 CpxRNTS

↳25 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳26 CpxRNTS

↳27 IntTrsBoundProof (UPPER BOUND(ID), 1455 ms)

↳28 CpxRNTS

↳29 IntTrsBoundProof (UPPER BOUND(ID), 499 ms)

↳30 CpxRNTS

↳31 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳32 CpxRNTS

↳33 IntTrsBoundProof (UPPER BOUND(ID), 152 ms)

↳34 CpxRNTS

↳35 IntTrsBoundProof (UPPER BOUND(ID), 17 ms)

↳36 CpxRNTS

↳37 FinalProof (⇔, 0 ms)

↳38 BOUNDS(1, n^1)

(0) Obligation:

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

The TRS R consists of the following rules:

member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)

The (relative) TRS S consists of the following rules:

!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0, S(y)) → False
!EQ(S(x), 0) → False
!EQ(0, 0) → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

Rewrite Strategy: INNERMOST

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed relative 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:

member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs)) [1]
member(x, Nil) → False [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
goal(x, xs) → member(x, xs) [1]
!EQ(S(x), S(y)) → !EQ(x, y) [0]
!EQ(0, S(y)) → False [0]
!EQ(S(x), 0) → False [0]
!EQ(0, 0) → True [0]
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs) [0]
member[Ite][True][Ite](True, x, xs) → True [0]

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:

member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs)) [1]
member(x, Nil) → False [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
goal(x, xs) → member(x, xs) [1]
!EQ(S(x), S(y)) → !EQ(x, y) [0]
!EQ(0, S(y)) → False [0]
!EQ(S(x), 0) → False [0]
!EQ(0, 0) → True [0]
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs) [0]
member[Ite][True][Ite](True, x, xs) → True [0]

The TRS has the following type information:

member :: S:0 → Cons:Nil → False:True
Cons :: S:0 → Cons:Nil → Cons:Nil
member[Ite][True][Ite] :: False:True → S:0 → Cons:Nil → False:True
!EQ :: S:0 → S:0 → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: S:0 → Cons:Nil → False:True
S :: S:0 → S:0
0 :: S:0

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:

member
notEmpty
goal

!EQ
member[Ite][True][Ite]

Due to the following rules being added:

!EQ(v0, v1) → null_!EQ [0]
member[Ite][True][Ite](v0, v1, v2) → null_member[Ite][True][Ite] [0]

And the following fresh constants:

null_!EQ, null_member[Ite][True][Ite]

(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:

member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs)) [1]
member(x, Nil) → False [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
goal(x, xs) → member(x, xs) [1]
!EQ(S(x), S(y)) → !EQ(x, y) [0]
!EQ(0, S(y)) → False [0]
!EQ(S(x), 0) → False [0]
!EQ(0, 0) → True [0]
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs) [0]
member[Ite][True][Ite](True, x, xs) → True [0]
!EQ(v0, v1) → null_!EQ [0]
member[Ite][True][Ite](v0, v1, v2) → null_member[Ite][True][Ite] [0]

The TRS has the following type information:

member :: S:0 → Cons:Nil → False:True:null_!EQ:null_member[Ite][True][Ite]
Cons :: S:0 → Cons:Nil → Cons:Nil
member[Ite][True][Ite] :: False:True:null_!EQ:null_member[Ite][True][Ite] → S:0 → Cons:Nil → False:True:null_!EQ:null_member[Ite][True][Ite]
!EQ :: S:0 → S:0 → False:True:null_!EQ:null_member[Ite][True][Ite]
Nil :: Cons:Nil
False :: False:True:null_!EQ:null_member[Ite][True][Ite]
notEmpty :: Cons:Nil → False:True:null_!EQ:null_member[Ite][True][Ite]
True :: False:True:null_!EQ:null_member[Ite][True][Ite]
goal :: S:0 → Cons:Nil → False:True:null_!EQ:null_member[Ite][True][Ite]
S :: S:0 → S:0
0 :: S:0
null_!EQ :: False:True:null_!EQ:null_member[Ite][True][Ite]
null_member[Ite][True][Ite] :: False:True:null_!EQ:null_member[Ite][True][Ite]

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:

member(S(x''), Cons(S(y'), xs)) → member[Ite][True][Ite](!EQ(x'', y'), S(x''), Cons(S(y'), xs)) [1]
member(0, Cons(S(y''), xs)) → member[Ite][True][Ite](False, 0, Cons(S(y''), xs)) [1]
member(S(x1), Cons(0, xs)) → member[Ite][True][Ite](False, S(x1), Cons(0, xs)) [1]
member(0, Cons(0, xs)) → member[Ite][True][Ite](True, 0, Cons(0, xs)) [1]
member(x', Cons(x, xs)) → member[Ite][True][Ite](null_!EQ, x', Cons(x, xs)) [1]
member(x, Nil) → False [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
goal(x, xs) → member(x, xs) [1]
!EQ(S(x), S(y)) → !EQ(x, y) [0]
!EQ(0, S(y)) → False [0]
!EQ(S(x), 0) → False [0]
!EQ(0, 0) → True [0]
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs) [0]
member[Ite][True][Ite](True, x, xs) → True [0]
!EQ(v0, v1) → null_!EQ [0]
member[Ite][True][Ite](v0, v1, v2) → null_member[Ite][True][Ite] [0]

The TRS has the following type information:

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
False => 1
True => 2
0 => 0
null_!EQ => 0
null_member[Ite][True][Ite] => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' = 1 + y, y >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: x >= 0, z = 1 + x, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
!EQ(z, z') -{ 0 }→ !EQ(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
goal(z, z') -{ 1 }→ member(x, xs) :|: xs >= 0, x >= 0, z' = xs, z = x
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + x1, 1 + 0 + xs) :|: xs >= 0, x1 >= 0, z' = 1 + 0 + xs, z = 1 + x1
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
member(z, z') -{ 1 }→ member[Ite][True][Ite](!EQ(x'', y'), 1 + x'', 1 + (1 + y') + xs) :|: z = 1 + x'', xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, x'' >= 0
member(z, z') -{ 1 }→ 1 :|: x >= 0, z = x, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, xs >= 0, z' = x, x >= 0, z'' = xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

(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:

!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
!EQ(z, z') -{ 0 }→ !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
goal(z, z') -{ 1 }→ member(z, z') :|: z' >= 0, z >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](!EQ(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

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

Found the following analysis order by SCC decomposition:

{ notEmpty }
{ !EQ }
{ member, member[Ite][True][Ite] }
{ goal }

(14) Obligation:

Complexity RNTS consisting of the following rules:

!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
!EQ(z, z') -{ 0 }→ !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
goal(z, z') -{ 1 }→ member(z, z') :|: z' >= 0, z >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](!EQ(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {!EQ}, {member,member[Ite][True][Ite]}, {goal}

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

Computed SIZE bound using CoFloCo for: notEmpty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(16) Obligation:

Complexity RNTS consisting of the following rules:

!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
!EQ(z, z') -{ 0 }→ !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
goal(z, z') -{ 1 }→ member(z, z') :|: z' >= 0, z >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](!EQ(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {!EQ}, {member,member[Ite][True][Ite]}, {goal}
Previous analysis results are:

notEmpty: runtime: ?, size: O(1) [2]

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

Computed RUNTIME bound using CoFloCo for: notEmpty
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:

!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
!EQ(z, z') -{ 0 }→ !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
goal(z, z') -{ 1 }→ member(z, z') :|: z' >= 0, z >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](!EQ(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {!EQ}, {member,member[Ite][True][Ite]}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [2]

(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:

!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
!EQ(z, z') -{ 0 }→ !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
goal(z, z') -{ 1 }→ member(z, z') :|: z' >= 0, z >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](!EQ(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {!EQ}, {member,member[Ite][True][Ite]}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [2]

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

Computed SIZE bound using CoFloCo for: !EQ
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(22) Obligation:

Complexity RNTS consisting of the following rules:

!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
!EQ(z, z') -{ 0 }→ !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
goal(z, z') -{ 1 }→ member(z, z') :|: z' >= 0, z >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](!EQ(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {!EQ}, {member,member[Ite][True][Ite]}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [2]
!EQ: runtime: ?, size: O(1) [2]

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
!EQ(z, z') -{ 0 }→ !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
goal(z, z') -{ 1 }→ member(z, z') :|: z' >= 0, z >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](!EQ(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [2]
!EQ: runtime: O(1) [0], size: O(1) [2]

(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:

!EQ(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
goal(z, z') -{ 1 }→ member(z, z') :|: z' >= 0, z >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs) :|: s >= 0, s <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [2]
!EQ: runtime: O(1) [0], size: O(1) [2]

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

Computed SIZE bound using CoFloCo for: member
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

Computed SIZE bound using CoFloCo for: member[Ite][True][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(28) Obligation:

Complexity RNTS consisting of the following rules:

!EQ(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
goal(z, z') -{ 1 }→ member(z, z') :|: z' >= 0, z >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs) :|: s >= 0, s <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [2]
!EQ: runtime: O(1) [0], size: O(1) [2]
member: runtime: ?, size: O(1) [2]
member[Ite][True][Ite]: runtime: ?, size: O(1) [2]

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

Computed RUNTIME bound using PUBS for: member
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z'

Computed RUNTIME bound using CoFloCo for: member[Ite][True][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z''

(30) Obligation:

Complexity RNTS consisting of the following rules:

!EQ(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
goal(z, z') -{ 1 }→ member(z, z') :|: z' >= 0, z >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs) :|: s >= 0, s <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [2]
!EQ: runtime: O(1) [0], size: O(1) [2]
member: runtime: O(n¹) [1 + z'], size: O(1) [2]
member[Ite][True][Ite]: runtime: O(n¹) [z''], size: O(1) [2]

(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:

!EQ(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
goal(z, z') -{ 2 + z' }→ s5 :|: s5 >= 0, s5 <= 2, z' >= 0, z >= 0
member(z, z') -{ 3 + xs + y' }→ s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 3 + xs + y'' }→ s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z = 0
member(z, z') -{ 2 + x + xs }→ s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 1 + xs }→ s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [2]
!EQ: runtime: O(1) [0], size: O(1) [2]
member: runtime: O(n¹) [1 + z'], size: O(1) [2]
member[Ite][True][Ite]: runtime: O(n¹) [z''], size: O(1) [2]

(33) 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: 2

(34) Obligation:

Complexity RNTS consisting of the following rules:

!EQ(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
goal(z, z') -{ 2 + z' }→ s5 :|: s5 >= 0, s5 <= 2, z' >= 0, z >= 0
member(z, z') -{ 3 + xs + y' }→ s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 3 + xs + y'' }→ s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z = 0
member(z, z') -{ 2 + x + xs }→ s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 1 + xs }→ s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [2]
!EQ: runtime: O(1) [0], size: O(1) [2]
member: runtime: O(n¹) [1 + z'], size: O(1) [2]
member[Ite][True][Ite]: runtime: O(n¹) [z''], size: O(1) [2]
goal: runtime: ?, size: O(1) [2]

(35) 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: 2 + z'

(36) Obligation:

Complexity RNTS consisting of the following rules:

!EQ(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
!EQ(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
!EQ(z, z') -{ 0 }→ 1 :|: z' - 1 >= 0, z = 0
!EQ(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
!EQ(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
goal(z, z') -{ 2 + z' }→ s5 :|: s5 >= 0, s5 <= 2, z' >= 0, z >= 0
member(z, z') -{ 3 + xs + y' }→ s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
member(z, z') -{ 3 + xs + y'' }→ s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
member(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z = 0
member(z, z') -{ 2 + x + xs }→ s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
member(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
member[Ite][True][Ite](z, z', z'') -{ 1 + xs }→ s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z'' >= 0, z' >= 0
member[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
notEmpty(z) -{ 1 }→ 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 0

Function symbols to be analyzed:
Previous analysis results are:

(37) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(0) Obligation:

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

(37) FinalProof (EQUIVALENT transformation)

(38) BOUNDS(1, n^1)