The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

overlap(Cons(x, xs), ys) → overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys)
overlap(Nil, ys) → False
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(xs, ys) → overlap(xs, ys)

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
overlap[Ite][True][Ite](False, Cons(x, xs), ys) → overlap(xs, ys)
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
overlap[Ite][True][Ite](True, xs, ys) → True
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^2).


The TRS R consists of the following rules:

overlap(Cons(x, xs), ys) → overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) [1]
overlap(Nil, ys) → False [1]
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(xs, ys) → overlap(xs, ys) [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]
overlap[Ite][True][Ite](False, Cons(x, xs), ys) → overlap(xs, ys) [0]
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs) [0]
overlap[Ite][True][Ite](True, xs, ys) → True [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:

overlap(Cons(x, xs), ys) → overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) [1]
overlap(Nil, ys) → False [1]
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(xs, ys) → overlap(xs, ys) [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]
overlap[Ite][True][Ite](False, Cons(x, xs), ys) → overlap(xs, ys) [0]
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs) [0]
overlap[Ite][True][Ite](True, xs, ys) → True [0]
member[Ite][True][Ite](True, x, xs) → True [0]

The TRS has the following type information:

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

overlap
notEmpty
goal

(c) The following functions are completely defined:

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

Due to the following rules being added:

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

And the following fresh constants:

null_!EQ, null_overlap[Ite][True][Ite], 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: overlap(Cons(x, xs), ys) → overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) [1] overlap(Nil, ys) → False [1] 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(xs, ys) → overlap(xs, ys) [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] overlap[Ite][True][Ite](False, Cons(x, xs), ys) → overlap(xs, ys) [0] member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs) [0] overlap[Ite][True][Ite](True, xs, ys) → True [0] member[Ite][True][Ite](True, x, xs) → True [0] !EQ(v0, v1) → null_!EQ [0] overlap[Ite][True][Ite](v0, v1, v2) → null_overlap[Ite][True][Ite] [0] member[Ite][True][Ite](v0, v1, v2) → null_member[Ite][True][Ite] [0] The TRS has the following type information: overlap :: Cons:Nil → Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] Cons :: S:0 → Cons:Nil → Cons:Nil overlap[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] → Cons:Nil → Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] member :: S:0 → Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] Nil :: Cons:Nil False :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] member[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] → S:0 → Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] !EQ :: S:0 → S:0 → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] notEmpty :: Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] True :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] goal :: Cons:Nil → Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] S :: S:0 → S:0 0 :: S:0 null_!EQ :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] null_overlap[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] null_member[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]: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: overlap(Cons(x, xs), Cons(x1, xs')) → overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, Cons(x1, xs')), Cons(x, xs), Cons(x1, xs')) [2] overlap(Cons(x, xs), Nil) → overlap[Ite][True][Ite](False, Cons(x, xs), Nil) [2] overlap(Nil, ys) → False [1] member(S(y'), Cons(S(x''), xs)) → member[Ite][True][Ite](!EQ(x'', y'), S(y'), Cons(S(x''), xs)) [1] member(S(y''), Cons(0, xs)) → member[Ite][True][Ite](False, S(y''), Cons(0, xs)) [1] member(0, Cons(S(x2), xs)) → member[Ite][True][Ite](False, 0, Cons(S(x2), 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(xs, ys) → overlap(xs, ys) [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] overlap[Ite][True][Ite](False, Cons(x, xs), ys) → overlap(xs, ys) [0] member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs) [0] overlap[Ite][True][Ite](True, xs, ys) → True [0] member[Ite][True][Ite](True, x, xs) → True [0] !EQ(v0, v1) → null_!EQ [0] overlap[Ite][True][Ite](v0, v1, v2) → null_overlap[Ite][True][Ite] [0] member[Ite][True][Ite](v0, v1, v2) → null_member[Ite][True][Ite] [0] The TRS has the following type information: overlap :: Cons:Nil → Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] Cons :: S:0 → Cons:Nil → Cons:Nil overlap[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] → Cons:Nil → Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] member :: S:0 → Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] Nil :: Cons:Nil False :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] member[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] → S:0 → Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] !EQ :: S:0 → S:0 → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] notEmpty :: Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] True :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] goal :: Cons:Nil → Cons:Nil → False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] S :: S:0 → S:0 0 :: S:0 null_!EQ :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] null_overlap[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] null_member[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] 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_overlap[Ite][True][Ite] => 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 }→ overlap(xs, ys) :|: xs >= 0, z = xs, z' = ys, ys >= 0
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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](1, 1 + y'', 1 + 0 + xs) :|: xs >= 0, z = 1 + y'', z' = 1 + 0 + xs, y'' >= 0
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 + y', 1 + (1 + x'') + xs) :|: xs >= 0, z = 1 + y', y' >= 0, z' = 1 + (1 + x'') + xs, 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' = ys, ys >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, ys) :|: xs >= 0, z' = 1 + x + xs, z = 1, ys >= 0, x >= 0, z'' = ys
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, xs >= 0, ys >= 0, z'' = ys, z' = xs
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 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 }→ overlap(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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 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(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

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

Found the following analysis order by SCC decomposition:

{ notEmpty } { !EQ } { member, member[Ite][True][Ite] } { overlap[Ite][True][Ite], overlap } { 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 }→ overlap(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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 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(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {notEmpty}, {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {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 }→ overlap(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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 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(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {notEmpty}, {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {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 }→ overlap(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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 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(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {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 }→ overlap(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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 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(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {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 }→ overlap(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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 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(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {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 }→ overlap(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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 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(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {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 }→ overlap(z, z') :|: z >= 0, z' >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](s', 1 + (z - 1), 1 + (1 + x'') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](s, x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {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 }→ overlap(z, z') :|: z >= 0, z' >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](s', 1 + (z - 1), 1 + (1 + x'') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](s, x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {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 }→ overlap(z, z') :|: z >= 0, z' >= 0
member(z, z') -{ 1 }→ member[Ite][True][Ite](s', 1 + (z - 1), 1 + (1 + x'') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 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 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 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
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](member[Ite][True][Ite](s, x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {overlap[Ite][True][Ite],overlap}, {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') -{ 1 }→ overlap(z, z') :|: z >= 0, z' >= 0
member(z, z') -{ 3 + x'' + xs }→ s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
member(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 3 + x2 + xs }→ s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
member(z, z') -{ 1 + z' }→ s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
member(z, z') -{ 2 + x + xs }→ s6 :|: s6 >= 0, s6 <= 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 }→ s7 :|: s7 >= 0, s7 <= 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
overlap(z, z') -{ 3 + x1 + xs' }→ overlap[Ite][True][Ite](s1, 1 + x + xs, 1 + x1 + xs') :|: s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {overlap[Ite][True][Ite],overlap}, {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: overlap[Ite][True][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

Computed SIZE bound using CoFloCo for: overlap
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') -{ 1 }→ overlap(z, z') :|: z >= 0, z' >= 0
member(z, z') -{ 3 + x'' + xs }→ s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
member(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 3 + x2 + xs }→ s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
member(z, z') -{ 1 + z' }→ s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
member(z, z') -{ 2 + x + xs }→ s6 :|: s6 >= 0, s6 <= 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 }→ s7 :|: s7 >= 0, s7 <= 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
overlap(z, z') -{ 3 + x1 + xs' }→ overlap[Ite][True][Ite](s1, 1 + x + xs, 1 + x1 + xs') :|: s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {overlap[Ite][True][Ite],overlap}, {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] overlap[Ite][True][Ite]: runtime: ?, size: O(1) [2] overlap: runtime: ?, size: O(1) [2]

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

Computed RUNTIME bound using CoFloCo for: overlap[Ite][True][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 3 + 2·z' + z'·z'' + z''

Computed RUNTIME bound using KoAT for: overlap
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 18 + 10·z + 4·z·z' + 6·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') -{ 1 }→ overlap(z, z') :|: z >= 0, z' >= 0
member(z, z') -{ 3 + x'' + xs }→ s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
member(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 3 + x2 + xs }→ s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
member(z, z') -{ 1 + z' }→ s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
member(z, z') -{ 2 + x + xs }→ s6 :|: s6 >= 0, s6 <= 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 }→ s7 :|: s7 >= 0, s7 <= 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
overlap(z, z') -{ 3 + x1 + xs' }→ overlap[Ite][True][Ite](s1, 1 + x + xs, 1 + x1 + xs') :|: s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 2 }→ overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, 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] overlap[Ite][True][Ite]: runtime: O(n2) [3 + 2·z' + z'·z'' + z''], size: O(1) [2] overlap: runtime: O(n2) [18 + 10·z + 4·z·z' + 6·z'], size: O(1) [2]

(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:

!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') -{ 19 + 10·z + 4·z·z' + 6·z' }→ s10 :|: s10 >= 0, s10 <= 2, z >= 0, z' >= 0
member(z, z') -{ 3 + x'' + xs }→ s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
member(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 3 + x2 + xs }→ s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
member(z, z') -{ 1 + z' }→ s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
member(z, z') -{ 2 + x + xs }→ s6 :|: s6 >= 0, s6 <= 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 }→ s7 :|: s7 >= 0, s7 <= 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
overlap(z, z') -{ 10 + 3·x + x·x1 + x·xs' + 3·x1 + x1·xs + 3·xs + xs·xs' + 3·xs' }→ s8 :|: s8 >= 0, s8 <= 2, s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 7 + 2·x + 2·xs }→ s9 :|: s9 >= 0, s9 <= 2, z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 18 + 10·xs + 4·xs·z'' + 6·z'' }→ s11 :|: s11 >= 0, s11 <= 2, xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, 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] overlap[Ite][True][Ite]: runtime: O(n2) [3 + 2·z' + z'·z'' + z''], size: O(1) [2] overlap: runtime: O(n2) [18 + 10·z + 4·z·z' + 6·z'], size: O(1) [2]

(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: 2

(40) 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') -{ 19 + 10·z + 4·z·z' + 6·z' }→ s10 :|: s10 >= 0, s10 <= 2, z >= 0, z' >= 0
member(z, z') -{ 3 + x'' + xs }→ s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
member(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 3 + x2 + xs }→ s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
member(z, z') -{ 1 + z' }→ s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
member(z, z') -{ 2 + x + xs }→ s6 :|: s6 >= 0, s6 <= 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 }→ s7 :|: s7 >= 0, s7 <= 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
overlap(z, z') -{ 10 + 3·x + x·x1 + x·xs' + 3·x1 + x1·xs + 3·xs + xs·xs' + 3·xs' }→ s8 :|: s8 >= 0, s8 <= 2, s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 7 + 2·x + 2·xs }→ s9 :|: s9 >= 0, s9 <= 2, z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 18 + 10·xs + 4·xs·z'' + 6·z'' }→ s11 :|: s11 >= 0, s11 <= 2, xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, 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] overlap[Ite][True][Ite]: runtime: O(n2) [3 + 2·z' + z'·z'' + z''], size: O(1) [2] overlap: runtime: O(n2) [18 + 10·z + 4·z·z' + 6·z'], size: O(1) [2] goal: runtime: ?, size: O(1) [2]

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

Computed RUNTIME bound using KoAT for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 19 + 10·z + 4·z·z' + 6·z'

(42) 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') -{ 19 + 10·z + 4·z·z' + 6·z' }→ s10 :|: s10 >= 0, s10 <= 2, z >= 0, z' >= 0
member(z, z') -{ 3 + x'' + xs }→ s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
member(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
member(z, z') -{ 3 + x2 + xs }→ s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
member(z, z') -{ 1 + z' }→ s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
member(z, z') -{ 2 + x + xs }→ s6 :|: s6 >= 0, s6 <= 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 }→ s7 :|: s7 >= 0, s7 <= 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
overlap(z, z') -{ 10 + 3·x + x·x1 + x·xs' + 3·x1 + x1·xs + 3·xs + xs·xs' + 3·xs' }→ s8 :|: s8 >= 0, s8 <= 2, s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0
overlap(z, z') -{ 7 + 2·x + 2·xs }→ s9 :|: s9 >= 0, s9 <= 2, z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
overlap(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
overlap[Ite][True][Ite](z, z', z'') -{ 18 + 10·xs + 4·xs·z'' + 6·z'' }→ s11 :|: s11 >= 0, s11 <= 2, xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 2 :|: z = 2, z' >= 0, z'' >= 0
overlap[Ite][True][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed:
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] overlap[Ite][True][Ite]: runtime: O(n2) [3 + 2·z' + z'·z'' + z''], size: O(1) [2] overlap: runtime: O(n2) [18 + 10·z + 4·z·z' + 6·z'], size: O(1) [2] goal: runtime: O(n2) [19 + 10·z + 4·z·z' + 6·z'], size: O(1) [2]

(43) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(44) BOUNDS(1, n^2)