WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 156 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 351 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 15 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 1068 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 305 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 927 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 382 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 158 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (46) CpxRNTS
    (47) FinalProof [FINISHED, 0 ms]
    (48) BOUNDS(1, n^2)
(49) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(50) CpxRelTRS
    (51) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) typed CpxTrs
    (53) OrderProof [LOWER BOUND(ID), 0 ms]
    (54) typed CpxTrs
    (55) RewriteLemmaProof [LOWER BOUND(ID), 288 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 34 ms]
    (58) BEST
        (59) proven lower bound
            (60) LowerBoundPropagationProof [FINISHED, 0 ms]
            (61) BOUNDS(n^1, INF)
        (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
            (64) BOUNDS(1, INF)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^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) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^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
----------------------------------------

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(4)
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
----------------------------------------

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(6)
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
----------------------------------------

(7) CompletionProof (UPPER BOUND(ID))
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_2
   notEmpty_1
   goal_2

(c) The following functions are completely defined:

   member_2
   !EQ_2
   overlap[Ite][True][Ite]_3
   member[Ite][True][Ite]_3

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]

----------------------------------------

(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), 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
----------------------------------------

(9) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(10)
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
----------------------------------------

(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
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

----------------------------------------

(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 + 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


----------------------------------------

(13) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(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


----------------------------------------

(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

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

----------------------------------------

(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}

----------------------------------------

(17) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

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

----------------------------------------

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(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: {notEmpty}, {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {goal}
Previous analysis results are:
  notEmpty: runtime: ?, size: O(1) [2]

----------------------------------------

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(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]

----------------------------------------

(23) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(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: {!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]

----------------------------------------

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(26)
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]

----------------------------------------

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(28)
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]

----------------------------------------

(29) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(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: {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]

----------------------------------------

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(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') -{ 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]

----------------------------------------

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: member
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

Computed RUNTIME bound using CoFloCo for: member[Ite][True][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z''

----------------------------------------

(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') -{ 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(n^1) [2 + z'], size: O(1) [2]
  member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2]

----------------------------------------

(35) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(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') -{ 4 + 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') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
   member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
   member(z, z') -{ 3 + 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'') -{ 2 + 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') -{ 4 + 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(n^1) [2 + z'], size: O(1) [2]
  member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2]

----------------------------------------

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(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') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0
   member(z, z') -{ 4 + 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') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
   member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
   member(z, z') -{ 3 + 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'') -{ 2 + 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') -{ 4 + 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(n^1) [2 + z'], size: O(1) [2]
  member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2]
  overlap[Ite][True][Ite]: runtime: ?, size: O(1) [2]
  overlap: runtime: ?, size: O(1) [2]

----------------------------------------

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: overlap[Ite][True][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + 3*z' + z'*z'' + z''

Computed RUNTIME bound using KoAT for: overlap
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 23 + 14*z + 4*z*z' + 6*z'

----------------------------------------

(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') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0
   member(z, z') -{ 4 + 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') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
   member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
   member(z, z') -{ 3 + 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'') -{ 2 + 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') -{ 4 + 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(n^1) [2 + z'], size: O(1) [2]
  member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2]
  overlap[Ite][True][Ite]: runtime: O(n^2) [4 + 3*z' + z'*z'' + z''], size: O(1) [2]
  overlap: runtime: O(n^2) [23 + 14*z + 4*z*z' + 6*z'], size: O(1) [2]

----------------------------------------

(41) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(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') -{ 24 + 14*z + 4*z*z' + 6*z' }-> s10 :|: s10 >= 0, s10 <= 2, z >= 0, z' >= 0
   member(z, z') -{ 4 + 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') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
   member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
   member(z, z') -{ 3 + 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'') -{ 2 + 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') -{ 13 + 4*x + x*x1 + x*xs' + 3*x1 + x1*xs + 4*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') -{ 9 + 3*x + 3*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'') -{ 23 + 14*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(n^1) [2 + z'], size: O(1) [2]
  member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2]
  overlap[Ite][True][Ite]: runtime: O(n^2) [4 + 3*z' + z'*z'' + z''], size: O(1) [2]
  overlap: runtime: O(n^2) [23 + 14*z + 4*z*z' + 6*z'], size: O(1) [2]

----------------------------------------

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(44)
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') -{ 24 + 14*z + 4*z*z' + 6*z' }-> s10 :|: s10 >= 0, s10 <= 2, z >= 0, z' >= 0
   member(z, z') -{ 4 + 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') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
   member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
   member(z, z') -{ 3 + 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'') -{ 2 + 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') -{ 13 + 4*x + x*x1 + x*xs' + 3*x1 + x1*xs + 4*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') -{ 9 + 3*x + 3*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'') -{ 23 + 14*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(n^1) [2 + z'], size: O(1) [2]
  member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2]
  overlap[Ite][True][Ite]: runtime: O(n^2) [4 + 3*z' + z'*z'' + z''], size: O(1) [2]
  overlap: runtime: O(n^2) [23 + 14*z + 4*z*z' + 6*z'], size: O(1) [2]
  goal: runtime: ?, size: O(1) [2]

----------------------------------------

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 24 + 14*z + 4*z*z' + 6*z'

----------------------------------------

(46)
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') -{ 24 + 14*z + 4*z*z' + 6*z' }-> s10 :|: s10 >= 0, s10 <= 2, z >= 0, z' >= 0
   member(z, z') -{ 4 + 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') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0
   member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0
   member(z, z') -{ 3 + 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'') -{ 2 + 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') -{ 13 + 4*x + x*x1 + x*xs' + 3*x1 + x1*xs + 4*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') -{ 9 + 3*x + 3*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'') -{ 23 + 14*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(n^1) [2 + z'], size: O(1) [2]
  member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2]
  overlap[Ite][True][Ite]: runtime: O(n^2) [4 + 3*z' + z'*z'' + z''], size: O(1) [2]
  overlap: runtime: O(n^2) [23 + 14*z + 4*z*z' + 6*z'], size: O(1) [2]
  goal: runtime: O(n^2) [24 + 14*z + 4*z*z' + 6*z'], size: O(1) [2]

----------------------------------------

(47) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(48)
BOUNDS(1, n^2)

----------------------------------------

(49) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(50)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


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
----------------------------------------

(51) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(52)
Obligation:
Innermost TRS:
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)
!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

Types:
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'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'

----------------------------------------

(53) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
overlap, member, !EQ

They will be analysed ascendingly in the following order:
member < overlap
!EQ < member

----------------------------------------

(54)
Obligation:
Innermost TRS:
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)
!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

Types:
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'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
!EQ, overlap, member

They will be analysed ascendingly in the following order:
member < overlap
!EQ < member

----------------------------------------

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> False, rt in Omega(0)

Induction Base:
!EQ(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) ->_R^Omega(0)
False

Induction Step:
!EQ(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) ->_R^Omega(0)
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) ->_IH
False

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(56)
Obligation:
Innermost TRS:
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)
!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

Types:
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'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> False, rt in Omega(0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
member, overlap

They will be analysed ascendingly in the following order:
member < overlap

----------------------------------------

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
member(gen_S:0'5_0(1), gen_Cons:Nil4_0(n286_0)) -> False, rt in Omega(1 + n286_0)

Induction Base:
member(gen_S:0'5_0(1), gen_Cons:Nil4_0(0)) ->_R^Omega(1)
False

Induction Step:
member(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(n286_0, 1))) ->_R^Omega(1)
member[Ite][True][Ite](!EQ(0', gen_S:0'5_0(1)), gen_S:0'5_0(1), Cons(0', gen_Cons:Nil4_0(n286_0))) ->_L^Omega(0)
member[Ite][True][Ite](False, gen_S:0'5_0(1), Cons(0', gen_Cons:Nil4_0(n286_0))) ->_R^Omega(0)
member(gen_S:0'5_0(1), gen_Cons:Nil4_0(n286_0)) ->_IH
False

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(58)
Complex Obligation (BEST)

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(59)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
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)
!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

Types:
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'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> False, rt in Omega(0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
member, overlap

They will be analysed ascendingly in the following order:
member < overlap

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(60) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(61)
BOUNDS(n^1, INF)

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(62)
Obligation:
Innermost TRS:
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)
!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

Types:
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'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> False, rt in Omega(0)
member(gen_S:0'5_0(1), gen_Cons:Nil4_0(n286_0)) -> False, rt in Omega(1 + n286_0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
overlap
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(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
overlap(gen_Cons:Nil4_0(n633_0), gen_Cons:Nil4_0(0)) -> False, rt in Omega(1 + n633_0)

Induction Base:
overlap(gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(0)) ->_R^Omega(1)
False

Induction Step:
overlap(gen_Cons:Nil4_0(+(n633_0, 1)), gen_Cons:Nil4_0(0)) ->_R^Omega(1)
overlap[Ite][True][Ite](member(0', gen_Cons:Nil4_0(0)), Cons(0', gen_Cons:Nil4_0(n633_0)), gen_Cons:Nil4_0(0)) ->_R^Omega(1)
overlap[Ite][True][Ite](False, Cons(0', gen_Cons:Nil4_0(n633_0)), gen_Cons:Nil4_0(0)) ->_R^Omega(0)
overlap(gen_Cons:Nil4_0(n633_0), gen_Cons:Nil4_0(0)) ->_IH
False

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(64)
BOUNDS(1, INF)