/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 278 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), 523 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 137 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 9046 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 4413 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) FinalProof [FINISHED, 0 ms]
    (36) BOUNDS(1, n^2)
(37) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(38) TRS for Loop Detection
    (39) DecreasingLoopProof [LOWER BOUND(ID), 40 ms]
    (40) BEST
        (41) proven lower bound
            (42) LowerBoundPropagationProof [FINISHED, 0 ms]
            (43) BOUNDS(n^1, INF)
        (44) TRS for Loop Detection


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

(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:

   loop(Cons(x, xs), Nil, pp, ss) -> False
   loop(Cons(x', xs'), Cons(x, xs), pp, ss) -> loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
   loop(Nil, s, pp, ss) -> True
   match1(p, s) -> loop(p, s, p, s)

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
   loop[Ite](False, p, s, pp, Cons(x, xs)) -> loop(pp, xs, pp, xs)
   loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) -> loop(xs', xs, pp, ss)

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:

   loop(Cons(x, xs), Nil, pp, ss) -> False
   loop(Cons(x', xs'), Cons(x, xs), pp, ss) -> loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
   loop(Nil, s, pp, ss) -> True
   match1(p, s) -> loop(p, s, p, s)

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
   loop[Ite](False, p, s, pp, Cons(x, xs)) -> loop(pp, xs, pp, xs)
   loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) -> loop(xs', xs, pp, ss)

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:

   loop(Cons(x, xs), Nil, pp, ss) -> False [1]
   loop(Cons(x', xs'), Cons(x, xs), pp, ss) -> loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss) [1]
   loop(Nil, s, pp, ss) -> True [1]
   match1(p, s) -> loop(p, s, p, s) [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]
   loop[Ite](False, p, s, pp, Cons(x, xs)) -> loop(pp, xs, pp, xs) [0]
   loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) -> loop(xs', xs, pp, ss) [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:

   loop(Cons(x, xs), Nil, pp, ss) -> False [1]
   loop(Cons(x', xs'), Cons(x, xs), pp, ss) -> loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss) [1]
   loop(Nil, s, pp, ss) -> True [1]
   match1(p, s) -> loop(p, s, p, s) [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]
   loop[Ite](False, p, s, pp, Cons(x, xs)) -> loop(pp, xs, pp, xs) [0]
   loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) -> loop(xs', xs, pp, ss) [0]

The TRS has the following type information:
loop :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil -> False:True
Cons :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True
loop[Ite] :: False:True -> Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil -> False:True
!EQ :: S:0 -> S:0 -> False:True
True :: False:True
match1 :: 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:

   loop_4
   match1_2

(c) The following functions are completely defined:

   !EQ_2
   loop[Ite]_5

Due to the following rules being added:

   !EQ(v0, v1) -> null_!EQ [0]
   loop[Ite](v0, v1, v2, v3, v4) -> null_loop[Ite] [0]

And the following fresh constants:    null_!EQ, null_loop[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:

   loop(Cons(x, xs), Nil, pp, ss) -> False [1]
   loop(Cons(x', xs'), Cons(x, xs), pp, ss) -> loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss) [1]
   loop(Nil, s, pp, ss) -> True [1]
   match1(p, s) -> loop(p, s, p, s) [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]
   loop[Ite](False, p, s, pp, Cons(x, xs)) -> loop(pp, xs, pp, xs) [0]
   loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) -> loop(xs', xs, pp, ss) [0]
   !EQ(v0, v1) -> null_!EQ [0]
   loop[Ite](v0, v1, v2, v3, v4) -> null_loop[Ite] [0]

The TRS has the following type information:
loop :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_loop[Ite]
Cons :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True:null_!EQ:null_loop[Ite]
loop[Ite] :: False:True:null_!EQ:null_loop[Ite] -> Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_loop[Ite]
!EQ :: S:0 -> S:0 -> False:True:null_!EQ:null_loop[Ite]
True :: False:True:null_!EQ:null_loop[Ite]
match1 :: Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_loop[Ite]
S :: S:0 -> S:0
0 :: S:0
null_!EQ :: False:True:null_!EQ:null_loop[Ite]
null_loop[Ite] :: False:True:null_!EQ:null_loop[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:

   loop(Cons(x, xs), Nil, pp, ss) -> False [1]
   loop(Cons(S(x''), xs'), Cons(S(y'), xs), pp, ss) -> loop[Ite](!EQ(x'', y'), Cons(S(x''), xs'), Cons(S(y'), xs), pp, ss) [1]
   loop(Cons(0, xs'), Cons(S(y''), xs), pp, ss) -> loop[Ite](False, Cons(0, xs'), Cons(S(y''), xs), pp, ss) [1]
   loop(Cons(S(x1), xs'), Cons(0, xs), pp, ss) -> loop[Ite](False, Cons(S(x1), xs'), Cons(0, xs), pp, ss) [1]
   loop(Cons(0, xs'), Cons(0, xs), pp, ss) -> loop[Ite](True, Cons(0, xs'), Cons(0, xs), pp, ss) [1]
   loop(Cons(x', xs'), Cons(x, xs), pp, ss) -> loop[Ite](null_!EQ, Cons(x', xs'), Cons(x, xs), pp, ss) [1]
   loop(Nil, s, pp, ss) -> True [1]
   match1(p, s) -> loop(p, s, p, s) [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]
   loop[Ite](False, p, s, pp, Cons(x, xs)) -> loop(pp, xs, pp, xs) [0]
   loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) -> loop(xs', xs, pp, ss) [0]
   !EQ(v0, v1) -> null_!EQ [0]
   loop[Ite](v0, v1, v2, v3, v4) -> null_loop[Ite] [0]

The TRS has the following type information:
loop :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_loop[Ite]
Cons :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True:null_!EQ:null_loop[Ite]
loop[Ite] :: False:True:null_!EQ:null_loop[Ite] -> Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_loop[Ite]
!EQ :: S:0 -> S:0 -> False:True:null_!EQ:null_loop[Ite]
True :: False:True:null_!EQ:null_loop[Ite]
match1 :: Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_loop[Ite]
S :: S:0 -> S:0
0 :: S:0
null_!EQ :: False:True:null_!EQ:null_loop[Ite]
null_loop[Ite] :: False:True:null_!EQ:null_loop[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_loop[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
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](2, 1 + 0 + xs', 1 + 0 + xs, pp, ss) :|: z'' = pp, xs >= 0, z' = 1 + 0 + xs, z = 1 + 0 + xs', xs' >= 0, z1 = ss, ss >= 0, pp >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + 0 + xs', 1 + (1 + y'') + xs, pp, ss) :|: z'' = pp, xs >= 0, z = 1 + 0 + xs', xs' >= 0, z' = 1 + (1 + y'') + xs, z1 = ss, y'' >= 0, ss >= 0, pp >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + xs, pp, ss) :|: z'' = pp, xs >= 0, x1 >= 0, z' = 1 + 0 + xs, z = 1 + (1 + x1) + xs', xs' >= 0, z1 = ss, ss >= 0, pp >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](0, 1 + x' + xs', 1 + x + xs, pp, ss) :|: z'' = pp, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 = ss, ss >= 0, z = 1 + x' + xs', pp >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](!EQ(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs, pp, ss) :|: z = 1 + (1 + x'') + xs', z'' = pp, xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, z1 = ss, y' >= 0, ss >= 0, x'' >= 0, pp >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z'' = pp, z1 = ss, ss >= 0, s >= 0, z = 0, z' = s, pp >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, z'' = pp, xs >= 0, x >= 0, z1 = ss, ss >= 0, pp >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(pp, xs, pp, xs) :|: z' = p, z2 = 1 + x + xs, xs >= 0, z = 1, z1 = pp, x >= 0, p >= 0, s >= 0, z'' = s, pp >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(xs', xs, pp, ss) :|: z = 2, xs >= 0, z1 = pp, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', ss >= 0, z'' = 1 + x + xs, z2 = ss, pp >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0
   match1(z, z') -{ 1 }-> loop(p, s, p, s) :|: p >= 0, s >= 0, z = p, z' = s


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

(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
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](2, 1 + 0 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + 0 + (z - 1), 1 + (1 + y'') + xs, z'', z1) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](0, 1 + x' + xs', 1 + x + xs, z'', z1) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](!EQ(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs, z'', z1) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(xs', xs, z1, z2) :|: z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(z1, xs, z1, xs) :|: z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 1 }-> loop(z, z', z, z') :|: z >= 0, z' >= 0


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

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

   { !EQ }
   { loop[Ite], loop }
   { match1 }

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

(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
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](2, 1 + 0 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + 0 + (z - 1), 1 + (1 + y'') + xs, z'', z1) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](0, 1 + x' + xs', 1 + x + xs, z'', z1) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](!EQ(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs, z'', z1) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(xs', xs, z1, z2) :|: z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(z1, xs, z1, xs) :|: z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 1 }-> loop(z, z', z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {!EQ}, {loop[Ite],loop}, {match1}

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

(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
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](2, 1 + 0 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + 0 + (z - 1), 1 + (1 + y'') + xs, z'', z1) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](0, 1 + x' + xs', 1 + x + xs, z'', z1) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](!EQ(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs, z'', z1) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(xs', xs, z1, z2) :|: z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(z1, xs, z1, xs) :|: z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 1 }-> loop(z, z', z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {!EQ}, {loop[Ite],loop}, {match1}

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

(19) 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

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

(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
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](2, 1 + 0 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + 0 + (z - 1), 1 + (1 + y'') + xs, z'', z1) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](0, 1 + x' + xs', 1 + x + xs, z'', z1) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](!EQ(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs, z'', z1) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(xs', xs, z1, z2) :|: z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(z1, xs, z1, xs) :|: z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 1 }-> loop(z, z', z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {!EQ}, {loop[Ite],loop}, {match1}
Previous analysis results are:
  !EQ: runtime: ?, size: O(1) [2]

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

(21) 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

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

(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
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](2, 1 + 0 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + 0 + (z - 1), 1 + (1 + y'') + xs, z'', z1) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](0, 1 + x' + xs', 1 + x + xs, z'', z1) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](!EQ(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs, z'', z1) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(xs', xs, z1, z2) :|: z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(z1, xs, z1, xs) :|: z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 1 }-> loop(z, z', z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {loop[Ite],loop}, {match1}
Previous analysis results are:
  !EQ: runtime: O(1) [0], 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 }-> 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
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](s, 1 + (1 + x'') + xs', 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](2, 1 + 0 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + 0 + (z - 1), 1 + (1 + y'') + xs, z'', z1) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](0, 1 + x' + xs', 1 + x + xs, z'', z1) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(xs', xs, z1, z2) :|: z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(z1, xs, z1, xs) :|: z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 1 }-> loop(z, z', z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {loop[Ite],loop}, {match1}
Previous analysis results are:
  !EQ: runtime: O(1) [0], size: O(1) [2]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: loop
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 }-> 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
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](s, 1 + (1 + x'') + xs', 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](2, 1 + 0 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + 0 + (z - 1), 1 + (1 + y'') + xs, z'', z1) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](0, 1 + x' + xs', 1 + x + xs, z'', z1) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(xs', xs, z1, z2) :|: z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(z1, xs, z1, xs) :|: z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 1 }-> loop(z, z', z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {loop[Ite],loop}, {match1}
Previous analysis results are:
  !EQ: runtime: O(1) [0], size: O(1) [2]
  loop[Ite]: runtime: ?, size: O(1) [2]
  loop: runtime: ?, size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: loop[Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + 3*z + 4*z*z1 + 4*z' + 8*z1*z2 + 6*z2

Computed RUNTIME bound using KoAT for: loop
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 65 + 32*z + 24*z'' + 40*z''*z1 + 30*z1

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

(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
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](s, 1 + (1 + x'') + xs', 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](2, 1 + 0 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + 0 + (z - 1), 1 + (1 + y'') + xs, z'', z1) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> loop[Ite](0, 1 + x' + xs', 1 + x + xs, z'', z1) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(xs', xs, z1, z2) :|: z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> loop(z1, xs, z1, xs) :|: z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 1 }-> loop(z, z', z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {match1}
Previous analysis results are:
  !EQ: runtime: O(1) [0], size: O(1) [2]
  loop[Ite]: runtime: O(n^2) [4 + 3*z + 4*z*z1 + 4*z' + 8*z1*z2 + 6*z2], size: O(1) [2]
  loop: runtime: O(n^2) [65 + 32*z + 24*z'' + 40*z''*z1 + 30*z1], 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
   loop(z, z', z'', z1) -{ 13 + 3*s + 4*s*z'' + 4*x'' + 4*xs' + 8*z''*z1 + 6*z1 }-> s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 8 + 4*z + 4*z'' + 8*z''*z1 + 6*z1 }-> s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 16 + 4*x1 + 4*xs' + 4*z'' + 8*z''*z1 + 6*z1 }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 11 + 4*z + 8*z'' + 8*z''*z1 + 6*z1 }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 9 + 4*x' + 4*xs' + 8*z''*z1 + 6*z1 }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 65 + 30*xs + 40*xs*z1 + 56*z1 }-> s6 :|: s6 >= 0, s6 <= 2, z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 65 + 32*xs' + 24*z1 + 40*z1*z2 + 30*z2 }-> s7 :|: s7 >= 0, s7 <= 2, z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 66 + 56*z + 40*z*z' + 30*z' }-> s5 :|: s5 >= 0, s5 <= 2, z >= 0, z' >= 0

Function symbols to be analyzed: {match1}
Previous analysis results are:
  !EQ: runtime: O(1) [0], size: O(1) [2]
  loop[Ite]: runtime: O(n^2) [4 + 3*z + 4*z*z1 + 4*z' + 8*z1*z2 + 6*z2], size: O(1) [2]
  loop: runtime: O(n^2) [65 + 32*z + 24*z'' + 40*z''*z1 + 30*z1], size: O(1) [2]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: match1
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
   loop(z, z', z'', z1) -{ 13 + 3*s + 4*s*z'' + 4*x'' + 4*xs' + 8*z''*z1 + 6*z1 }-> s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 8 + 4*z + 4*z'' + 8*z''*z1 + 6*z1 }-> s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 16 + 4*x1 + 4*xs' + 4*z'' + 8*z''*z1 + 6*z1 }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 11 + 4*z + 8*z'' + 8*z''*z1 + 6*z1 }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 9 + 4*x' + 4*xs' + 8*z''*z1 + 6*z1 }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 65 + 30*xs + 40*xs*z1 + 56*z1 }-> s6 :|: s6 >= 0, s6 <= 2, z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 65 + 32*xs' + 24*z1 + 40*z1*z2 + 30*z2 }-> s7 :|: s7 >= 0, s7 <= 2, z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 66 + 56*z + 40*z*z' + 30*z' }-> s5 :|: s5 >= 0, s5 <= 2, z >= 0, z' >= 0

Function symbols to be analyzed: {match1}
Previous analysis results are:
  !EQ: runtime: O(1) [0], size: O(1) [2]
  loop[Ite]: runtime: O(n^2) [4 + 3*z + 4*z*z1 + 4*z' + 8*z1*z2 + 6*z2], size: O(1) [2]
  loop: runtime: O(n^2) [65 + 32*z + 24*z'' + 40*z''*z1 + 30*z1], size: O(1) [2]
  match1: runtime: ?, size: O(1) [2]

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(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: match1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 66 + 56*z + 40*z*z' + 30*z'

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(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
   loop(z, z', z'', z1) -{ 13 + 3*s + 4*s*z'' + 4*x'' + 4*xs' + 8*z''*z1 + 6*z1 }-> s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, z1 >= 0, x'' >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 8 + 4*z + 4*z'' + 8*z''*z1 + 6*z1 }-> s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 16 + 4*x1 + 4*xs' + 4*z'' + 8*z''*z1 + 6*z1 }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 11 + 4*z + 8*z'' + 8*z''*z1 + 6*z1 }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   loop(z, z', z'', z1) -{ 9 + 4*x' + 4*xs' + 8*z''*z1 + 6*z1 }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z1 >= 0, z = 1 + x' + xs', z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 2 :|: z1 >= 0, z' >= 0, z = 0, z'' >= 0
   loop(z, z', z'', z1) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z1 >= 0, z'' >= 0, z' = 0
   loop[Ite](z, z', z'', z1, z2) -{ 65 + 30*xs + 40*xs*z1 + 56*z1 }-> s6 :|: s6 >= 0, s6 <= 2, z2 = 1 + x + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' >= 0, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 65 + 32*xs' + 24*z1 + 40*z1*z2 + 30*z2 }-> s7 :|: s7 >= 0, s7 <= 2, z = 2, xs >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z2 >= 0, z'' = 1 + x + xs, z1 >= 0
   loop[Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   match1(z, z') -{ 66 + 56*z + 40*z*z' + 30*z' }-> s5 :|: s5 >= 0, s5 <= 2, z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  !EQ: runtime: O(1) [0], size: O(1) [2]
  loop[Ite]: runtime: O(n^2) [4 + 3*z + 4*z*z1 + 4*z' + 8*z1*z2 + 6*z2], size: O(1) [2]
  loop: runtime: O(n^2) [65 + 32*z + 24*z'' + 40*z''*z1 + 30*z1], size: O(1) [2]
  match1: runtime: O(n^2) [66 + 56*z + 40*z*z' + 30*z'], size: O(1) [2]

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(35) FinalProof (FINISHED)
Computed overall runtime complexity
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(36)
BOUNDS(1, n^2)

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(37) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(38)
Obligation:
Analyzing the following TRS for decreasing loops:

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:

   loop(Cons(x, xs), Nil, pp, ss) -> False
   loop(Cons(x', xs'), Cons(x, xs), pp, ss) -> loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
   loop(Nil, s, pp, ss) -> True
   match1(p, s) -> loop(p, s, p, s)

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
   loop[Ite](False, p, s, pp, Cons(x, xs)) -> loop(pp, xs, pp, xs)
   loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) -> loop(xs', xs, pp, ss)

Rewrite Strategy: INNERMOST
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(39) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

loop(Cons(0, xs'), Cons(0, xs), pp, ss) ->^+ loop(xs', xs, pp, ss)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [xs' / Cons(0, xs'), xs / Cons(0, xs)].

The result substitution is [ ].




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(40)
Complex Obligation (BEST)

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(41)
Obligation:
Proved the lower bound n^1 for the following 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:

   loop(Cons(x, xs), Nil, pp, ss) -> False
   loop(Cons(x', xs'), Cons(x, xs), pp, ss) -> loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
   loop(Nil, s, pp, ss) -> True
   match1(p, s) -> loop(p, s, p, s)

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
   loop[Ite](False, p, s, pp, Cons(x, xs)) -> loop(pp, xs, pp, xs)
   loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) -> loop(xs', xs, pp, ss)

Rewrite Strategy: INNERMOST
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(42) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(43)
BOUNDS(n^1, INF)

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(44)
Obligation:
Analyzing the following TRS for decreasing loops:

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:

   loop(Cons(x, xs), Nil, pp, ss) -> False
   loop(Cons(x', xs'), Cons(x, xs), pp, ss) -> loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
   loop(Nil, s, pp, ss) -> True
   match1(p, s) -> loop(p, s, p, s)

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
   loop[Ite](False, p, s, pp, Cons(x, xs)) -> loop(pp, xs, pp, xs)
   loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) -> loop(xs', xs, pp, ss)

Rewrite Strategy: INNERMOST