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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 2 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 335 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 100 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 70 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 246 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 279 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 1752 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 2137 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^2)
(47) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(48) TRS for Loop Detection
    (49) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (50) BEST
        (51) proven lower bound
            (52) LowerBoundPropagationProof [FINISHED, 0 ms]
            (53) BOUNDS(n^1, INF)
        (54) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   cond1(true, x) -> cond2(even(x), x)
   cond2(true, x) -> cond1(neq(x, 0), div2(x))
   cond2(false, x) -> cond1(neq(x, 0), p(x))
   neq(0, 0) -> false
   neq(0, s(x)) -> true
   neq(s(x), 0) -> true
   neq(s(x), s(y)) -> neq(x, y)
   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   div2(0) -> 0
   div2(s(0)) -> 0
   div2(s(s(x))) -> s(div2(x))
   p(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

   cond1(true, x) -> cond2(even(x), x) [1]
   cond2(true, x) -> cond1(neq(x, 0), div2(x)) [1]
   cond2(false, x) -> cond1(neq(x, 0), p(x)) [1]
   neq(0, 0) -> false [1]
   neq(0, s(x)) -> true [1]
   neq(s(x), 0) -> true [1]
   neq(s(x), s(y)) -> neq(x, y) [1]
   even(0) -> true [1]
   even(s(0)) -> false [1]
   even(s(s(x))) -> even(x) [1]
   div2(0) -> 0 [1]
   div2(s(0)) -> 0 [1]
   div2(s(s(x))) -> s(div2(x)) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

Rewrite Strategy: INNERMOST
----------------------------------------

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   cond1(true, x) -> cond2(even(x), x) [1]
   cond2(true, x) -> cond1(neq(x, 0), div2(x)) [1]
   cond2(false, x) -> cond1(neq(x, 0), p(x)) [1]
   neq(0, 0) -> false [1]
   neq(0, s(x)) -> true [1]
   neq(s(x), 0) -> true [1]
   neq(s(x), s(y)) -> neq(x, y) [1]
   even(0) -> true [1]
   even(s(0)) -> false [1]
   even(s(s(x))) -> even(x) [1]
   div2(0) -> 0 [1]
   div2(s(0)) -> 0 [1]
   div2(s(s(x))) -> s(div2(x)) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond1 :: true:false -> 0:s:y -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0:s:y -> cond1:cond2
even :: 0:s:y -> true:false
neq :: 0:s:y -> 0:s:y -> true:false
0 :: 0:s:y
div2 :: 0:s:y -> 0:s:y
false :: true:false
p :: 0:s:y -> 0:s:y
s :: 0:s:y -> 0:s:y
y :: 0:s:y

Rewrite Strategy: INNERMOST
----------------------------------------

(5) 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:

   cond1_2
   cond2_2

(c) The following functions are completely defined:

   neq_2
   div2_1
   p_1
   even_1

Due to the following rules being added:

   neq(v0, v1) -> null_neq [0]
   div2(v0) -> null_div2 [0]
   p(v0) -> null_p [0]
   even(v0) -> null_even [0]

And the following fresh constants:    null_neq, null_div2, null_p, null_even, const

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

(6)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   cond1(true, x) -> cond2(even(x), x) [1]
   cond2(true, x) -> cond1(neq(x, 0), div2(x)) [1]
   cond2(false, x) -> cond1(neq(x, 0), p(x)) [1]
   neq(0, 0) -> false [1]
   neq(0, s(x)) -> true [1]
   neq(s(x), 0) -> true [1]
   neq(s(x), s(y)) -> neq(x, y) [1]
   even(0) -> true [1]
   even(s(0)) -> false [1]
   even(s(s(x))) -> even(x) [1]
   div2(0) -> 0 [1]
   div2(s(0)) -> 0 [1]
   div2(s(s(x))) -> s(div2(x)) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   neq(v0, v1) -> null_neq [0]
   div2(v0) -> null_div2 [0]
   p(v0) -> null_p [0]
   even(v0) -> null_even [0]

The TRS has the following type information:
cond1 :: true:false:null_neq:null_even -> 0:s:y:null_div2:null_p -> cond1:cond2
true :: true:false:null_neq:null_even
cond2 :: true:false:null_neq:null_even -> 0:s:y:null_div2:null_p -> cond1:cond2
even :: 0:s:y:null_div2:null_p -> true:false:null_neq:null_even
neq :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p -> true:false:null_neq:null_even
0 :: 0:s:y:null_div2:null_p
div2 :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p
false :: true:false:null_neq:null_even
p :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p
s :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p
y :: 0:s:y:null_div2:null_p
null_neq :: true:false:null_neq:null_even
null_div2 :: 0:s:y:null_div2:null_p
null_p :: 0:s:y:null_div2:null_p
null_even :: true:false:null_neq:null_even
const :: cond1:cond2

Rewrite Strategy: INNERMOST
----------------------------------------

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

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   cond1(true, 0) -> cond2(true, 0) [2]
   cond1(true, s(0)) -> cond2(false, s(0)) [2]
   cond1(true, s(s(x'))) -> cond2(even(x'), s(s(x'))) [2]
   cond1(true, x) -> cond2(null_even, x) [1]
   cond2(true, 0) -> cond1(false, 0) [3]
   cond2(true, 0) -> cond1(false, null_div2) [2]
   cond2(true, s(0)) -> cond1(true, 0) [3]
   cond2(true, s(s(x1))) -> cond1(true, s(div2(x1))) [3]
   cond2(true, s(x'')) -> cond1(true, null_div2) [2]
   cond2(true, 0) -> cond1(null_neq, 0) [2]
   cond2(true, s(0)) -> cond1(null_neq, 0) [2]
   cond2(true, s(s(x2))) -> cond1(null_neq, s(div2(x2))) [2]
   cond2(true, x) -> cond1(null_neq, null_div2) [1]
   cond2(false, 0) -> cond1(false, 0) [3]
   cond2(false, 0) -> cond1(false, null_p) [2]
   cond2(false, s(x3)) -> cond1(true, x3) [3]
   cond2(false, s(x3)) -> cond1(true, null_p) [2]
   cond2(false, 0) -> cond1(null_neq, 0) [2]
   cond2(false, s(x4)) -> cond1(null_neq, x4) [2]
   cond2(false, x) -> cond1(null_neq, null_p) [1]
   neq(0, 0) -> false [1]
   neq(0, s(x)) -> true [1]
   neq(s(x), 0) -> true [1]
   neq(s(x), s(y)) -> neq(x, y) [1]
   even(0) -> true [1]
   even(s(0)) -> false [1]
   even(s(s(x))) -> even(x) [1]
   div2(0) -> 0 [1]
   div2(s(0)) -> 0 [1]
   div2(s(s(x))) -> s(div2(x)) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   neq(v0, v1) -> null_neq [0]
   div2(v0) -> null_div2 [0]
   p(v0) -> null_p [0]
   even(v0) -> null_even [0]

The TRS has the following type information:
cond1 :: true:false:null_neq:null_even -> 0:s:y:null_div2:null_p -> cond1:cond2
true :: true:false:null_neq:null_even
cond2 :: true:false:null_neq:null_even -> 0:s:y:null_div2:null_p -> cond1:cond2
even :: 0:s:y:null_div2:null_p -> true:false:null_neq:null_even
neq :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p -> true:false:null_neq:null_even
0 :: 0:s:y:null_div2:null_p
div2 :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p
false :: true:false:null_neq:null_even
p :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p
s :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p
y :: 0:s:y:null_div2:null_p
null_neq :: true:false:null_neq:null_even
null_div2 :: 0:s:y:null_div2:null_p
null_p :: 0:s:y:null_div2:null_p
null_even :: true:false:null_neq:null_even
const :: cond1:cond2

Rewrite Strategy: INNERMOST
----------------------------------------

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

   true => 2
   0 => 0
   false => 1
   y => 1
   null_neq => 0
   null_div2 => 0
   null_p => 0
   null_even => 0
   const => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(x'), 1 + (1 + x')) :|: z = 2, z' = 1 + (1 + x'), x' >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, x) :|: z = 2, z' = x, x >= 0
   cond2(z, z') -{ 3 }-> cond1(2, x3) :|: z = 1, z' = 1 + x3, x3 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' = 1 + x'', x'' >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' = 1 + x3, x3 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(x1)) :|: z = 2, x1 >= 0, z' = 1 + (1 + x1)
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, x4) :|: x4 >= 0, z = 1, z' = 1 + x4
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' = x, x >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z' = x, z = 1, x >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(x2)) :|: z = 2, z' = 1 + (1 + x2), x2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   div2(z) -{ 1 }-> 1 + div2(x) :|: x >= 0, z = 1 + (1 + x)
   even(z) -{ 1 }-> even(x) :|: x >= 0, z = 1 + (1 + x)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   neq(z, z') -{ 1 }-> neq(x, 1) :|: z' = 1 + 1, x >= 0, z = 1 + x
   neq(z, z') -{ 1 }-> 2 :|: z' = 1 + x, x >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: x >= 0, z = 1 + x, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 1 }-> neq(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0


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

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

   { neq }
   { p }
   { div2 }
   { even }
   { cond2, cond1 }

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 1 }-> neq(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {neq}, {p}, {div2}, {even}, {cond2,cond1}

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

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 1 }-> neq(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {neq}, {p}, {div2}, {even}, {cond2,cond1}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 1 }-> neq(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {neq}, {p}, {div2}, {even}, {cond2,cond1}
Previous analysis results are:
  neq: runtime: ?, size: O(1) [2]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: neq
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:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 1 }-> neq(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {div2}, {even}, {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]

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

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

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {div2}, {even}, {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: p
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {div2}, {even}, {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: ?, size: O(n^1) [z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {div2}, {even}, {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {div2}, {even}, {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: div2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {div2}, {even}, {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  div2: runtime: ?, size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: div2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 }-> cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 }-> 1 + div2(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {even}, {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  div2: runtime: O(n^1) [2 + z], size: O(n^1) [z]

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

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 + z' }-> cond1(2, 1 + s') :|: s' >= 0, s' <= z' - 2, z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 + z' }-> cond1(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 2, z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {even}, {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  div2: runtime: O(n^1) [2 + z], size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 + z' }-> cond1(2, 1 + s') :|: s' >= 0, s' <= z' - 2, z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 + z' }-> cond1(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 2, z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {even}, {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  div2: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  even: runtime: ?, size: O(1) [2]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 }-> cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 + z' }-> cond1(2, 1 + s') :|: s' >= 0, s' <= z' - 2, z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 + z' }-> cond1(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 2, z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  div2: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]

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

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 + z' }-> cond2(s2, 1 + (1 + (z' - 2))) :|: s2 >= 0, s2 <= 2, z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 + z' }-> cond1(2, 1 + s') :|: s' >= 0, s' <= z' - 2, z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 + z' }-> cond1(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 2, z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   even(z) -{ 1 + z }-> s3 :|: s3 >= 0, s3 <= 2, z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  div2: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 + z' }-> cond2(s2, 1 + (1 + (z' - 2))) :|: s2 >= 0, s2 <= 2, z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 + z' }-> cond1(2, 1 + s') :|: s' >= 0, s' <= z' - 2, z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 + z' }-> cond1(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 2, z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   even(z) -{ 1 + z }-> s3 :|: s3 >= 0, s3 <= 2, z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  div2: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]
  cond2: runtime: ?, size: O(1) [0]
  cond1: runtime: ?, size: O(1) [0]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 45 + 69*z' + 6*z'^2

Computed RUNTIME bound using KoAT for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 262 + 139*z' + 12*z'^2

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z') -{ 2 + z' }-> cond2(s2, 1 + (1 + (z' - 2))) :|: s2 >= 0, s2 <= 2, z = 2, z' - 2 >= 0
   cond1(z, z') -{ 2 }-> cond2(2, 0) :|: z = 2, z' = 0
   cond1(z, z') -{ 2 }-> cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
   cond1(z, z') -{ 1 }-> cond2(0, z') :|: z = 2, z' >= 0
   cond2(z, z') -{ 3 }-> cond1(2, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 2, z' - 1 >= 0
   cond2(z, z') -{ 2 }-> cond1(2, 0) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 }-> cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
   cond2(z, z') -{ 3 + z' }-> cond1(2, 1 + s') :|: s' >= 0, s' <= z' - 2, z = 2, z' - 2 >= 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 3 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(1, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 2, z' = 1 + 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 2, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, 0) :|: z = 1, z' = 0
   cond2(z, z') -{ 1 }-> cond1(0, 0) :|: z = 1, z' >= 0
   cond2(z, z') -{ 2 }-> cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
   cond2(z, z') -{ 2 + z' }-> cond1(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 2, z = 2, z' - 2 >= 0
   div2(z) -{ 1 }-> 0 :|: z = 0
   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
   div2(z) -{ 0 }-> 0 :|: z >= 0
   div2(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   even(z) -{ 1 + z }-> s3 :|: s3 >= 0, s3 <= 2, z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   neq(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   neq(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   neq(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   neq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  neq: runtime: O(1) [2], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  div2: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]
  cond2: runtime: O(n^2) [45 + 69*z' + 6*z'^2], size: O(1) [0]
  cond1: runtime: O(n^2) [262 + 139*z' + 12*z'^2], size: O(1) [0]

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

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

(46)
BOUNDS(1, n^2)

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

(47) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(48)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   cond1(true, x) -> cond2(even(x), x)
   cond2(true, x) -> cond1(neq(x, 0), div2(x))
   cond2(false, x) -> cond1(neq(x, 0), p(x))
   neq(0, 0) -> false
   neq(0, s(x)) -> true
   neq(s(x), 0) -> true
   neq(s(x), s(y)) -> neq(x, y)
   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   div2(0) -> 0
   div2(s(0)) -> 0
   div2(s(s(x))) -> s(div2(x))
   p(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

The rewrite sequence

even(s(s(x))) ->^+ even(x)

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

The pumping substitution is [x / s(s(x))].

The result substitution is [ ].




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

(50)
Complex Obligation (BEST)

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

(51)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   cond1(true, x) -> cond2(even(x), x)
   cond2(true, x) -> cond1(neq(x, 0), div2(x))
   cond2(false, x) -> cond1(neq(x, 0), p(x))
   neq(0, 0) -> false
   neq(0, s(x)) -> true
   neq(s(x), 0) -> true
   neq(s(x), s(y)) -> neq(x, y)
   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   div2(0) -> 0
   div2(s(0)) -> 0
   div2(s(s(x))) -> s(div2(x))
   p(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(52) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(53)
BOUNDS(n^1, INF)

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

(54)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   cond1(true, x) -> cond2(even(x), x)
   cond2(true, x) -> cond1(neq(x, 0), div2(x))
   cond2(false, x) -> cond1(neq(x, 0), p(x))
   neq(0, 0) -> false
   neq(0, s(x)) -> true
   neq(s(x), 0) -> true
   neq(s(x), s(y)) -> neq(x, y)
   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   div2(0) -> 0
   div2(s(0)) -> 0
   div2(s(s(x))) -> s(div2(x))
   p(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: INNERMOST