/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 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), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 2 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 409 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 120 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 231 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 85 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 259 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 2022 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 2066 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^2)
(47) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CpxTRS
    (49) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) typed CpxTrs
    (51) OrderProof [LOWER BOUND(ID), 0 ms]
    (52) typed CpxTrs
    (53) RewriteLemmaProof [LOWER BOUND(ID), 226 ms]
    (54) BEST
        (55) proven lower bound
            (56) LowerBoundPropagationProof [FINISHED, 0 ms]
            (57) BOUNDS(n^1, INF)
        (58) typed CpxTrs
            (59) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
            (60) typed CpxTrs


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

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

(48)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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

(50)
Obligation:
Innermost TRS:
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

Types:
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
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat -> 0':s:y

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

(51) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond1, cond2, even, neq, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
even < cond1
neq < cond2
div2 < cond2

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

(52)
Obligation:
Innermost TRS:
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

Types:
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
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat -> 0':s:y


Generator Equations:
gen_0':s:y4_0(0) <=> 0'
gen_0':s:y4_0(+(x, 1)) <=> s(gen_0':s:y4_0(x))


The following defined symbols remain to be analysed:
even, cond1, cond2, neq, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
even < cond1
neq < cond2
div2 < cond2

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_0':s:y4_0(*(2, n6_0))) -> true, rt in Omega(1 + n6_0)

Induction Base:
even(gen_0':s:y4_0(*(2, 0))) ->_R^Omega(1)
true

Induction Step:
even(gen_0':s:y4_0(*(2, +(n6_0, 1)))) ->_R^Omega(1)
even(gen_0':s:y4_0(*(2, n6_0))) ->_IH
true

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

(54)
Complex Obligation (BEST)

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

(55)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
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

Types:
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
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat -> 0':s:y


Generator Equations:
gen_0':s:y4_0(0) <=> 0'
gen_0':s:y4_0(+(x, 1)) <=> s(gen_0':s:y4_0(x))


The following defined symbols remain to be analysed:
even, cond1, cond2, neq, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
even < cond1
neq < cond2
div2 < cond2

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

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

(57)
BOUNDS(n^1, INF)

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

(58)
Obligation:
Innermost TRS:
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

Types:
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
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat -> 0':s:y


Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) -> true, rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s:y4_0(0) <=> 0'
gen_0':s:y4_0(+(x, 1)) <=> s(gen_0':s:y4_0(x))


The following defined symbols remain to be analysed:
neq, cond1, cond2, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
neq < cond2
div2 < cond2

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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
div2(gen_0':s:y4_0(*(2, n208_0))) -> gen_0':s:y4_0(n208_0), rt in Omega(1 + n208_0)

Induction Base:
div2(gen_0':s:y4_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
div2(gen_0':s:y4_0(*(2, +(n208_0, 1)))) ->_R^Omega(1)
s(div2(gen_0':s:y4_0(*(2, n208_0)))) ->_IH
s(gen_0':s:y4_0(c209_0))

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

(60)
Obligation:
Innermost TRS:
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

Types:
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
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat -> 0':s:y


Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) -> true, rt in Omega(1 + n6_0)
div2(gen_0':s:y4_0(*(2, n208_0))) -> gen_0':s:y4_0(n208_0), rt in Omega(1 + n208_0)


Generator Equations:
gen_0':s:y4_0(0) <=> 0'
gen_0':s:y4_0(+(x, 1)) <=> s(gen_0':s:y4_0(x))


The following defined symbols remain to be analysed:
cond2, cond1

They will be analysed ascendingly in the following order:
cond1 = cond2