WORST_CASE(Omega(n^1),O(n^2))
proof of input_Nux5B4dfSx.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(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), 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), 376 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 80 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 6 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 247 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), 82 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 1902 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 2239 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^2)
(47) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
    (49) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CpxRelTRS
    (51) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CpxRelTRS
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 3 ms]
    (54) typed CpxTrs
    (55) OrderProof [LOWER BOUND(ID), 0 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 286 ms]
    (58) typed CpxTrs
    (59) RewriteLemmaProof [LOWER BOUND(ID), 137 ms]
    (60) BEST
        (61) proven lower bound
            (62) LowerBoundPropagationProof [FINISHED, 0 ms]
            (63) BOUNDS(n^1, INF)
        (64) typed CpxTrs
            (65) RewriteLemmaProof [LOWER BOUND(ID), 64 ms]
            (66) typed CpxTrs
            (67) RewriteLemmaProof [LOWER BOUND(ID), 117 ms]
            (68) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(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 CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 22 + 10*z' + 2*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: 107 + 21*z' + 4*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) [22 + 10*z' + 2*z'^2], size: O(1) [0]
  cond1: runtime: O(n^2) [107 + 21*z' + 4*z'^2], size: O(1) [0]

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

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

(46)
BOUNDS(1, n^2)

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

(47) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond1(true, z0) -> cond2(even(z0), z0)
   cond2(true, z0) -> cond1(neq(z0, 0), div2(z0))
   cond2(false, z0) -> cond1(neq(z0, 0), p(z0))
   neq(0, 0) -> false
   neq(0, s(z0)) -> true
   neq(s(z0), 0) -> true
   neq(s(z0), s(y)) -> neq(z0, y)
   even(0) -> true
   even(s(0)) -> false
   even(s(s(z0))) -> even(z0)
   div2(0) -> 0
   div2(s(0)) -> 0
   div2(s(s(z0))) -> s(div2(z0))
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
   COND2(true, z0) -> c1(COND1(neq(z0, 0), div2(z0)), NEQ(z0, 0))
   COND2(true, z0) -> c2(COND1(neq(z0, 0), div2(z0)), DIV2(z0))
   COND2(false, z0) -> c3(COND1(neq(z0, 0), p(z0)), NEQ(z0, 0))
   COND2(false, z0) -> c4(COND1(neq(z0, 0), p(z0)), P(z0))
   NEQ(0, 0) -> c5
   NEQ(0, s(z0)) -> c6
   NEQ(s(z0), 0) -> c7
   NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
   EVEN(0) -> c9
   EVEN(s(0)) -> c10
   EVEN(s(s(z0))) -> c11(EVEN(z0))
   DIV2(0) -> c12
   DIV2(s(0)) -> c13
   DIV2(s(s(z0))) -> c14(DIV2(z0))
   P(0) -> c15
   P(s(z0)) -> c16
S tuples:
   COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
   COND2(true, z0) -> c1(COND1(neq(z0, 0), div2(z0)), NEQ(z0, 0))
   COND2(true, z0) -> c2(COND1(neq(z0, 0), div2(z0)), DIV2(z0))
   COND2(false, z0) -> c3(COND1(neq(z0, 0), p(z0)), NEQ(z0, 0))
   COND2(false, z0) -> c4(COND1(neq(z0, 0), p(z0)), P(z0))
   NEQ(0, 0) -> c5
   NEQ(0, s(z0)) -> c6
   NEQ(s(z0), 0) -> c7
   NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
   EVEN(0) -> c9
   EVEN(s(0)) -> c10
   EVEN(s(s(z0))) -> c11(EVEN(z0))
   DIV2(0) -> c12
   DIV2(s(0)) -> c13
   DIV2(s(s(z0))) -> c14(DIV2(z0))
   P(0) -> c15
   P(s(z0)) -> c16
K tuples:none
Defined Rule Symbols:   cond1_2, cond2_2, neq_2, even_1, div2_1, p_1

Defined Pair Symbols:   COND1_2, COND2_2, NEQ_2, EVEN_1, DIV2_1, P_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4_2, c5, c6, c7, c8_1, c9, c10, c11_1, c12, c13, c14_1, c15, c16


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

(49) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

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


The TRS R consists of the following rules:

   COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
   COND2(true, z0) -> c1(COND1(neq(z0, 0), div2(z0)), NEQ(z0, 0))
   COND2(true, z0) -> c2(COND1(neq(z0, 0), div2(z0)), DIV2(z0))
   COND2(false, z0) -> c3(COND1(neq(z0, 0), p(z0)), NEQ(z0, 0))
   COND2(false, z0) -> c4(COND1(neq(z0, 0), p(z0)), P(z0))
   NEQ(0, 0) -> c5
   NEQ(0, s(z0)) -> c6
   NEQ(s(z0), 0) -> c7
   NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
   EVEN(0) -> c9
   EVEN(s(0)) -> c10
   EVEN(s(s(z0))) -> c11(EVEN(z0))
   DIV2(0) -> c12
   DIV2(s(0)) -> c13
   DIV2(s(s(z0))) -> c14(DIV2(z0))
   P(0) -> c15
   P(s(z0)) -> c16

The (relative) TRS S consists of the following rules:

   cond1(true, z0) -> cond2(even(z0), z0)
   cond2(true, z0) -> cond1(neq(z0, 0), div2(z0))
   cond2(false, z0) -> cond1(neq(z0, 0), p(z0))
   neq(0, 0) -> false
   neq(0, s(z0)) -> true
   neq(s(z0), 0) -> true
   neq(s(z0), s(y)) -> neq(z0, y)
   even(0) -> true
   even(s(0)) -> false
   even(s(s(z0))) -> even(z0)
   div2(0) -> 0
   div2(s(0)) -> 0
   div2(s(s(z0))) -> s(div2(z0))
   p(0) -> 0
   p(s(z0)) -> z0

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

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

(52)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
   COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0'))
   COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0))
   COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0'))
   COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0))
   NEQ(0', 0') -> c5
   NEQ(0', s(z0)) -> c6
   NEQ(s(z0), 0') -> c7
   NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
   EVEN(0') -> c9
   EVEN(s(0')) -> c10
   EVEN(s(s(z0))) -> c11(EVEN(z0))
   DIV2(0') -> c12
   DIV2(s(0')) -> c13
   DIV2(s(s(z0))) -> c14(DIV2(z0))
   P(0') -> c15
   P(s(z0)) -> c16

The (relative) TRS S consists of the following rules:

   cond1(true, z0) -> cond2(even(z0), z0)
   cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0))
   cond2(false, z0) -> cond1(neq(z0, 0'), p(z0))
   neq(0', 0') -> false
   neq(0', s(z0)) -> true
   neq(s(z0), 0') -> true
   neq(s(z0), s(y)) -> neq(z0, y)
   even(0') -> true
   even(s(0')) -> false
   even(s(s(z0))) -> even(z0)
   div2(0') -> 0'
   div2(s(0')) -> 0'
   div2(s(s(z0))) -> s(div2(z0))
   p(0') -> 0'
   p(s(z0)) -> z0

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

(53) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(54)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0'))
COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0))
COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0'))
COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0))
NEQ(0', 0') -> c5
NEQ(0', s(z0)) -> c6
NEQ(s(z0), 0') -> c7
NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
EVEN(0') -> c9
EVEN(s(0')) -> c10
EVEN(s(s(z0))) -> c11(EVEN(z0))
DIV2(0') -> c12
DIV2(s(0')) -> c13
DIV2(s(s(z0))) -> c14(DIV2(z0))
P(0') -> c15
P(s(z0)) -> c16
cond1(true, z0) -> cond2(even(z0), z0)
cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0))
cond2(false, z0) -> cond1(neq(z0, 0'), p(z0))
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(y)) -> neq(z0, y)
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(z0))) -> s(div2(z0))
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s:y -> c
true :: true:false
c :: c1:c2:c3:c4 -> c9:c10:c11 -> c
COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4
even :: 0':s:y -> true:false
EVEN :: 0':s:y -> c9:c10:c11
c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8
c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4
DIV2 :: 0':s:y -> c12:c13:c14
false :: true:false
c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
p :: 0':s:y -> 0':s:y
c4 :: c -> c15:c16 -> c1:c2:c3:c4
P :: 0':s:y -> c15:c16
c5 :: c5:c6:c7:c8
s :: 0':s:y -> 0':s:y
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
y :: 0':s:y
c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14
c13 :: c12:c13:c14
c14 :: c12:c13:c14 -> c12:c13:c14
c15 :: c15:c16
c16 :: c15:c16
cond1 :: true:false -> 0':s:y -> cond1:cond2
cond2 :: true:false -> 0':s:y -> cond1:cond2
hole_c1_17 :: c
hole_true:false2_17 :: true:false
hole_0':s:y3_17 :: 0':s:y
hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4
hole_c9:c10:c115_17 :: c9:c10:c11
hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8
hole_c12:c13:c147_17 :: c12:c13:c14
hole_c15:c168_17 :: c15:c16
hole_cond1:cond29_17 :: cond1:cond2
gen_0':s:y10_17 :: Nat -> 0':s:y
gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11
gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8
gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14

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

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
COND1, COND2, even, EVEN, neq, div2, NEQ, DIV2, cond1, cond2

They will be analysed ascendingly in the following order:
COND1 = COND2
even < COND1
EVEN < COND1
neq < COND2
div2 < COND2
NEQ < COND2
DIV2 < COND2
even < cond1
neq < cond2
div2 < cond2
cond1 = cond2

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

(56)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0'))
COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0))
COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0'))
COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0))
NEQ(0', 0') -> c5
NEQ(0', s(z0)) -> c6
NEQ(s(z0), 0') -> c7
NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
EVEN(0') -> c9
EVEN(s(0')) -> c10
EVEN(s(s(z0))) -> c11(EVEN(z0))
DIV2(0') -> c12
DIV2(s(0')) -> c13
DIV2(s(s(z0))) -> c14(DIV2(z0))
P(0') -> c15
P(s(z0)) -> c16
cond1(true, z0) -> cond2(even(z0), z0)
cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0))
cond2(false, z0) -> cond1(neq(z0, 0'), p(z0))
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(y)) -> neq(z0, y)
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(z0))) -> s(div2(z0))
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s:y -> c
true :: true:false
c :: c1:c2:c3:c4 -> c9:c10:c11 -> c
COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4
even :: 0':s:y -> true:false
EVEN :: 0':s:y -> c9:c10:c11
c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8
c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4
DIV2 :: 0':s:y -> c12:c13:c14
false :: true:false
c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
p :: 0':s:y -> 0':s:y
c4 :: c -> c15:c16 -> c1:c2:c3:c4
P :: 0':s:y -> c15:c16
c5 :: c5:c6:c7:c8
s :: 0':s:y -> 0':s:y
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
y :: 0':s:y
c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14
c13 :: c12:c13:c14
c14 :: c12:c13:c14 -> c12:c13:c14
c15 :: c15:c16
c16 :: c15:c16
cond1 :: true:false -> 0':s:y -> cond1:cond2
cond2 :: true:false -> 0':s:y -> cond1:cond2
hole_c1_17 :: c
hole_true:false2_17 :: true:false
hole_0':s:y3_17 :: 0':s:y
hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4
hole_c9:c10:c115_17 :: c9:c10:c11
hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8
hole_c12:c13:c147_17 :: c12:c13:c14
hole_c15:c168_17 :: c15:c16
hole_cond1:cond29_17 :: cond1:cond2
gen_0':s:y10_17 :: Nat -> 0':s:y
gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11
gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8
gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14


Generator Equations:
gen_0':s:y10_17(0) <=> 0'
gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x))
gen_c9:c10:c1111_17(0) <=> c9
gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x))
gen_c5:c6:c7:c812_17(0) <=> c5
gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x))
gen_c12:c13:c1413_17(0) <=> c12
gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x))


The following defined symbols remain to be analysed:
even, COND1, COND2, EVEN, neq, div2, NEQ, DIV2, cond1, cond2

They will be analysed ascendingly in the following order:
COND1 = COND2
even < COND1
EVEN < COND1
neq < COND2
div2 < COND2
NEQ < COND2
DIV2 < COND2
even < cond1
neq < cond2
div2 < cond2
cond1 = cond2

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0)

Induction Base:
even(gen_0':s:y10_17(*(2, 0))) ->_R^Omega(0)
true

Induction Step:
even(gen_0':s:y10_17(*(2, +(n15_17, 1)))) ->_R^Omega(0)
even(gen_0':s:y10_17(*(2, n15_17))) ->_IH
true

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

(58)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0'))
COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0))
COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0'))
COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0))
NEQ(0', 0') -> c5
NEQ(0', s(z0)) -> c6
NEQ(s(z0), 0') -> c7
NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
EVEN(0') -> c9
EVEN(s(0')) -> c10
EVEN(s(s(z0))) -> c11(EVEN(z0))
DIV2(0') -> c12
DIV2(s(0')) -> c13
DIV2(s(s(z0))) -> c14(DIV2(z0))
P(0') -> c15
P(s(z0)) -> c16
cond1(true, z0) -> cond2(even(z0), z0)
cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0))
cond2(false, z0) -> cond1(neq(z0, 0'), p(z0))
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(y)) -> neq(z0, y)
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(z0))) -> s(div2(z0))
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s:y -> c
true :: true:false
c :: c1:c2:c3:c4 -> c9:c10:c11 -> c
COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4
even :: 0':s:y -> true:false
EVEN :: 0':s:y -> c9:c10:c11
c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8
c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4
DIV2 :: 0':s:y -> c12:c13:c14
false :: true:false
c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
p :: 0':s:y -> 0':s:y
c4 :: c -> c15:c16 -> c1:c2:c3:c4
P :: 0':s:y -> c15:c16
c5 :: c5:c6:c7:c8
s :: 0':s:y -> 0':s:y
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
y :: 0':s:y
c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14
c13 :: c12:c13:c14
c14 :: c12:c13:c14 -> c12:c13:c14
c15 :: c15:c16
c16 :: c15:c16
cond1 :: true:false -> 0':s:y -> cond1:cond2
cond2 :: true:false -> 0':s:y -> cond1:cond2
hole_c1_17 :: c
hole_true:false2_17 :: true:false
hole_0':s:y3_17 :: 0':s:y
hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4
hole_c9:c10:c115_17 :: c9:c10:c11
hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8
hole_c12:c13:c147_17 :: c12:c13:c14
hole_c15:c168_17 :: c15:c16
hole_cond1:cond29_17 :: cond1:cond2
gen_0':s:y10_17 :: Nat -> 0':s:y
gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11
gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8
gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14


Lemmas:
even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0)


Generator Equations:
gen_0':s:y10_17(0) <=> 0'
gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x))
gen_c9:c10:c1111_17(0) <=> c9
gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x))
gen_c5:c6:c7:c812_17(0) <=> c5
gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x))
gen_c12:c13:c1413_17(0) <=> c12
gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x))


The following defined symbols remain to be analysed:
EVEN, COND1, COND2, neq, div2, NEQ, DIV2, cond1, cond2

They will be analysed ascendingly in the following order:
COND1 = COND2
EVEN < COND1
neq < COND2
div2 < COND2
NEQ < COND2
DIV2 < COND2
neq < cond2
div2 < cond2
cond1 = cond2

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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
EVEN(gen_0':s:y10_17(*(2, n243_17))) -> gen_c9:c10:c1111_17(n243_17), rt in Omega(1 + n243_17)

Induction Base:
EVEN(gen_0':s:y10_17(*(2, 0))) ->_R^Omega(1)
c9

Induction Step:
EVEN(gen_0':s:y10_17(*(2, +(n243_17, 1)))) ->_R^Omega(1)
c11(EVEN(gen_0':s:y10_17(*(2, n243_17)))) ->_IH
c11(gen_c9:c10:c1111_17(c244_17))

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

(60)
Complex Obligation (BEST)

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

(61)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0'))
COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0))
COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0'))
COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0))
NEQ(0', 0') -> c5
NEQ(0', s(z0)) -> c6
NEQ(s(z0), 0') -> c7
NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
EVEN(0') -> c9
EVEN(s(0')) -> c10
EVEN(s(s(z0))) -> c11(EVEN(z0))
DIV2(0') -> c12
DIV2(s(0')) -> c13
DIV2(s(s(z0))) -> c14(DIV2(z0))
P(0') -> c15
P(s(z0)) -> c16
cond1(true, z0) -> cond2(even(z0), z0)
cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0))
cond2(false, z0) -> cond1(neq(z0, 0'), p(z0))
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(y)) -> neq(z0, y)
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(z0))) -> s(div2(z0))
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s:y -> c
true :: true:false
c :: c1:c2:c3:c4 -> c9:c10:c11 -> c
COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4
even :: 0':s:y -> true:false
EVEN :: 0':s:y -> c9:c10:c11
c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8
c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4
DIV2 :: 0':s:y -> c12:c13:c14
false :: true:false
c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
p :: 0':s:y -> 0':s:y
c4 :: c -> c15:c16 -> c1:c2:c3:c4
P :: 0':s:y -> c15:c16
c5 :: c5:c6:c7:c8
s :: 0':s:y -> 0':s:y
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
y :: 0':s:y
c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14
c13 :: c12:c13:c14
c14 :: c12:c13:c14 -> c12:c13:c14
c15 :: c15:c16
c16 :: c15:c16
cond1 :: true:false -> 0':s:y -> cond1:cond2
cond2 :: true:false -> 0':s:y -> cond1:cond2
hole_c1_17 :: c
hole_true:false2_17 :: true:false
hole_0':s:y3_17 :: 0':s:y
hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4
hole_c9:c10:c115_17 :: c9:c10:c11
hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8
hole_c12:c13:c147_17 :: c12:c13:c14
hole_c15:c168_17 :: c15:c16
hole_cond1:cond29_17 :: cond1:cond2
gen_0':s:y10_17 :: Nat -> 0':s:y
gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11
gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8
gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14


Lemmas:
even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0)


Generator Equations:
gen_0':s:y10_17(0) <=> 0'
gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x))
gen_c9:c10:c1111_17(0) <=> c9
gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x))
gen_c5:c6:c7:c812_17(0) <=> c5
gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x))
gen_c12:c13:c1413_17(0) <=> c12
gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x))


The following defined symbols remain to be analysed:
EVEN, COND1, COND2, neq, div2, NEQ, DIV2, cond1, cond2

They will be analysed ascendingly in the following order:
COND1 = COND2
EVEN < COND1
neq < COND2
div2 < COND2
NEQ < COND2
DIV2 < COND2
neq < cond2
div2 < cond2
cond1 = cond2

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

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

(63)
BOUNDS(n^1, INF)

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

(64)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0'))
COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0))
COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0'))
COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0))
NEQ(0', 0') -> c5
NEQ(0', s(z0)) -> c6
NEQ(s(z0), 0') -> c7
NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
EVEN(0') -> c9
EVEN(s(0')) -> c10
EVEN(s(s(z0))) -> c11(EVEN(z0))
DIV2(0') -> c12
DIV2(s(0')) -> c13
DIV2(s(s(z0))) -> c14(DIV2(z0))
P(0') -> c15
P(s(z0)) -> c16
cond1(true, z0) -> cond2(even(z0), z0)
cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0))
cond2(false, z0) -> cond1(neq(z0, 0'), p(z0))
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(y)) -> neq(z0, y)
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(z0))) -> s(div2(z0))
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s:y -> c
true :: true:false
c :: c1:c2:c3:c4 -> c9:c10:c11 -> c
COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4
even :: 0':s:y -> true:false
EVEN :: 0':s:y -> c9:c10:c11
c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8
c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4
DIV2 :: 0':s:y -> c12:c13:c14
false :: true:false
c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
p :: 0':s:y -> 0':s:y
c4 :: c -> c15:c16 -> c1:c2:c3:c4
P :: 0':s:y -> c15:c16
c5 :: c5:c6:c7:c8
s :: 0':s:y -> 0':s:y
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
y :: 0':s:y
c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14
c13 :: c12:c13:c14
c14 :: c12:c13:c14 -> c12:c13:c14
c15 :: c15:c16
c16 :: c15:c16
cond1 :: true:false -> 0':s:y -> cond1:cond2
cond2 :: true:false -> 0':s:y -> cond1:cond2
hole_c1_17 :: c
hole_true:false2_17 :: true:false
hole_0':s:y3_17 :: 0':s:y
hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4
hole_c9:c10:c115_17 :: c9:c10:c11
hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8
hole_c12:c13:c147_17 :: c12:c13:c14
hole_c15:c168_17 :: c15:c16
hole_cond1:cond29_17 :: cond1:cond2
gen_0':s:y10_17 :: Nat -> 0':s:y
gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11
gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8
gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14


Lemmas:
even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0)
EVEN(gen_0':s:y10_17(*(2, n243_17))) -> gen_c9:c10:c1111_17(n243_17), rt in Omega(1 + n243_17)


Generator Equations:
gen_0':s:y10_17(0) <=> 0'
gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x))
gen_c9:c10:c1111_17(0) <=> c9
gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x))
gen_c5:c6:c7:c812_17(0) <=> c5
gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x))
gen_c12:c13:c1413_17(0) <=> c12
gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x))


The following defined symbols remain to be analysed:
neq, COND1, COND2, div2, NEQ, DIV2, cond1, cond2

They will be analysed ascendingly in the following order:
COND1 = COND2
neq < COND2
div2 < COND2
NEQ < COND2
DIV2 < COND2
neq < cond2
div2 < cond2
cond1 = cond2

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
div2(gen_0':s:y10_17(*(2, n779_17))) -> gen_0':s:y10_17(n779_17), rt in Omega(0)

Induction Base:
div2(gen_0':s:y10_17(*(2, 0))) ->_R^Omega(0)
0'

Induction Step:
div2(gen_0':s:y10_17(*(2, +(n779_17, 1)))) ->_R^Omega(0)
s(div2(gen_0':s:y10_17(*(2, n779_17)))) ->_IH
s(gen_0':s:y10_17(c780_17))

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

(66)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0'))
COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0))
COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0'))
COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0))
NEQ(0', 0') -> c5
NEQ(0', s(z0)) -> c6
NEQ(s(z0), 0') -> c7
NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
EVEN(0') -> c9
EVEN(s(0')) -> c10
EVEN(s(s(z0))) -> c11(EVEN(z0))
DIV2(0') -> c12
DIV2(s(0')) -> c13
DIV2(s(s(z0))) -> c14(DIV2(z0))
P(0') -> c15
P(s(z0)) -> c16
cond1(true, z0) -> cond2(even(z0), z0)
cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0))
cond2(false, z0) -> cond1(neq(z0, 0'), p(z0))
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(y)) -> neq(z0, y)
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(z0))) -> s(div2(z0))
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s:y -> c
true :: true:false
c :: c1:c2:c3:c4 -> c9:c10:c11 -> c
COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4
even :: 0':s:y -> true:false
EVEN :: 0':s:y -> c9:c10:c11
c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8
c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4
DIV2 :: 0':s:y -> c12:c13:c14
false :: true:false
c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
p :: 0':s:y -> 0':s:y
c4 :: c -> c15:c16 -> c1:c2:c3:c4
P :: 0':s:y -> c15:c16
c5 :: c5:c6:c7:c8
s :: 0':s:y -> 0':s:y
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
y :: 0':s:y
c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14
c13 :: c12:c13:c14
c14 :: c12:c13:c14 -> c12:c13:c14
c15 :: c15:c16
c16 :: c15:c16
cond1 :: true:false -> 0':s:y -> cond1:cond2
cond2 :: true:false -> 0':s:y -> cond1:cond2
hole_c1_17 :: c
hole_true:false2_17 :: true:false
hole_0':s:y3_17 :: 0':s:y
hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4
hole_c9:c10:c115_17 :: c9:c10:c11
hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8
hole_c12:c13:c147_17 :: c12:c13:c14
hole_c15:c168_17 :: c15:c16
hole_cond1:cond29_17 :: cond1:cond2
gen_0':s:y10_17 :: Nat -> 0':s:y
gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11
gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8
gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14


Lemmas:
even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0)
EVEN(gen_0':s:y10_17(*(2, n243_17))) -> gen_c9:c10:c1111_17(n243_17), rt in Omega(1 + n243_17)
div2(gen_0':s:y10_17(*(2, n779_17))) -> gen_0':s:y10_17(n779_17), rt in Omega(0)


Generator Equations:
gen_0':s:y10_17(0) <=> 0'
gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x))
gen_c9:c10:c1111_17(0) <=> c9
gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x))
gen_c5:c6:c7:c812_17(0) <=> c5
gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x))
gen_c12:c13:c1413_17(0) <=> c12
gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x))


The following defined symbols remain to be analysed:
NEQ, COND1, COND2, DIV2, cond1, cond2

They will be analysed ascendingly in the following order:
COND1 = COND2
NEQ < COND2
DIV2 < COND2
cond1 = cond2

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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
DIV2(gen_0':s:y10_17(*(2, n1244_17))) -> gen_c12:c13:c1413_17(n1244_17), rt in Omega(1 + n1244_17)

Induction Base:
DIV2(gen_0':s:y10_17(*(2, 0))) ->_R^Omega(1)
c12

Induction Step:
DIV2(gen_0':s:y10_17(*(2, +(n1244_17, 1)))) ->_R^Omega(1)
c14(DIV2(gen_0':s:y10_17(*(2, n1244_17)))) ->_IH
c14(gen_c12:c13:c1413_17(c1245_17))

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

(68)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0))
COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0'))
COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0))
COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0'))
COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0))
NEQ(0', 0') -> c5
NEQ(0', s(z0)) -> c6
NEQ(s(z0), 0') -> c7
NEQ(s(z0), s(y)) -> c8(NEQ(z0, y))
EVEN(0') -> c9
EVEN(s(0')) -> c10
EVEN(s(s(z0))) -> c11(EVEN(z0))
DIV2(0') -> c12
DIV2(s(0')) -> c13
DIV2(s(s(z0))) -> c14(DIV2(z0))
P(0') -> c15
P(s(z0)) -> c16
cond1(true, z0) -> cond2(even(z0), z0)
cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0))
cond2(false, z0) -> cond1(neq(z0, 0'), p(z0))
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(y)) -> neq(z0, y)
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(z0))) -> s(div2(z0))
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s:y -> c
true :: true:false
c :: c1:c2:c3:c4 -> c9:c10:c11 -> c
COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4
even :: 0':s:y -> true:false
EVEN :: 0':s:y -> c9:c10:c11
c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8
c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4
DIV2 :: 0':s:y -> c12:c13:c14
false :: true:false
c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4
p :: 0':s:y -> 0':s:y
c4 :: c -> c15:c16 -> c1:c2:c3:c4
P :: 0':s:y -> c15:c16
c5 :: c5:c6:c7:c8
s :: 0':s:y -> 0':s:y
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
y :: 0':s:y
c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14
c13 :: c12:c13:c14
c14 :: c12:c13:c14 -> c12:c13:c14
c15 :: c15:c16
c16 :: c15:c16
cond1 :: true:false -> 0':s:y -> cond1:cond2
cond2 :: true:false -> 0':s:y -> cond1:cond2
hole_c1_17 :: c
hole_true:false2_17 :: true:false
hole_0':s:y3_17 :: 0':s:y
hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4
hole_c9:c10:c115_17 :: c9:c10:c11
hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8
hole_c12:c13:c147_17 :: c12:c13:c14
hole_c15:c168_17 :: c15:c16
hole_cond1:cond29_17 :: cond1:cond2
gen_0':s:y10_17 :: Nat -> 0':s:y
gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11
gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8
gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14


Lemmas:
even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0)
EVEN(gen_0':s:y10_17(*(2, n243_17))) -> gen_c9:c10:c1111_17(n243_17), rt in Omega(1 + n243_17)
div2(gen_0':s:y10_17(*(2, n779_17))) -> gen_0':s:y10_17(n779_17), rt in Omega(0)
DIV2(gen_0':s:y10_17(*(2, n1244_17))) -> gen_c12:c13:c1413_17(n1244_17), rt in Omega(1 + n1244_17)


Generator Equations:
gen_0':s:y10_17(0) <=> 0'
gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x))
gen_c9:c10:c1111_17(0) <=> c9
gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x))
gen_c5:c6:c7:c812_17(0) <=> c5
gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x))
gen_c12:c13:c1413_17(0) <=> c12
gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x))


The following defined symbols remain to be analysed:
cond2, COND1, COND2, cond1

They will be analysed ascendingly in the following order:
COND1 = COND2
cond1 = cond2