WORST_CASE(Omega(n^1),O(n^2))
proof of input_J5Kbzx9vBr.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# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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), 6 ms]
    (10) CpxRNTS
    (11) InliningProof [UPPER BOUND(ID), 1613 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 126 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 90 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 171 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 313 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 7 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 260 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 1438 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 1673 ms]
    (46) CpxRNTS
    (47) FinalProof [FINISHED, 0 ms]
    (48) BOUNDS(1, n^2)
(49) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(50) CdtProblem
    (51) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CpxRelTRS
    (53) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CpxRelTRS
    (55) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) typed CpxTrs
    (57) OrderProof [LOWER BOUND(ID), 0 ms]
    (58) typed CpxTrs
    (59) RewriteLemmaProof [LOWER BOUND(ID), 275 ms]
    (60) typed CpxTrs
    (61) RewriteLemmaProof [LOWER BOUND(ID), 99 ms]
    (62) BEST
        (63) proven lower bound
            (64) LowerBoundPropagationProof [FINISHED, 0 ms]
            (65) BOUNDS(n^1, INF)
        (66) 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:

   cond(true, x) -> cond(and(even(x), gr(x, 0)), p(x))
   and(x, false) -> false
   and(false, x) -> false
   and(true, true) -> true
   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   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:

   cond(true, x) -> cond(and(even(x), gr(x, 0)), p(x)) [1]
   and(x, false) -> false [1]
   and(false, x) -> false [1]
   and(true, true) -> true [1]
   even(0) -> true [1]
   even(s(0)) -> false [1]
   even(s(s(x))) -> even(x) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [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:

   cond(true, x) -> cond(and(even(x), gr(x, 0)), p(x)) [1]
   and(x, false) -> false [1]
   and(false, x) -> false [1]
   and(true, true) -> true [1]
   even(0) -> true [1]
   even(s(0)) -> false [1]
   even(s(s(x))) -> even(x) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond :: true:false -> 0:s:y -> cond
true :: true:false
and :: true:false -> true:false -> true:false
even :: 0:s:y -> true:false
gr :: 0:s:y -> 0:s:y -> true:false
0 :: 0:s:y
p :: 0:s:y -> 0:s:y
false :: true:false
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:

   cond_2

(c) The following functions are completely defined:

   and_2
   even_1
   gr_2
   p_1

Due to the following rules being added:

   even(v0) -> null_even [0]
   gr(v0, v1) -> null_gr [0]
   p(v0) -> null_p [0]
   and(v0, v1) -> null_and [0]

And the following fresh constants:    null_even, null_gr, null_p, null_and, 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:

   cond(true, x) -> cond(and(even(x), gr(x, 0)), p(x)) [1]
   and(x, false) -> false [1]
   and(false, x) -> false [1]
   and(true, true) -> true [1]
   even(0) -> true [1]
   even(s(0)) -> false [1]
   even(s(s(x))) -> even(x) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   even(v0) -> null_even [0]
   gr(v0, v1) -> null_gr [0]
   p(v0) -> null_p [0]
   and(v0, v1) -> null_and [0]

The TRS has the following type information:
cond :: true:false:null_even:null_gr:null_and -> 0:s:y:null_p -> cond
true :: true:false:null_even:null_gr:null_and
and :: true:false:null_even:null_gr:null_and -> true:false:null_even:null_gr:null_and -> true:false:null_even:null_gr:null_and
even :: 0:s:y:null_p -> true:false:null_even:null_gr:null_and
gr :: 0:s:y:null_p -> 0:s:y:null_p -> true:false:null_even:null_gr:null_and
0 :: 0:s:y:null_p
p :: 0:s:y:null_p -> 0:s:y:null_p
false :: true:false:null_even:null_gr:null_and
s :: 0:s:y:null_p -> 0:s:y:null_p
y :: 0:s:y:null_p
null_even :: true:false:null_even:null_gr:null_and
null_gr :: true:false:null_even:null_gr:null_and
null_p :: 0:s:y:null_p
null_and :: true:false:null_even:null_gr:null_and
const :: cond

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:

   cond(true, 0) -> cond(and(true, false), 0) [4]
   cond(true, 0) -> cond(and(true, false), null_p) [3]
   cond(true, 0) -> cond(and(true, null_gr), 0) [3]
   cond(true, 0) -> cond(and(true, null_gr), null_p) [2]
   cond(true, s(0)) -> cond(and(false, true), 0) [4]
   cond(true, s(0)) -> cond(and(false, true), null_p) [3]
   cond(true, s(0)) -> cond(and(false, null_gr), 0) [3]
   cond(true, s(0)) -> cond(and(false, null_gr), null_p) [2]
   cond(true, s(s(x'))) -> cond(and(even(x'), true), s(x')) [4]
   cond(true, s(s(x'))) -> cond(and(even(x'), true), null_p) [3]
   cond(true, s(s(x'))) -> cond(and(even(x'), null_gr), s(x')) [3]
   cond(true, s(s(x'))) -> cond(and(even(x'), null_gr), null_p) [2]
   cond(true, 0) -> cond(and(null_even, false), 0) [3]
   cond(true, 0) -> cond(and(null_even, false), null_p) [2]
   cond(true, s(x'')) -> cond(and(null_even, true), x'') [3]
   cond(true, s(x'')) -> cond(and(null_even, true), null_p) [2]
   cond(true, 0) -> cond(and(null_even, null_gr), 0) [2]
   cond(true, s(x1)) -> cond(and(null_even, null_gr), x1) [2]
   cond(true, x) -> cond(and(null_even, null_gr), null_p) [1]
   and(x, false) -> false [1]
   and(false, x) -> false [1]
   and(true, true) -> true [1]
   even(0) -> true [1]
   even(s(0)) -> false [1]
   even(s(s(x))) -> even(x) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   even(v0) -> null_even [0]
   gr(v0, v1) -> null_gr [0]
   p(v0) -> null_p [0]
   and(v0, v1) -> null_and [0]

The TRS has the following type information:
cond :: true:false:null_even:null_gr:null_and -> 0:s:y:null_p -> cond
true :: true:false:null_even:null_gr:null_and
and :: true:false:null_even:null_gr:null_and -> true:false:null_even:null_gr:null_and -> true:false:null_even:null_gr:null_and
even :: 0:s:y:null_p -> true:false:null_even:null_gr:null_and
gr :: 0:s:y:null_p -> 0:s:y:null_p -> true:false:null_even:null_gr:null_and
0 :: 0:s:y:null_p
p :: 0:s:y:null_p -> 0:s:y:null_p
false :: true:false:null_even:null_gr:null_and
s :: 0:s:y:null_p -> 0:s:y:null_p
y :: 0:s:y:null_p
null_even :: true:false:null_even:null_gr:null_and
null_gr :: true:false:null_even:null_gr:null_and
null_p :: 0:s:y:null_p
null_and :: true:false:null_even:null_gr:null_and
const :: cond

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_even => 0
   null_gr => 0
   null_p => 0
   null_and => 0
   const => 0

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1, z = x
   and(z, z') -{ 1 }-> 1 :|: z' = x, z = 1, x >= 0
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   cond(z, z') -{ 3 }-> cond(and(even(x'), 2), 0) :|: z = 2, z' = 1 + (1 + x'), x' >= 0
   cond(z, z') -{ 4 }-> cond(and(even(x'), 2), 1 + x') :|: z = 2, z' = 1 + (1 + x'), x' >= 0
   cond(z, z') -{ 2 }-> cond(and(even(x'), 0), 0) :|: z = 2, z' = 1 + (1 + x'), x' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(x'), 0), 1 + x') :|: z = 2, z' = 1 + (1 + x'), x' >= 0
   cond(z, z') -{ 4 }-> cond(and(2, 1), 0) :|: z = 2, z' = 0
   cond(z, z') -{ 3 }-> cond(and(2, 1), 0) :|: z = 2, z' = 0
   cond(z, z') -{ 3 }-> cond(and(2, 0), 0) :|: z = 2, z' = 0
   cond(z, z') -{ 2 }-> cond(and(2, 0), 0) :|: z = 2, z' = 0
   cond(z, z') -{ 4 }-> cond(and(1, 2), 0) :|: z = 2, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(and(1, 2), 0) :|: z = 2, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(and(1, 0), 0) :|: z = 2, z' = 1 + 0
   cond(z, z') -{ 2 }-> cond(and(1, 0), 0) :|: z = 2, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(and(0, 2), x'') :|: z = 2, z' = 1 + x'', x'' >= 0
   cond(z, z') -{ 2 }-> cond(and(0, 2), 0) :|: z = 2, z' = 1 + x'', x'' >= 0
   cond(z, z') -{ 3 }-> cond(and(0, 1), 0) :|: z = 2, z' = 0
   cond(z, z') -{ 2 }-> cond(and(0, 1), 0) :|: z = 2, z' = 0
   cond(z, z') -{ 2 }-> cond(and(0, 0), x1) :|: z = 2, x1 >= 0, z' = 1 + x1
   cond(z, z') -{ 2 }-> cond(and(0, 0), 0) :|: z = 2, z' = 0
   cond(z, z') -{ 1 }-> cond(and(0, 0), 0) :|: z = 2, z' = x, x >= 0
   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
   gr(z, z') -{ 1 }-> gr(x, 1) :|: z' = 1 + 1, x >= 0, z = 1 + x
   gr(z, z') -{ 1 }-> 2 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' = x, x >= 0, z = 0
   gr(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) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1, z = x
   and(z, z') -{ 1 }-> 1 :|: z' = x, z = 1, x >= 0
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1, z = x
   and(z, z') -{ 1 }-> 1 :|: z' = x, z = 1, x >= 0
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   cond(z, z') -{ 3 }-> cond(and(even(x'), 2), 0) :|: z = 2, z' = 1 + (1 + x'), x' >= 0
   cond(z, z') -{ 4 }-> cond(and(even(x'), 2), 1 + x') :|: z = 2, z' = 1 + (1 + x'), x' >= 0
   cond(z, z') -{ 2 }-> cond(and(even(x'), 0), 0) :|: z = 2, z' = 1 + (1 + x'), x' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(x'), 0), 1 + x') :|: z = 2, z' = 1 + (1 + x'), x' >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(0, x'') :|: z = 2, z' = 1 + x'', x'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, x1) :|: z = 2, x1 >= 0, z' = 1 + x1, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + x'', x'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' = x, x >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(x, 1) :|: z' = 1 + 1, x >= 0, z = 1 + x
   gr(z, z') -{ 1 }-> 2 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' = x, x >= 0, z = 0
   gr(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


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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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


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

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

   { and }
   { p }
   { gr }
   { even }
   { cond }

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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: {and}, {p}, {gr}, {even}, {cond}

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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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: {and}, {p}, {gr}, {even}, {cond}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: and
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:

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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: {and}, {p}, {gr}, {even}, {cond}
Previous analysis results are:
  and: runtime: ?, size: O(1) [2]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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}, {gr}, {even}, {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]

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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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}, {gr}, {even}, {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]

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

(25) 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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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}, {gr}, {even}, {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: ?, size: O(n^1) [z]

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

(27) 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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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: {gr}, {even}, {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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: {gr}, {even}, {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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: {gr}, {even}, {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: ?, size: O(1) [2]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 1 }-> gr(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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}, {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: O(1) [2], size: O(1) [2]

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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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}, {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: O(1) [2], size: O(1) [2]

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

(37) 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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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}, {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: O(1) [2], size: O(1) [2]
  even: runtime: ?, size: O(1) [2]

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

(39) 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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 2), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 }-> cond(and(even(z' - 2), 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 2 }-> cond(and(even(z' - 2), 0), 0) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(and(even(z' - 2), 0), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   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
   gr(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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: {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: O(1) [2], size: O(1) [2]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]

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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 5 + z' }-> cond(s1, 1 + (z' - 2)) :|: s'' >= 0, s'' <= 2, s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 + z' }-> cond(s3, 0) :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 + z' }-> cond(s5, 1 + (z' - 2)) :|: s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 + z' }-> cond(s7, 0) :|: s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   even(z) -{ 1 + z }-> s' :|: s' >= 0, s' <= 2, z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   gr(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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: {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: O(1) [2], size: O(1) [2]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 5 + z' }-> cond(s1, 1 + (z' - 2)) :|: s'' >= 0, s'' <= 2, s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 + z' }-> cond(s3, 0) :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 + z' }-> cond(s5, 1 + (z' - 2)) :|: s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 + z' }-> cond(s7, 0) :|: s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   even(z) -{ 1 + z }-> s' :|: s' >= 0, s' <= 2, z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   gr(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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: {cond}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: O(1) [2], size: O(1) [2]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]
  cond: runtime: ?, size: O(1) [0]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: cond
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 25 + 9*z' + z'^2

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1
   and(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cond(z, z') -{ 5 + z' }-> cond(s1, 1 + (z' - 2)) :|: s'' >= 0, s'' <= 2, s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 + z' }-> cond(s3, 0) :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 4 + z' }-> cond(s5, 1 + (z' - 2)) :|: s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 3 + z' }-> cond(s7, 0) :|: s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, z = 2, z' - 2 >= 0
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 2 = x
   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 2 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 1 + 0, 0 = x, 1 = 1, x >= 0
   cond(z, z') -{ 4 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0, x >= 0, 1 = 1, 0 = x
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1
   cond(z, z') -{ 4 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 1 + 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   cond(z, z') -{ 3 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1
   cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1
   even(z) -{ 1 + z }-> s' :|: s' >= 0, s' <= 2, z - 2 >= 0
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z = 1 + 0
   even(z) -{ 0 }-> 0 :|: z >= 0
   gr(z, z') -{ 3 }-> s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
   gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   gr(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:
  and: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: O(1) [2], size: O(1) [2]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]
  cond: runtime: O(n^2) [25 + 9*z' + z'^2], size: O(1) [0]

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

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

(48)
BOUNDS(1, n^2)

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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond(true, z0) -> cond(and(even(z0), gr(z0, 0)), p(z0))
   and(z0, false) -> false
   and(false, z0) -> false
   and(true, true) -> true
   even(0) -> true
   even(s(0)) -> false
   even(s(s(z0))) -> even(z0)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(y)) -> gr(z0, y)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND(true, z0) -> c(COND(and(even(z0), gr(z0, 0)), p(z0)), AND(even(z0), gr(z0, 0)), EVEN(z0))
   COND(true, z0) -> c1(COND(and(even(z0), gr(z0, 0)), p(z0)), AND(even(z0), gr(z0, 0)), GR(z0, 0))
   COND(true, z0) -> c2(COND(and(even(z0), gr(z0, 0)), p(z0)), P(z0))
   AND(z0, false) -> c3
   AND(false, z0) -> c4
   AND(true, true) -> c5
   EVEN(0) -> c6
   EVEN(s(0)) -> c7
   EVEN(s(s(z0))) -> c8(EVEN(z0))
   GR(0, z0) -> c9
   GR(s(z0), 0) -> c10
   GR(s(z0), s(y)) -> c11(GR(z0, y))
   P(0) -> c12
   P(s(z0)) -> c13
S tuples:
   COND(true, z0) -> c(COND(and(even(z0), gr(z0, 0)), p(z0)), AND(even(z0), gr(z0, 0)), EVEN(z0))
   COND(true, z0) -> c1(COND(and(even(z0), gr(z0, 0)), p(z0)), AND(even(z0), gr(z0, 0)), GR(z0, 0))
   COND(true, z0) -> c2(COND(and(even(z0), gr(z0, 0)), p(z0)), P(z0))
   AND(z0, false) -> c3
   AND(false, z0) -> c4
   AND(true, true) -> c5
   EVEN(0) -> c6
   EVEN(s(0)) -> c7
   EVEN(s(s(z0))) -> c8(EVEN(z0))
   GR(0, z0) -> c9
   GR(s(z0), 0) -> c10
   GR(s(z0), s(y)) -> c11(GR(z0, y))
   P(0) -> c12
   P(s(z0)) -> c13
K tuples:none
Defined Rule Symbols:   cond_2, and_2, even_1, gr_2, p_1

Defined Pair Symbols:   COND_2, AND_2, EVEN_1, GR_2, P_1

Compound Symbols:   c_3, c1_3, c2_2, c3, c4, c5, c6, c7, c8_1, c9, c10, c11_1, c12, c13


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

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

(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:

   COND(true, z0) -> c(COND(and(even(z0), gr(z0, 0)), p(z0)), AND(even(z0), gr(z0, 0)), EVEN(z0))
   COND(true, z0) -> c1(COND(and(even(z0), gr(z0, 0)), p(z0)), AND(even(z0), gr(z0, 0)), GR(z0, 0))
   COND(true, z0) -> c2(COND(and(even(z0), gr(z0, 0)), p(z0)), P(z0))
   AND(z0, false) -> c3
   AND(false, z0) -> c4
   AND(true, true) -> c5
   EVEN(0) -> c6
   EVEN(s(0)) -> c7
   EVEN(s(s(z0))) -> c8(EVEN(z0))
   GR(0, z0) -> c9
   GR(s(z0), 0) -> c10
   GR(s(z0), s(y)) -> c11(GR(z0, y))
   P(0) -> c12
   P(s(z0)) -> c13

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

   cond(true, z0) -> cond(and(even(z0), gr(z0, 0)), p(z0))
   and(z0, false) -> false
   and(false, z0) -> false
   and(true, true) -> true
   even(0) -> true
   even(s(0)) -> false
   even(s(s(z0))) -> even(z0)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(y)) -> gr(z0, y)
   p(0) -> 0
   p(s(z0)) -> z0

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

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

(54)
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:

   COND(true, z0) -> c(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), EVEN(z0))
   COND(true, z0) -> c1(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), GR(z0, 0'))
   COND(true, z0) -> c2(COND(and(even(z0), gr(z0, 0')), p(z0)), P(z0))
   AND(z0, false) -> c3
   AND(false, z0) -> c4
   AND(true, true) -> c5
   EVEN(0') -> c6
   EVEN(s(0')) -> c7
   EVEN(s(s(z0))) -> c8(EVEN(z0))
   GR(0', z0) -> c9
   GR(s(z0), 0') -> c10
   GR(s(z0), s(y)) -> c11(GR(z0, y))
   P(0') -> c12
   P(s(z0)) -> c13

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

   cond(true, z0) -> cond(and(even(z0), gr(z0, 0')), p(z0))
   and(z0, false) -> false
   and(false, z0) -> false
   and(true, true) -> true
   even(0') -> true
   even(s(0')) -> false
   even(s(s(z0))) -> even(z0)
   gr(0', z0) -> false
   gr(s(z0), 0') -> true
   gr(s(z0), s(y)) -> gr(z0, y)
   p(0') -> 0'
   p(s(z0)) -> z0

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

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

(56)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), EVEN(z0))
COND(true, z0) -> c1(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), GR(z0, 0'))
COND(true, z0) -> c2(COND(and(even(z0), gr(z0, 0')), p(z0)), P(z0))
AND(z0, false) -> c3
AND(false, z0) -> c4
AND(true, true) -> c5
EVEN(0') -> c6
EVEN(s(0')) -> c7
EVEN(s(s(z0))) -> c8(EVEN(z0))
GR(0', z0) -> c9
GR(s(z0), 0') -> c10
GR(s(z0), s(y)) -> c11(GR(z0, y))
P(0') -> c12
P(s(z0)) -> c13
cond(true, z0) -> cond(and(even(z0), gr(z0, 0')), p(z0))
and(z0, false) -> false
and(false, z0) -> false
and(true, true) -> true
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(y)) -> gr(z0, y)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s:y -> c:c1:c2
true :: true:false
c :: c:c1:c2 -> c3:c4:c5 -> c6:c7:c8 -> c:c1:c2
and :: true:false -> true:false -> true:false
even :: 0':s:y -> true:false
gr :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
p :: 0':s:y -> 0':s:y
AND :: true:false -> true:false -> c3:c4:c5
EVEN :: 0':s:y -> c6:c7:c8
c1 :: c:c1:c2 -> c3:c4:c5 -> c9:c10:c11 -> c:c1:c2
GR :: 0':s:y -> 0':s:y -> c9:c10:c11
c2 :: c:c1:c2 -> c12:c13 -> c:c1:c2
P :: 0':s:y -> c12:c13
false :: true:false
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5
c6 :: c6:c7:c8
s :: 0':s:y -> 0':s:y
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
y :: 0':s:y
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13
c13 :: c12:c13
cond :: true:false -> 0':s:y -> cond
hole_c:c1:c21_14 :: c:c1:c2
hole_true:false2_14 :: true:false
hole_0':s:y3_14 :: 0':s:y
hole_c3:c4:c54_14 :: c3:c4:c5
hole_c6:c7:c85_14 :: c6:c7:c8
hole_c9:c10:c116_14 :: c9:c10:c11
hole_c12:c137_14 :: c12:c13
hole_cond8_14 :: cond
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s:y10_14 :: Nat -> 0':s:y
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c9:c10:c1112_14 :: Nat -> c9:c10:c11

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

(57) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
COND, even, gr, EVEN, GR, cond

They will be analysed ascendingly in the following order:
even < COND
gr < COND
EVEN < COND
GR < COND
even < cond
gr < cond

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

(58)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), EVEN(z0))
COND(true, z0) -> c1(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), GR(z0, 0'))
COND(true, z0) -> c2(COND(and(even(z0), gr(z0, 0')), p(z0)), P(z0))
AND(z0, false) -> c3
AND(false, z0) -> c4
AND(true, true) -> c5
EVEN(0') -> c6
EVEN(s(0')) -> c7
EVEN(s(s(z0))) -> c8(EVEN(z0))
GR(0', z0) -> c9
GR(s(z0), 0') -> c10
GR(s(z0), s(y)) -> c11(GR(z0, y))
P(0') -> c12
P(s(z0)) -> c13
cond(true, z0) -> cond(and(even(z0), gr(z0, 0')), p(z0))
and(z0, false) -> false
and(false, z0) -> false
and(true, true) -> true
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(y)) -> gr(z0, y)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s:y -> c:c1:c2
true :: true:false
c :: c:c1:c2 -> c3:c4:c5 -> c6:c7:c8 -> c:c1:c2
and :: true:false -> true:false -> true:false
even :: 0':s:y -> true:false
gr :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
p :: 0':s:y -> 0':s:y
AND :: true:false -> true:false -> c3:c4:c5
EVEN :: 0':s:y -> c6:c7:c8
c1 :: c:c1:c2 -> c3:c4:c5 -> c9:c10:c11 -> c:c1:c2
GR :: 0':s:y -> 0':s:y -> c9:c10:c11
c2 :: c:c1:c2 -> c12:c13 -> c:c1:c2
P :: 0':s:y -> c12:c13
false :: true:false
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5
c6 :: c6:c7:c8
s :: 0':s:y -> 0':s:y
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
y :: 0':s:y
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13
c13 :: c12:c13
cond :: true:false -> 0':s:y -> cond
hole_c:c1:c21_14 :: c:c1:c2
hole_true:false2_14 :: true:false
hole_0':s:y3_14 :: 0':s:y
hole_c3:c4:c54_14 :: c3:c4:c5
hole_c6:c7:c85_14 :: c6:c7:c8
hole_c9:c10:c116_14 :: c9:c10:c11
hole_c12:c137_14 :: c12:c13
hole_cond8_14 :: cond
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s:y10_14 :: Nat -> 0':s:y
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c9:c10:c1112_14 :: Nat -> c9:c10:c11


Generator Equations:
gen_c:c1:c29_14(0) <=> hole_c:c1:c21_14
gen_c:c1:c29_14(+(x, 1)) <=> c(gen_c:c1:c29_14(x), c3, c6)
gen_0':s:y10_14(0) <=> 0'
gen_0':s:y10_14(+(x, 1)) <=> s(gen_0':s:y10_14(x))
gen_c6:c7:c811_14(0) <=> c6
gen_c6:c7:c811_14(+(x, 1)) <=> c8(gen_c6:c7:c811_14(x))
gen_c9:c10:c1112_14(0) <=> c9
gen_c9:c10:c1112_14(+(x, 1)) <=> c11(gen_c9:c10:c1112_14(x))


The following defined symbols remain to be analysed:
even, COND, gr, EVEN, GR, cond

They will be analysed ascendingly in the following order:
even < COND
gr < COND
EVEN < COND
GR < COND
even < cond
gr < cond

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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_0':s:y10_14(*(2, n14_14))) -> true, rt in Omega(0)

Induction Base:
even(gen_0':s:y10_14(*(2, 0))) ->_R^Omega(0)
true

Induction Step:
even(gen_0':s:y10_14(*(2, +(n14_14, 1)))) ->_R^Omega(0)
even(gen_0':s:y10_14(*(2, n14_14))) ->_IH
true

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

(60)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), EVEN(z0))
COND(true, z0) -> c1(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), GR(z0, 0'))
COND(true, z0) -> c2(COND(and(even(z0), gr(z0, 0')), p(z0)), P(z0))
AND(z0, false) -> c3
AND(false, z0) -> c4
AND(true, true) -> c5
EVEN(0') -> c6
EVEN(s(0')) -> c7
EVEN(s(s(z0))) -> c8(EVEN(z0))
GR(0', z0) -> c9
GR(s(z0), 0') -> c10
GR(s(z0), s(y)) -> c11(GR(z0, y))
P(0') -> c12
P(s(z0)) -> c13
cond(true, z0) -> cond(and(even(z0), gr(z0, 0')), p(z0))
and(z0, false) -> false
and(false, z0) -> false
and(true, true) -> true
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(y)) -> gr(z0, y)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s:y -> c:c1:c2
true :: true:false
c :: c:c1:c2 -> c3:c4:c5 -> c6:c7:c8 -> c:c1:c2
and :: true:false -> true:false -> true:false
even :: 0':s:y -> true:false
gr :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
p :: 0':s:y -> 0':s:y
AND :: true:false -> true:false -> c3:c4:c5
EVEN :: 0':s:y -> c6:c7:c8
c1 :: c:c1:c2 -> c3:c4:c5 -> c9:c10:c11 -> c:c1:c2
GR :: 0':s:y -> 0':s:y -> c9:c10:c11
c2 :: c:c1:c2 -> c12:c13 -> c:c1:c2
P :: 0':s:y -> c12:c13
false :: true:false
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5
c6 :: c6:c7:c8
s :: 0':s:y -> 0':s:y
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
y :: 0':s:y
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13
c13 :: c12:c13
cond :: true:false -> 0':s:y -> cond
hole_c:c1:c21_14 :: c:c1:c2
hole_true:false2_14 :: true:false
hole_0':s:y3_14 :: 0':s:y
hole_c3:c4:c54_14 :: c3:c4:c5
hole_c6:c7:c85_14 :: c6:c7:c8
hole_c9:c10:c116_14 :: c9:c10:c11
hole_c12:c137_14 :: c12:c13
hole_cond8_14 :: cond
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s:y10_14 :: Nat -> 0':s:y
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c9:c10:c1112_14 :: Nat -> c9:c10:c11


Lemmas:
even(gen_0':s:y10_14(*(2, n14_14))) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c29_14(0) <=> hole_c:c1:c21_14
gen_c:c1:c29_14(+(x, 1)) <=> c(gen_c:c1:c29_14(x), c3, c6)
gen_0':s:y10_14(0) <=> 0'
gen_0':s:y10_14(+(x, 1)) <=> s(gen_0':s:y10_14(x))
gen_c6:c7:c811_14(0) <=> c6
gen_c6:c7:c811_14(+(x, 1)) <=> c8(gen_c6:c7:c811_14(x))
gen_c9:c10:c1112_14(0) <=> c9
gen_c9:c10:c1112_14(+(x, 1)) <=> c11(gen_c9:c10:c1112_14(x))


The following defined symbols remain to be analysed:
gr, COND, EVEN, GR, cond

They will be analysed ascendingly in the following order:
gr < COND
EVEN < COND
GR < COND
gr < cond

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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
EVEN(gen_0':s:y10_14(*(2, n231_14))) -> gen_c6:c7:c811_14(n231_14), rt in Omega(1 + n231_14)

Induction Base:
EVEN(gen_0':s:y10_14(*(2, 0))) ->_R^Omega(1)
c6

Induction Step:
EVEN(gen_0':s:y10_14(*(2, +(n231_14, 1)))) ->_R^Omega(1)
c8(EVEN(gen_0':s:y10_14(*(2, n231_14)))) ->_IH
c8(gen_c6:c7:c811_14(c232_14))

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

(62)
Complex Obligation (BEST)

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

(63)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
COND(true, z0) -> c(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), EVEN(z0))
COND(true, z0) -> c1(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), GR(z0, 0'))
COND(true, z0) -> c2(COND(and(even(z0), gr(z0, 0')), p(z0)), P(z0))
AND(z0, false) -> c3
AND(false, z0) -> c4
AND(true, true) -> c5
EVEN(0') -> c6
EVEN(s(0')) -> c7
EVEN(s(s(z0))) -> c8(EVEN(z0))
GR(0', z0) -> c9
GR(s(z0), 0') -> c10
GR(s(z0), s(y)) -> c11(GR(z0, y))
P(0') -> c12
P(s(z0)) -> c13
cond(true, z0) -> cond(and(even(z0), gr(z0, 0')), p(z0))
and(z0, false) -> false
and(false, z0) -> false
and(true, true) -> true
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(y)) -> gr(z0, y)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s:y -> c:c1:c2
true :: true:false
c :: c:c1:c2 -> c3:c4:c5 -> c6:c7:c8 -> c:c1:c2
and :: true:false -> true:false -> true:false
even :: 0':s:y -> true:false
gr :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
p :: 0':s:y -> 0':s:y
AND :: true:false -> true:false -> c3:c4:c5
EVEN :: 0':s:y -> c6:c7:c8
c1 :: c:c1:c2 -> c3:c4:c5 -> c9:c10:c11 -> c:c1:c2
GR :: 0':s:y -> 0':s:y -> c9:c10:c11
c2 :: c:c1:c2 -> c12:c13 -> c:c1:c2
P :: 0':s:y -> c12:c13
false :: true:false
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5
c6 :: c6:c7:c8
s :: 0':s:y -> 0':s:y
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
y :: 0':s:y
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13
c13 :: c12:c13
cond :: true:false -> 0':s:y -> cond
hole_c:c1:c21_14 :: c:c1:c2
hole_true:false2_14 :: true:false
hole_0':s:y3_14 :: 0':s:y
hole_c3:c4:c54_14 :: c3:c4:c5
hole_c6:c7:c85_14 :: c6:c7:c8
hole_c9:c10:c116_14 :: c9:c10:c11
hole_c12:c137_14 :: c12:c13
hole_cond8_14 :: cond
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s:y10_14 :: Nat -> 0':s:y
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c9:c10:c1112_14 :: Nat -> c9:c10:c11


Lemmas:
even(gen_0':s:y10_14(*(2, n14_14))) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c29_14(0) <=> hole_c:c1:c21_14
gen_c:c1:c29_14(+(x, 1)) <=> c(gen_c:c1:c29_14(x), c3, c6)
gen_0':s:y10_14(0) <=> 0'
gen_0':s:y10_14(+(x, 1)) <=> s(gen_0':s:y10_14(x))
gen_c6:c7:c811_14(0) <=> c6
gen_c6:c7:c811_14(+(x, 1)) <=> c8(gen_c6:c7:c811_14(x))
gen_c9:c10:c1112_14(0) <=> c9
gen_c9:c10:c1112_14(+(x, 1)) <=> c11(gen_c9:c10:c1112_14(x))


The following defined symbols remain to be analysed:
EVEN, COND, GR, cond

They will be analysed ascendingly in the following order:
EVEN < COND
GR < COND

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

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

(65)
BOUNDS(n^1, INF)

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

(66)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), EVEN(z0))
COND(true, z0) -> c1(COND(and(even(z0), gr(z0, 0')), p(z0)), AND(even(z0), gr(z0, 0')), GR(z0, 0'))
COND(true, z0) -> c2(COND(and(even(z0), gr(z0, 0')), p(z0)), P(z0))
AND(z0, false) -> c3
AND(false, z0) -> c4
AND(true, true) -> c5
EVEN(0') -> c6
EVEN(s(0')) -> c7
EVEN(s(s(z0))) -> c8(EVEN(z0))
GR(0', z0) -> c9
GR(s(z0), 0') -> c10
GR(s(z0), s(y)) -> c11(GR(z0, y))
P(0') -> c12
P(s(z0)) -> c13
cond(true, z0) -> cond(and(even(z0), gr(z0, 0')), p(z0))
and(z0, false) -> false
and(false, z0) -> false
and(true, true) -> true
even(0') -> true
even(s(0')) -> false
even(s(s(z0))) -> even(z0)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(y)) -> gr(z0, y)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s:y -> c:c1:c2
true :: true:false
c :: c:c1:c2 -> c3:c4:c5 -> c6:c7:c8 -> c:c1:c2
and :: true:false -> true:false -> true:false
even :: 0':s:y -> true:false
gr :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
p :: 0':s:y -> 0':s:y
AND :: true:false -> true:false -> c3:c4:c5
EVEN :: 0':s:y -> c6:c7:c8
c1 :: c:c1:c2 -> c3:c4:c5 -> c9:c10:c11 -> c:c1:c2
GR :: 0':s:y -> 0':s:y -> c9:c10:c11
c2 :: c:c1:c2 -> c12:c13 -> c:c1:c2
P :: 0':s:y -> c12:c13
false :: true:false
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5
c6 :: c6:c7:c8
s :: 0':s:y -> 0':s:y
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
c9 :: c9:c10:c11
c10 :: c9:c10:c11
y :: 0':s:y
c11 :: c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13
c13 :: c12:c13
cond :: true:false -> 0':s:y -> cond
hole_c:c1:c21_14 :: c:c1:c2
hole_true:false2_14 :: true:false
hole_0':s:y3_14 :: 0':s:y
hole_c3:c4:c54_14 :: c3:c4:c5
hole_c6:c7:c85_14 :: c6:c7:c8
hole_c9:c10:c116_14 :: c9:c10:c11
hole_c12:c137_14 :: c12:c13
hole_cond8_14 :: cond
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s:y10_14 :: Nat -> 0':s:y
gen_c6:c7:c811_14 :: Nat -> c6:c7:c8
gen_c9:c10:c1112_14 :: Nat -> c9:c10:c11


Lemmas:
even(gen_0':s:y10_14(*(2, n14_14))) -> true, rt in Omega(0)
EVEN(gen_0':s:y10_14(*(2, n231_14))) -> gen_c6:c7:c811_14(n231_14), rt in Omega(1 + n231_14)


Generator Equations:
gen_c:c1:c29_14(0) <=> hole_c:c1:c21_14
gen_c:c1:c29_14(+(x, 1)) <=> c(gen_c:c1:c29_14(x), c3, c6)
gen_0':s:y10_14(0) <=> 0'
gen_0':s:y10_14(+(x, 1)) <=> s(gen_0':s:y10_14(x))
gen_c6:c7:c811_14(0) <=> c6
gen_c6:c7:c811_14(+(x, 1)) <=> c8(gen_c6:c7:c811_14(x))
gen_c9:c10:c1112_14(0) <=> c9
gen_c9:c10:c1112_14(+(x, 1)) <=> c11(gen_c9:c10:c1112_14(x))


The following defined symbols remain to be analysed:
GR, COND, cond

They will be analysed ascendingly in the following order:
GR < COND