WORST_CASE(Omega(n^1),O(n^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 506 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 1 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 153 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) InliningProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 339 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 139 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 404 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 302 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 113 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 101 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 281 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 113 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 2450 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 698 ms]
    (60) CpxRNTS
    (61) ResultPropagationProof [UPPER BOUND(ID), 2 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 532 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
    (66) CpxRNTS
    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 4964 ms]
    (70) CpxRNTS
    (71) IntTrsBoundProof [UPPER BOUND(ID), 896 ms]
    (72) CpxRNTS
    (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 500 ms]
    (76) CpxRNTS
    (77) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (78) CpxRNTS
    (79) FinalProof [FINISHED, 0 ms]
    (80) BOUNDS(1, n^3)
(81) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(82) TRS for Loop Detection
    (83) DecreasingLoopProof [LOWER BOUND(ID), 1 ms]
    (84) BEST
        (85) proven lower bound
            (86) LowerBoundPropagationProof [FINISHED, 0 ms]
            (87) BOUNDS(n^1, INF)
        (88) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))

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

   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))

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

   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   MINUS(0, z0) -> c [1]
   MINUS(s(z0), 0) -> c1 [1]
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   LE(0, z0) -> c3 [1]
   LE(s(z0), 0) -> c4 [1]
   LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1]
   IF(true, z0, z1) -> c6 [1]
   IF(false, z0, z1) -> c7 [1]
   PERFECTP(0) -> c8 [1]
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0))) [1]
   F(0, z0, 0, z1) -> c10 [1]
   F(0, z0, s(z1), z2) -> c11 [1]
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), 0) -> s(z0) [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   if(true, z0, z1) -> z0 [0]
   if(false, z0, z1) -> z1 [0]
   perfectp(0) -> false [0]
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0)) [0]
   f(0, z0, 0, z1) -> true [0]
   f(0, z0, s(z1), z2) -> false [0]
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2) [0]
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) [0]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MINUS(0, z0) -> c [1]
   MINUS(s(z0), 0) -> c1 [1]
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   LE(0, z0) -> c3 [1]
   LE(s(z0), 0) -> c4 [1]
   LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1]
   IF(true, z0, z1) -> c6 [1]
   IF(false, z0, z1) -> c7 [1]
   PERFECTP(0) -> c8 [1]
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0))) [1]
   F(0, z0, 0, z1) -> c10 [1]
   F(0, z0, s(z1), z2) -> c11 [1]
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), 0) -> s(z0) [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   if(true, z0, z1) -> z0 [0]
   if(false, z0, z1) -> z1 [0]
   perfectp(0) -> false [0]
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0)) [0]
   f(0, z0, 0, z1) -> true [0]
   f(0, z0, s(z1), z2) -> false [0]
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2) [0]
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) [0]

The TRS has the following type information:
MINUS :: 0:s -> 0:s -> c:c1:c2
0 :: 0:s
c :: c:c1:c2
s :: 0:s -> 0:s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LE :: 0:s -> 0:s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
IF :: true:false -> true:false -> true:false -> c6:c7
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c6:c7
PERFECTP :: 0:s -> c8:c9
c8 :: c8:c9
c9 :: c10:c11:c12:c13:c14:c15 -> c8:c9
F :: 0:s -> 0:s -> 0:s -> 0:s -> c10:c11:c12:c13:c14:c15
c10 :: c10:c11:c12:c13:c14:c15
c11 :: c10:c11:c12:c13:c14:c15
c12 :: c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
minus :: 0:s -> 0:s -> 0:s
c13 :: c6:c7 -> c3:c4:c5 -> c10:c11:c12:c13:c14:c15
le :: 0:s -> 0:s -> true:false
f :: 0:s -> 0:s -> 0:s -> 0:s -> true:false
c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
c15 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c10:c11:c12:c13:c14:c15
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0:s -> true:false

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

(7) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   MINUS_2
   LE_2
   IF_3
   PERFECTP_1
   F_4

(c) The following functions are completely defined:

   minus_2
   le_2
   if_3
   perfectp_1
   f_4

Due to the following rules being added:

   minus(v0, v1) -> 0 [0]
   le(v0, v1) -> null_le [0]
   if(v0, v1, v2) -> null_if [0]
   perfectp(v0) -> null_perfectp [0]
   f(v0, v1, v2, v3) -> null_f [0]

And the following fresh constants:    null_le, null_if, null_perfectp, null_f

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

(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:

   MINUS(0, z0) -> c [1]
   MINUS(s(z0), 0) -> c1 [1]
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   LE(0, z0) -> c3 [1]
   LE(s(z0), 0) -> c4 [1]
   LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1]
   IF(true, z0, z1) -> c6 [1]
   IF(false, z0, z1) -> c7 [1]
   PERFECTP(0) -> c8 [1]
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0))) [1]
   F(0, z0, 0, z1) -> c10 [1]
   F(0, z0, s(z1), z2) -> c11 [1]
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), 0) -> s(z0) [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   if(true, z0, z1) -> z0 [0]
   if(false, z0, z1) -> z1 [0]
   perfectp(0) -> false [0]
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0)) [0]
   f(0, z0, 0, z1) -> true [0]
   f(0, z0, s(z1), z2) -> false [0]
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2) [0]
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) [0]
   minus(v0, v1) -> 0 [0]
   le(v0, v1) -> null_le [0]
   if(v0, v1, v2) -> null_if [0]
   perfectp(v0) -> null_perfectp [0]
   f(v0, v1, v2, v3) -> null_f [0]

The TRS has the following type information:
MINUS :: 0:s -> 0:s -> c:c1:c2
0 :: 0:s
c :: c:c1:c2
s :: 0:s -> 0:s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LE :: 0:s -> 0:s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
IF :: true:false:null_le:null_if:null_perfectp:null_f -> true:false:null_le:null_if:null_perfectp:null_f -> true:false:null_le:null_if:null_perfectp:null_f -> c6:c7
true :: true:false:null_le:null_if:null_perfectp:null_f
c6 :: c6:c7
false :: true:false:null_le:null_if:null_perfectp:null_f
c7 :: c6:c7
PERFECTP :: 0:s -> c8:c9
c8 :: c8:c9
c9 :: c10:c11:c12:c13:c14:c15 -> c8:c9
F :: 0:s -> 0:s -> 0:s -> 0:s -> c10:c11:c12:c13:c14:c15
c10 :: c10:c11:c12:c13:c14:c15
c11 :: c10:c11:c12:c13:c14:c15
c12 :: c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
minus :: 0:s -> 0:s -> 0:s
c13 :: c6:c7 -> c3:c4:c5 -> c10:c11:c12:c13:c14:c15
le :: 0:s -> 0:s -> true:false:null_le:null_if:null_perfectp:null_f
f :: 0:s -> 0:s -> 0:s -> 0:s -> true:false:null_le:null_if:null_perfectp:null_f
c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
c15 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c10:c11:c12:c13:c14:c15
if :: true:false:null_le:null_if:null_perfectp:null_f -> true:false:null_le:null_if:null_perfectp:null_f -> true:false:null_le:null_if:null_perfectp:null_f -> true:false:null_le:null_if:null_perfectp:null_f
perfectp :: 0:s -> true:false:null_le:null_if:null_perfectp:null_f
null_le :: true:false:null_le:null_if:null_perfectp:null_f
null_if :: true:false:null_le:null_if:null_perfectp:null_f
null_perfectp :: true:false:null_le:null_if:null_perfectp:null_f
null_f :: true:false:null_le:null_if:null_perfectp:null_f

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   MINUS(0, z0) -> c [1]
   MINUS(s(z0), 0) -> c1 [1]
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   LE(0, z0) -> c3 [1]
   LE(s(z0), 0) -> c4 [1]
   LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1]
   IF(true, z0, z1) -> c6 [1]
   IF(false, z0, z1) -> c7 [1]
   PERFECTP(0) -> c8 [1]
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0))) [1]
   F(0, z0, 0, z1) -> c10 [1]
   F(0, z0, s(z1), z2) -> c11 [1]
   F(s(z0), 0, 0, z2) -> c12(F(z0, z2, 0, z2), MINUS(0, s(z0))) [1]
   F(s(z0), 0, s(z0'), z2) -> c12(F(z0, z2, minus(z0', z0), z2), MINUS(s(z0'), s(z0))) [1]
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, 0, z2), MINUS(z1, s(z0))) [1]
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1)) [1]
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) [1]
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), 0) -> s(z0) [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   if(true, z0, z1) -> z0 [0]
   if(false, z0, z1) -> z1 [0]
   perfectp(0) -> false [0]
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0)) [0]
   f(0, z0, 0, z1) -> true [0]
   f(0, z0, s(z1), z2) -> false [0]
   f(s(z0), 0, 0, z2) -> f(z0, z2, 0, z2) [0]
   f(s(z0), 0, s(z036), z2) -> f(z0, z2, minus(z036, z0), z2) [0]
   f(s(z0), 0, z1, z2) -> f(z0, z2, 0, z2) [0]
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) [0]
   minus(v0, v1) -> 0 [0]
   le(v0, v1) -> null_le [0]
   if(v0, v1, v2) -> null_if [0]
   perfectp(v0) -> null_perfectp [0]
   f(v0, v1, v2, v3) -> null_f [0]

The TRS has the following type information:
MINUS :: 0:s -> 0:s -> c:c1:c2
0 :: 0:s
c :: c:c1:c2
s :: 0:s -> 0:s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LE :: 0:s -> 0:s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
IF :: true:false:null_le:null_if:null_perfectp:null_f -> true:false:null_le:null_if:null_perfectp:null_f -> true:false:null_le:null_if:null_perfectp:null_f -> c6:c7
true :: true:false:null_le:null_if:null_perfectp:null_f
c6 :: c6:c7
false :: true:false:null_le:null_if:null_perfectp:null_f
c7 :: c6:c7
PERFECTP :: 0:s -> c8:c9
c8 :: c8:c9
c9 :: c10:c11:c12:c13:c14:c15 -> c8:c9
F :: 0:s -> 0:s -> 0:s -> 0:s -> c10:c11:c12:c13:c14:c15
c10 :: c10:c11:c12:c13:c14:c15
c11 :: c10:c11:c12:c13:c14:c15
c12 :: c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
minus :: 0:s -> 0:s -> 0:s
c13 :: c6:c7 -> c3:c4:c5 -> c10:c11:c12:c13:c14:c15
le :: 0:s -> 0:s -> true:false:null_le:null_if:null_perfectp:null_f
f :: 0:s -> 0:s -> 0:s -> 0:s -> true:false:null_le:null_if:null_perfectp:null_f
c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c:c1:c2 -> c10:c11:c12:c13:c14:c15
c15 :: c6:c7 -> c10:c11:c12:c13:c14:c15 -> c10:c11:c12:c13:c14:c15
if :: true:false:null_le:null_if:null_perfectp:null_f -> true:false:null_le:null_if:null_perfectp:null_f -> true:false:null_le:null_if:null_perfectp:null_f -> true:false:null_le:null_if:null_perfectp:null_f
perfectp :: 0:s -> true:false:null_le:null_if:null_perfectp:null_f
null_le :: true:false:null_le:null_if:null_perfectp:null_f
null_if :: true:false:null_le:null_if:null_perfectp:null_f
null_perfectp :: true:false:null_le:null_if:null_perfectp:null_f
null_f :: true:false:null_le:null_if:null_perfectp:null_f

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

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

   0 => 0
   c => 0
   c1 => 1
   c3 => 0
   c4 => 1
   true => 2
   c6 => 0
   false => 1
   c7 => 1
   c8 => 0
   c10 => 0
   c11 => 1
   null_le => 0
   null_if => 0
   null_perfectp => 0
   null_f => 0

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z'' = 1 + z1, z = 0, z' = z0, z2 >= 0, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z1 >= 0, z0 >= 0, z = 0, z4 = z1, z' = z0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z0, z1), f(1 + z0, minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) + LE(z0, z1) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z0, z1), f(1 + z0, minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) + F(z0, z3, z2, z3) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z2, minus(z0', z0), z2) + MINUS(1 + z0', 1 + z0) :|: z0' >= 0, z = 1 + z0, z0 >= 0, z'' = 1 + z0', z2 >= 0, z' = 0, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z2, 0, z2) + MINUS(z1, 1 + z0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z'' = z1, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z2, 0, z2) + MINUS(0, 1 + z0) :|: z'' = 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z0, z1), f(1 + z0, minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) + F(1 + z0, minus(z1, z0), z2, z3) + MINUS(z1, z0) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   LE(z, z') -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   LE(z, z') -{ 1 }-> 1 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MINUS(z, z') -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z0, 1 + 0, 1 + z0, 1 + z0) :|: z = 1 + z0, z0 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(le(z0, z1), f(1 + z0, minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z0, z2, minus(z036, z0), z2) :|: z036 >= 0, z'' = 1 + z036, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z4 = z2
   f(z, z', z'', z4) -{ 0 }-> f(z0, z2, 0, z2) :|: z'' = 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z4 = z2
   f(z, z', z'', z4) -{ 0 }-> f(z0, z2, 0, z2) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z'' = z1, z4 = z2
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z1 >= 0, z0 >= 0, z = 0, z4 = z1, z' = z0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z'' = 1 + z1, z = 0, z' = z0, z2 >= 0, z4 = z2
   f(z, z', z'', z4) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z4 = v3, v2 >= 0, v3 >= 0
   if(z, z', z'') -{ 0 }-> z0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> z1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   le(z, z') -{ 0 }-> le(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   le(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0
   le(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 0 }-> 1 + z0 :|: z = 1 + z0, z0 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z0, 1 + 0, 1 + z0, 1 + z0) :|: z = 1 + z0, z0 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

(13) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   IF(z, z', z'') -{ 1 }-> 1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> z0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> z1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z'' = 1 + z1, z = 0, z' = z0, z2 >= 0, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z1 >= 0, z0 >= 0, z = 0, z4 = z1, z' = z0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z0, z1), f(1 + z0, minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) + LE(z0, z1) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z0, z1), f(1 + z0, minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) + F(z0, z3, z2, z3) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z2, minus(z0', z0), z2) + MINUS(1 + z0', 1 + z0) :|: z0' >= 0, z = 1 + z0, z0 >= 0, z'' = 1 + z0', z2 >= 0, z' = 0, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z2, 0, z2) + MINUS(z1, 1 + z0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z'' = z1, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z2, 0, z2) + MINUS(0, 1 + z0) :|: z'' = 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z0, z1), f(1 + z0, minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) + F(1 + z0, minus(z1, z0), z2, z3) + MINUS(z1, z0) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   LE(z, z') -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   LE(z, z') -{ 1 }-> 1 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MINUS(z, z') -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z0, 1 + 0, 1 + z0, 1 + z0) :|: z = 1 + z0, z0 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(le(z0, z1), f(1 + z0, minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z0, z2, minus(z036, z0), z2) :|: z036 >= 0, z'' = 1 + z036, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z4 = z2
   f(z, z', z'', z4) -{ 0 }-> f(z0, z2, 0, z2) :|: z'' = 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z4 = z2
   f(z, z', z'', z4) -{ 0 }-> f(z0, z2, 0, z2) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z'' = z1, z4 = z2
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z1 >= 0, z0 >= 0, z = 0, z4 = z1, z' = z0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z'' = 1 + z1, z = 0, z' = z0, z2 >= 0, z4 = z2
   f(z, z', z'', z4) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z4 = v3, v2 >= 0, v3 >= 0
   if(z, z', z'') -{ 0 }-> z0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> z1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   le(z, z') -{ 0 }-> le(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   le(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0
   le(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 0 }-> 1 + z0 :|: z = 1 + z0, z0 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z0, 1 + 0, 1 + z0, 1 + z0) :|: z = 1 + z0, z0 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), minus(z' - 1, z - 1), z'', z4) + MINUS(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, minus(z'' - 1, z - 1), z4) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { minus }
   { le }
   { LE }
   { IF }
   { if }
   { MINUS }
   { f }
   { perfectp }
   { F }
   { PERFECTP }

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), minus(z' - 1, z - 1), z'', z4) + MINUS(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, minus(z'' - 1, z - 1), z4) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {minus}, {le}, {LE}, {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}

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

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), minus(z' - 1, z - 1), z'', z4) + MINUS(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, minus(z'' - 1, z - 1), z4) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {minus}, {le}, {LE}, {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), minus(z' - 1, z - 1), z'', z4) + MINUS(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, minus(z'' - 1, z - 1), z4) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {minus}, {le}, {LE}, {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: ?, size: O(n^1) [z]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, minus(z'' - 1, z - 1), z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), minus(z' - 1, z - 1), z'', z4) + MINUS(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(le(z - 1, z' - 1), f(1 + (z - 1), minus(z' - 1, z - 1), z'', z4), f(z - 1, z4, z'', z4)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, minus(z'' - 1, z - 1), z4) :|: z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {le}, {LE}, {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(le(z - 1, z' - 1), f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {le}, {LE}, {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(le(z - 1, z' - 1), f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {le}, {LE}, {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: ?, size: O(1) [2]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(le(z - 1, z' - 1), f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(le(z - 1, z' - 1), f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {LE}, {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {LE}, {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {LE}, {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: ?, size: O(n^1) [1 + z']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + LE(z - 1, z' - 1) :|: s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IF}, {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: ?, size: O(1) [1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if}, {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: ?, size: O(n^1) [z' + z'']

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']

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

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {MINUS}, {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: ?, size: O(n^1) [1 + z']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, s, z4) + MINUS(1 + (z'' - 1), 1 + (z - 1)) :|: s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(z'', 1 + (z - 1)) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 0, z4) + MINUS(0, 1 + (z - 1)) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + MINUS(z' - 1, z - 1) :|: s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']

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

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, s, z4) + s15 :|: s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s14 :|: s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s16 :|: s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 2 + z }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + s17 :|: s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

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

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, s, z4) + s15 :|: s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s14 :|: s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s16 :|: s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 2 + z }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + s17 :|: s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {f}, {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: ?, size: EXP

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 2 + z' }-> 1 + IF(s7, f(1 + (z - 1), s1, z'', z4), f(z - 1, z4, z'', z4)) + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + IF(s9, f(1 + (z - 1), s4, z'', z4), f(z - 1, z4, z'', z4)) + F(z - 1, z4, z'', z4) :|: s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, s, z4) + s15 :|: s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s14 :|: s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s16 :|: s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 2 + z }-> 1 + IF(s8, f(1 + (z - 1), s2, z'', z4), f(z - 1, z4, z'', z4)) + F(1 + (z - 1), s3, z'', z4) + s17 :|: s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> if(s10, f(1 + (z - 1), s5, z'', z4), f(z - 1, z4, z'', z4)) :|: s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, s'', z4) :|: s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> f(z - 1, z4, 0, z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> f(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: O(1) [0], size: EXP

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

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 3 + z' }-> 1 + s24 + s12 :|: s22 >= 0, s22 <= inf2, s23 >= 0, s23 <= inf3, s24 >= 0, s24 <= 1, s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 2 }-> 1 + s30 + F(z - 1, z4, z'', z4) :|: s28 >= 0, s28 <= inf6, s29 >= 0, s29 <= inf7, s30 >= 0, s30 <= 1, s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, s, z4) + s15 :|: s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s14 :|: s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s16 :|: s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + s27 + F(1 + (z - 1), s3, z'', z4) + s17 :|: s25 >= 0, s25 <= inf4, s26 >= 0, s26 <= inf5, s27 >= 0, s27 <= 1, s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> s19 :|: s19 >= 0, s19 <= inf', z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s20 :|: s20 >= 0, s20 <= inf'', s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s21 :|: s21 >= 0, s21 <= inf1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s33 :|: s31 >= 0, s31 <= inf8, s32 >= 0, s32 <= inf9, s33 >= 0, s33 <= s32 + s31, s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> s18 :|: s18 >= 0, s18 <= inf, z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: O(1) [0], size: EXP

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: perfectp
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 3 + z' }-> 1 + s24 + s12 :|: s22 >= 0, s22 <= inf2, s23 >= 0, s23 <= inf3, s24 >= 0, s24 <= 1, s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 2 }-> 1 + s30 + F(z - 1, z4, z'', z4) :|: s28 >= 0, s28 <= inf6, s29 >= 0, s29 <= inf7, s30 >= 0, s30 <= 1, s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, s, z4) + s15 :|: s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s14 :|: s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s16 :|: s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + s27 + F(1 + (z - 1), s3, z'', z4) + s17 :|: s25 >= 0, s25 <= inf4, s26 >= 0, s26 <= inf5, s27 >= 0, s27 <= 1, s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> s19 :|: s19 >= 0, s19 <= inf', z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s20 :|: s20 >= 0, s20 <= inf'', s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s21 :|: s21 >= 0, s21 <= inf1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s33 :|: s31 >= 0, s31 <= inf8, s32 >= 0, s32 <= inf9, s33 >= 0, s33 <= s32 + s31, s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> s18 :|: s18 >= 0, s18 <= inf, z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {perfectp}, {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: O(1) [0], size: EXP
  perfectp: runtime: ?, size: INF

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 3 + z' }-> 1 + s24 + s12 :|: s22 >= 0, s22 <= inf2, s23 >= 0, s23 <= inf3, s24 >= 0, s24 <= 1, s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 2 }-> 1 + s30 + F(z - 1, z4, z'', z4) :|: s28 >= 0, s28 <= inf6, s29 >= 0, s29 <= inf7, s30 >= 0, s30 <= 1, s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, s, z4) + s15 :|: s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s14 :|: s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s16 :|: s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + s27 + F(1 + (z - 1), s3, z'', z4) + s17 :|: s25 >= 0, s25 <= inf4, s26 >= 0, s26 <= inf5, s27 >= 0, s27 <= 1, s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> s19 :|: s19 >= 0, s19 <= inf', z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s20 :|: s20 >= 0, s20 <= inf'', s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s21 :|: s21 >= 0, s21 <= inf1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s33 :|: s31 >= 0, s31 <= inf8, s32 >= 0, s32 <= inf9, s33 >= 0, s33 <= s32 + s31, s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> s18 :|: s18 >= 0, s18 <= inf, z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: O(1) [0], size: EXP
  perfectp: runtime: O(1) [0], size: INF

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

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 3 + z' }-> 1 + s24 + s12 :|: s22 >= 0, s22 <= inf2, s23 >= 0, s23 <= inf3, s24 >= 0, s24 <= 1, s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 2 }-> 1 + s30 + F(z - 1, z4, z'', z4) :|: s28 >= 0, s28 <= inf6, s29 >= 0, s29 <= inf7, s30 >= 0, s30 <= 1, s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, s, z4) + s15 :|: s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s14 :|: s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s16 :|: s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + s27 + F(1 + (z - 1), s3, z'', z4) + s17 :|: s25 >= 0, s25 <= inf4, s26 >= 0, s26 <= inf5, s27 >= 0, s27 <= 1, s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> s19 :|: s19 >= 0, s19 <= inf', z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s20 :|: s20 >= 0, s20 <= inf'', s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s21 :|: s21 >= 0, s21 <= inf1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s33 :|: s31 >= 0, s31 <= inf8, s32 >= 0, s32 <= inf9, s33 >= 0, s33 <= s32 + s31, s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> s18 :|: s18 >= 0, s18 <= inf, z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: O(1) [0], size: EXP
  perfectp: runtime: O(1) [0], size: INF

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: F
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 3 + 8*z + z*z' + 8*z*z4 + 3*z^2 + 4*z^2*z4 + 3*z' + z4

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 3 + z' }-> 1 + s24 + s12 :|: s22 >= 0, s22 <= inf2, s23 >= 0, s23 <= inf3, s24 >= 0, s24 <= 1, s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 2 }-> 1 + s30 + F(z - 1, z4, z'', z4) :|: s28 >= 0, s28 <= inf6, s29 >= 0, s29 <= inf7, s30 >= 0, s30 <= 1, s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, s, z4) + s15 :|: s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s14 :|: s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s16 :|: s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + s27 + F(1 + (z - 1), s3, z'', z4) + s17 :|: s25 >= 0, s25 <= inf4, s26 >= 0, s26 <= inf5, s27 >= 0, s27 <= 1, s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> s19 :|: s19 >= 0, s19 <= inf', z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s20 :|: s20 >= 0, s20 <= inf'', s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s21 :|: s21 >= 0, s21 <= inf1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s33 :|: s31 >= 0, s31 <= inf8, s32 >= 0, s32 <= inf9, s33 >= 0, s33 <= s32 + s31, s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> s18 :|: s18 >= 0, s18 <= inf, z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {F}, {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: O(1) [0], size: EXP
  perfectp: runtime: O(1) [0], size: INF
  F: runtime: ?, size: O(n^3) [3 + 8*z + z*z' + 8*z*z4 + 3*z^2 + 4*z^2*z4 + 3*z' + z4]

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: F
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 13 + 14*z + z*z' + 4*z*z4 + 3*z^2 + z^2*z4 + 4*z' + 4*z4

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 3 + z' }-> 1 + s24 + s12 :|: s22 >= 0, s22 <= inf2, s23 >= 0, s23 <= inf3, s24 >= 0, s24 <= 1, s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 2 }-> 1 + s30 + F(z - 1, z4, z'', z4) :|: s28 >= 0, s28 <= inf6, s29 >= 0, s29 <= inf7, s30 >= 0, s30 <= 1, s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, s, z4) + s15 :|: s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s14 :|: s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + F(z - 1, z4, 0, z4) + s16 :|: s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 3 + z }-> 1 + s27 + F(1 + (z - 1), s3, z'', z4) + s17 :|: s25 >= 0, s25 <= inf4, s26 >= 0, s26 <= inf5, s27 >= 0, s27 <= 1, s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 1 }-> 1 + F(z - 1, 1 + 0, 1 + (z - 1), 1 + (z - 1)) :|: z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> s19 :|: s19 >= 0, s19 <= inf', z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s20 :|: s20 >= 0, s20 <= inf'', s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s21 :|: s21 >= 0, s21 <= inf1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s33 :|: s31 >= 0, s31 <= inf8, s32 >= 0, s32 <= inf9, s33 >= 0, s33 <= s32 + s31, s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> s18 :|: s18 >= 0, s18 <= inf, z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: O(1) [0], size: EXP
  perfectp: runtime: O(1) [0], size: INF
  F: runtime: O(n^3) [13 + 14*z + z*z' + 4*z*z4 + 3*z^2 + z^2*z4 + 4*z' + 4*z4], size: O(n^3) [3 + 8*z + z*z' + 8*z*z4 + 3*z^2 + 4*z^2*z4 + 3*z' + z4]

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

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 3 + z' }-> 1 + s24 + s12 :|: s22 >= 0, s22 <= inf2, s23 >= 0, s23 <= inf3, s24 >= 0, s24 <= 1, s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 4 + 8*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s30 + s39 :|: s39 >= 0, s39 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s28 >= 0, s28 <= inf6, s29 >= 0, s29 <= inf7, s30 >= 0, s30 <= 1, s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 5 + 9*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s35 + s14 :|: s35 >= 0, s35 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 5 + 9*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s36 + s15 :|: s36 >= 0, s36 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 5 + 9*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s37 + s16 :|: s37 >= 0, s37 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 16 + 4*s3 + s3*z + 15*z + 4*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s27 + s38 + s17 :|: s38 >= 0, s38 <= 3 * ((1 + (z - 1)) * (1 + (z - 1))) + 8 * (1 + (z - 1)) + 3 * s3 + z4 + s3 * (1 + (z - 1)) + 4 * (z4 * ((1 + (z - 1)) * (1 + (z - 1)))) + 8 * (z4 * (1 + (z - 1))) + 3, s25 >= 0, s25 <= inf4, s26 >= 0, s26 <= inf5, s27 >= 0, s27 <= 1, s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 6 + 10*z + 5*z^2 + z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * (1 + 0) + (1 + (z - 1)) + (1 + 0) * (z - 1) + 4 * ((1 + (z - 1)) * ((z - 1) * (z - 1))) + 8 * ((1 + (z - 1)) * (z - 1)) + 3, z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> s19 :|: s19 >= 0, s19 <= inf', z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s20 :|: s20 >= 0, s20 <= inf'', s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s21 :|: s21 >= 0, s21 <= inf1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s33 :|: s31 >= 0, s31 <= inf8, s32 >= 0, s32 <= inf9, s33 >= 0, s33 <= s32 + s31, s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> s18 :|: s18 >= 0, s18 <= inf, z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: O(1) [0], size: EXP
  perfectp: runtime: O(1) [0], size: INF
  F: runtime: O(n^3) [13 + 14*z + z*z' + 4*z*z4 + 3*z^2 + z^2*z4 + 4*z' + 4*z4], size: O(n^3) [3 + 8*z + z*z' + 8*z*z4 + 3*z^2 + 4*z^2*z4 + 3*z' + z4]

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: PERFECTP
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1 + 3*z^2 + 4*z^3

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 3 + z' }-> 1 + s24 + s12 :|: s22 >= 0, s22 <= inf2, s23 >= 0, s23 <= inf3, s24 >= 0, s24 <= 1, s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 4 + 8*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s30 + s39 :|: s39 >= 0, s39 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s28 >= 0, s28 <= inf6, s29 >= 0, s29 <= inf7, s30 >= 0, s30 <= 1, s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 5 + 9*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s35 + s14 :|: s35 >= 0, s35 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 5 + 9*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s36 + s15 :|: s36 >= 0, s36 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 5 + 9*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s37 + s16 :|: s37 >= 0, s37 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 16 + 4*s3 + s3*z + 15*z + 4*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s27 + s38 + s17 :|: s38 >= 0, s38 <= 3 * ((1 + (z - 1)) * (1 + (z - 1))) + 8 * (1 + (z - 1)) + 3 * s3 + z4 + s3 * (1 + (z - 1)) + 4 * (z4 * ((1 + (z - 1)) * (1 + (z - 1)))) + 8 * (z4 * (1 + (z - 1))) + 3, s25 >= 0, s25 <= inf4, s26 >= 0, s26 <= inf5, s27 >= 0, s27 <= 1, s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 6 + 10*z + 5*z^2 + z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * (1 + 0) + (1 + (z - 1)) + (1 + 0) * (z - 1) + 4 * ((1 + (z - 1)) * ((z - 1) * (z - 1))) + 8 * ((1 + (z - 1)) * (z - 1)) + 3, z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> s19 :|: s19 >= 0, s19 <= inf', z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s20 :|: s20 >= 0, s20 <= inf'', s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s21 :|: s21 >= 0, s21 <= inf1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s33 :|: s31 >= 0, s31 <= inf8, s32 >= 0, s32 <= inf9, s33 >= 0, s33 <= s32 + s31, s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> s18 :|: s18 >= 0, s18 <= inf, z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {PERFECTP}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: O(1) [0], size: EXP
  perfectp: runtime: O(1) [0], size: INF
  F: runtime: O(n^3) [13 + 14*z + z*z' + 4*z*z4 + 3*z^2 + z^2*z4 + 4*z' + 4*z4], size: O(n^3) [3 + 8*z + z*z' + 8*z*z4 + 3*z^2 + 4*z^2*z4 + 3*z' + z4]
  PERFECTP: runtime: ?, size: O(n^3) [1 + 3*z^2 + 4*z^3]

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

(77) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: PERFECTP
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 7 + 10*z + 5*z^2 + z^3

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

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z', z'', z4) -{ 1 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 0 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   F(z, z', z'', z4) -{ 3 + z' }-> 1 + s24 + s12 :|: s22 >= 0, s22 <= inf2, s23 >= 0, s23 <= inf3, s24 >= 0, s24 <= 1, s12 >= 0, s12 <= z' - 1 + 1, s7 >= 0, s7 <= 2, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 4 + 8*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s30 + s39 :|: s39 >= 0, s39 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s28 >= 0, s28 <= inf6, s29 >= 0, s29 <= inf7, s30 >= 0, s30 <= 1, s9 >= 0, s9 <= 2, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 5 + 9*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s35 + s14 :|: s35 >= 0, s35 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s14 >= 0, s14 <= 1 + (z - 1) + 1, z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 5 + 9*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s36 + s15 :|: s36 >= 0, s36 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s15 >= 0, s15 <= 1 + (z - 1) + 1, s >= 0, s <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 5 + 9*z + 3*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s37 + s16 :|: s37 >= 0, s37 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * z4 + z4 + z4 * (z - 1) + 4 * (z4 * ((z - 1) * (z - 1))) + 8 * (z4 * (z - 1)) + 3, s16 >= 0, s16 <= 1 + (z - 1) + 1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   F(z, z', z'', z4) -{ 16 + 4*s3 + s3*z + 15*z + 4*z*z4 + 3*z^2 + z^2*z4 + 4*z4 }-> 1 + s27 + s38 + s17 :|: s38 >= 0, s38 <= 3 * ((1 + (z - 1)) * (1 + (z - 1))) + 8 * (1 + (z - 1)) + 3 * s3 + z4 + s3 * (1 + (z - 1)) + 4 * (z4 * ((1 + (z - 1)) * (1 + (z - 1)))) + 8 * (z4 * (1 + (z - 1))) + 3, s25 >= 0, s25 <= inf4, s26 >= 0, s26 <= inf5, s27 >= 0, s27 <= 1, s17 >= 0, s17 <= z - 1 + 1, s8 >= 0, s8 <= 2, s2 >= 0, s2 <= z' - 1, s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   IF(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   PERFECTP(z) -{ 1 }-> 0 :|: z = 0
   PERFECTP(z) -{ 6 + 10*z + 5*z^2 + z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * ((z - 1) * (z - 1)) + 8 * (z - 1) + 3 * (1 + 0) + (1 + (z - 1)) + (1 + 0) * (z - 1) + 4 * ((1 + (z - 1)) * ((z - 1) * (z - 1))) + 8 * ((1 + (z - 1)) * (z - 1)) + 3, z - 1 >= 0
   f(z, z', z'', z4) -{ 0 }-> s19 :|: s19 >= 0, s19 <= inf', z'' = 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s20 :|: s20 >= 0, s20 <= inf'', s'' >= 0, s'' <= z'' - 1, z'' - 1 >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s21 :|: s21 >= 0, s21 <= inf1, z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0
   f(z, z', z'', z4) -{ 0 }-> s33 :|: s31 >= 0, s31 <= inf8, s32 >= 0, s32 <= inf9, s33 >= 0, s33 <= s32 + s31, s10 >= 0, s10 <= 2, s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 2 :|: z'' = 0, z4 >= 0, z' >= 0, z = 0
   f(z, z', z'', z4) -{ 0 }-> 1 :|: z'' - 1 >= 0, z' >= 0, z = 0, z4 >= 0
   f(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if(z, z', z'') -{ 0 }-> z' :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> z'' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   le(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   perfectp(z) -{ 0 }-> s18 :|: s18 >= 0, s18 <= inf, z - 1 >= 0
   perfectp(z) -{ 0 }-> 1 :|: z = 0
   perfectp(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  IF: runtime: O(1) [1], size: O(1) [1]
  if: runtime: O(1) [0], size: O(n^1) [z' + z'']
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  f: runtime: O(1) [0], size: EXP
  perfectp: runtime: O(1) [0], size: INF
  F: runtime: O(n^3) [13 + 14*z + z*z' + 4*z*z4 + 3*z^2 + z^2*z4 + 4*z' + 4*z4], size: O(n^3) [3 + 8*z + z*z' + 8*z*z4 + 3*z^2 + 4*z^2*z4 + 3*z' + z4]
  PERFECTP: runtime: O(n^3) [7 + 10*z + 5*z^2 + z^3], size: O(n^3) [1 + 3*z^2 + 4*z^3]

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

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

(80)
BOUNDS(1, n^3)

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

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

(82)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))

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

   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

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

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

The rewrite sequence

LE(s(z0), s(z1)) ->^+ c5(LE(z0, z1))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [z0 / s(z0), z1 / s(z1)].

The result substitution is [ ].




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

(84)
Complex Obligation (BEST)

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

(85)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))

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

   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

Rewrite Strategy: INNERMOST
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(86) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(87)
BOUNDS(n^1, INF)

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

(88)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))

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

   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

Rewrite Strategy: INNERMOST