WORST_CASE(Omega(n^1),O(n^2))
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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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^2).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 641 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), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) InliningProof [UPPER BOUND(ID), 594 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 15 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 4 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 96 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 17 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 377 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 89 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 231 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 3 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 309 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 180 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 22 ms]
    (60) CpxRNTS
    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 360 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 113 ms]
    (66) CpxRNTS
    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 4984 ms]
    (70) CpxRNTS
    (71) IntTrsBoundProof [UPPER BOUND(ID), 1102 ms]
    (72) CpxRNTS
    (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 8324 ms]
    (76) CpxRNTS
    (77) IntTrsBoundProof [UPPER BOUND(ID), 3338 ms]
    (78) CpxRNTS
    (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (80) CpxRNTS
    (81) IntTrsBoundProof [UPPER BOUND(ID), 58 ms]
    (82) CpxRNTS
    (83) IntTrsBoundProof [UPPER BOUND(ID), 13 ms]
    (84) CpxRNTS
    (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (86) CpxRNTS
    (87) IntTrsBoundProof [UPPER BOUND(ID), 627 ms]
    (88) CpxRNTS
    (89) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (90) CpxRNTS
    (91) FinalProof [FINISHED, 0 ms]
    (92) BOUNDS(1, n^2)
(93) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(94) CpxRelTRS
    (95) SlicingProof [LOWER BOUND(ID), 0 ms]
    (96) CpxRelTRS
    (97) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) typed CpxTrs
    (99) OrderProof [LOWER BOUND(ID), 6 ms]
    (100) typed CpxTrs
    (101) RewriteLemmaProof [LOWER BOUND(ID), 324 ms]
    (102) BEST
        (103) proven lower bound
            (104) LowerBoundPropagationProof [FINISHED, 0 ms]
            (105) BOUNDS(n^1, INF)
        (106) typed CpxTrs
            (107) RewriteLemmaProof [LOWER BOUND(ID), 38 ms]
            (108) typed CpxTrs
            (109) RewriteLemmaProof [LOWER BOUND(ID), 34 ms]
            (110) typed CpxTrs
            (111) RewriteLemmaProof [LOWER BOUND(ID), 55 ms]
            (112) typed CpxTrs


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

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


The TRS R consists of the following rules:

   MINUS(0, z0) -> c
   MINUS(z0, 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   PLUS(0, z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
   ZERO(s(z0)) -> c5
   ZERO(0) -> c6
   P(s(z0)) -> c7
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0))
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0))), ZERO(z0))
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0))), PLUS(z2, s(0)))
   IF(true, z0, z1, z2) -> c11(P(z2))
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))

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

   minus(0, z0) -> 0
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   zero(s(z0)) -> false
   zero(0) -> true
   p(s(z0)) -> z0
   div(z0, z1) -> quot(z0, z1, 0)
   quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0)))
   if(true, z0, z1, z2) -> p(z2)
   if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

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^2).


The TRS R consists of the following rules:

   MINUS(0, z0) -> c
   MINUS(z0, 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   PLUS(0, z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
   ZERO(s(z0)) -> c5
   ZERO(0) -> c6
   P(s(z0)) -> c7
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0))
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0))), ZERO(z0))
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0))), PLUS(z2, s(0)))
   IF(true, z0, z1, z2) -> c11(P(z2))
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))

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

   minus(0, z0) -> 0
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   zero(s(z0)) -> false
   zero(0) -> true
   p(s(z0)) -> z0
   div(z0, z1) -> quot(z0, z1, 0)
   quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0)))
   if(true, z0, z1, z2) -> p(z2)
   if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

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^2).


The TRS R consists of the following rules:

   MINUS(0, z0) -> c [1]
   MINUS(z0, 0) -> c1 [1]
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   PLUS(0, z0) -> c3 [1]
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1))) [1]
   ZERO(s(z0)) -> c5 [1]
   ZERO(0) -> c6 [1]
   P(s(z0)) -> c7 [1]
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0)) [1]
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0))), ZERO(z0)) [1]
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0))), PLUS(z2, s(0))) [1]
   IF(true, z0, z1, z2) -> c11(P(z2)) [1]
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) [1]
   minus(0, z0) -> 0 [0]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> plus(z0, s(z1)) [0]
   zero(s(z0)) -> false [0]
   zero(0) -> true [0]
   p(s(z0)) -> z0 [0]
   div(z0, z1) -> quot(z0, z1, 0) [0]
   quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0))) [0]
   if(true, z0, z1, z2) -> p(z2) [0]
   if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2) [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(z0, 0) -> c1 [1]
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   PLUS(0, z0) -> c3 [1]
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1))) [1]
   ZERO(s(z0)) -> c5 [1]
   ZERO(0) -> c6 [1]
   P(s(z0)) -> c7 [1]
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0)) [1]
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0))), ZERO(z0)) [1]
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0))), PLUS(z2, s(0))) [1]
   IF(true, z0, z1, z2) -> c11(P(z2)) [1]
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) [1]
   minus(0, z0) -> 0 [0]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> plus(z0, s(z1)) [0]
   zero(s(z0)) -> false [0]
   zero(0) -> true [0]
   p(s(z0)) -> z0 [0]
   div(z0, z1) -> quot(z0, z1, 0) [0]
   quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0))) [0]
   if(true, z0, z1, z2) -> p(z2) [0]
   if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2) [0]

The TRS has the following type information:
MINUS :: 0:s -> 0:s -> c:c1:c2
0 :: 0:s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0:s -> 0:s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0:s -> 0:s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0:s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0:s -> c7
c7 :: c7
DIV :: 0:s -> 0:s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0:s -> 0:s -> 0:s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0:s -> 0:s -> 0:s -> c11:c12
zero :: 0:s -> true:false
plus :: 0:s -> 0:s -> 0:s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s

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
   PLUS_2
   ZERO_1
   P_1
   DIV_2
   QUOT_3
   IF_4

(c) The following functions are completely defined:

   minus_2
   plus_2
   zero_1
   p_1
   div_2
   quot_3
   if_4

Due to the following rules being added:

   minus(v0, v1) -> 0 [0]
   plus(v0, v1) -> 0 [0]
   zero(v0) -> null_zero [0]
   p(v0) -> 0 [0]
   div(v0, v1) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]
   if(v0, v1, v2, v3) -> 0 [0]

And the following fresh constants:    null_zero, const, const1, const2

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

(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(z0, 0) -> c1 [1]
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   PLUS(0, z0) -> c3 [1]
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1))) [1]
   ZERO(s(z0)) -> c5 [1]
   ZERO(0) -> c6 [1]
   P(s(z0)) -> c7 [1]
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0)) [1]
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0))), ZERO(z0)) [1]
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0))), PLUS(z2, s(0))) [1]
   IF(true, z0, z1, z2) -> c11(P(z2)) [1]
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) [1]
   minus(0, z0) -> 0 [0]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> plus(z0, s(z1)) [0]
   zero(s(z0)) -> false [0]
   zero(0) -> true [0]
   p(s(z0)) -> z0 [0]
   div(z0, z1) -> quot(z0, z1, 0) [0]
   quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0))) [0]
   if(true, z0, z1, z2) -> p(z2) [0]
   if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2) [0]
   minus(v0, v1) -> 0 [0]
   plus(v0, v1) -> 0 [0]
   zero(v0) -> null_zero [0]
   p(v0) -> 0 [0]
   div(v0, v1) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]
   if(v0, v1, v2, v3) -> 0 [0]

The TRS has the following type information:
MINUS :: 0:s -> 0:s -> c:c1:c2
0 :: 0:s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0:s -> 0:s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0:s -> 0:s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0:s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0:s -> c7
c7 :: c7
DIV :: 0:s -> 0:s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0:s -> 0:s -> 0:s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false:null_zero -> 0:s -> 0:s -> 0:s -> c11:c12
zero :: 0:s -> true:false:null_zero
plus :: 0:s -> 0:s -> 0:s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false:null_zero
c11 :: c7 -> c11:c12
false :: true:false:null_zero
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s -> 0:s
if :: true:false:null_zero -> 0:s -> 0:s -> 0:s -> 0:s
null_zero :: true:false:null_zero
const :: c8
const1 :: c9:c10
const2 :: c11:c12

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(z0, 0) -> c1 [1]
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) [1]
   PLUS(0, z0) -> c3 [1]
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1))) [1]
   ZERO(s(z0)) -> c5 [1]
   ZERO(0) -> c6 [1]
   P(s(z0)) -> c7 [1]
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0)) [1]
   QUOT(s(z0'), z1, 0) -> c9(IF(false, s(z0'), z1, s(0)), ZERO(s(z0'))) [1]
   QUOT(s(z0'), z1, s(z0'')) -> c9(IF(false, s(z0'), z1, plus(z0'', s(s(0)))), ZERO(s(z0'))) [1]
   QUOT(s(z0'), z1, z2) -> c9(IF(false, s(z0'), z1, 0), ZERO(s(z0'))) [1]
   QUOT(0, z1, 0) -> c9(IF(true, 0, z1, s(0)), ZERO(0)) [1]
   QUOT(0, z1, s(z01)) -> c9(IF(true, 0, z1, plus(z01, s(s(0)))), ZERO(0)) [1]
   QUOT(0, z1, z2) -> c9(IF(true, 0, z1, 0), ZERO(0)) [1]
   QUOT(z0, z1, 0) -> c9(IF(null_zero, z0, z1, s(0)), ZERO(z0)) [1]
   QUOT(z0, z1, s(z02)) -> c9(IF(null_zero, z0, z1, plus(z02, s(s(0)))), ZERO(z0)) [1]
   QUOT(z0, z1, z2) -> c9(IF(null_zero, z0, z1, 0), ZERO(z0)) [1]
   QUOT(s(z03), z1, 0) -> c10(IF(false, s(z03), z1, s(0)), PLUS(0, s(0))) [1]
   QUOT(s(z03), z1, s(z04)) -> c10(IF(false, s(z03), z1, plus(z04, s(s(0)))), PLUS(s(z04), s(0))) [1]
   QUOT(s(z03), z1, z2) -> c10(IF(false, s(z03), z1, 0), PLUS(z2, s(0))) [1]
   QUOT(0, z1, 0) -> c10(IF(true, 0, z1, s(0)), PLUS(0, s(0))) [1]
   QUOT(0, z1, s(z05)) -> c10(IF(true, 0, z1, plus(z05, s(s(0)))), PLUS(s(z05), s(0))) [1]
   QUOT(0, z1, z2) -> c10(IF(true, 0, z1, 0), PLUS(z2, s(0))) [1]
   QUOT(z0, z1, 0) -> c10(IF(null_zero, z0, z1, s(0)), PLUS(0, s(0))) [1]
   QUOT(z0, z1, s(z06)) -> c10(IF(null_zero, z0, z1, plus(z06, s(s(0)))), PLUS(s(z06), s(0))) [1]
   QUOT(z0, z1, z2) -> c10(IF(null_zero, z0, z1, 0), PLUS(z2, s(0))) [1]
   IF(true, z0, z1, z2) -> c11(P(z2)) [1]
   IF(false, 0, s(z1), z2) -> c12(QUOT(0, s(z1), z2), MINUS(0, s(z1))) [1]
   IF(false, s(z07), s(z1), z2) -> c12(QUOT(minus(z07, z1), s(z1), z2), MINUS(s(z07), s(z1))) [1]
   IF(false, z0, s(z1), z2) -> c12(QUOT(0, s(z1), z2), MINUS(z0, s(z1))) [1]
   minus(0, z0) -> 0 [0]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> plus(z0, s(z1)) [0]
   zero(s(z0)) -> false [0]
   zero(0) -> true [0]
   p(s(z0)) -> z0 [0]
   div(z0, z1) -> quot(z0, z1, 0) [0]
   quot(s(z08), z1, 0) -> if(false, s(z08), z1, s(0)) [0]
   quot(s(z08), z1, s(z09)) -> if(false, s(z08), z1, plus(z09, s(s(0)))) [0]
   quot(s(z08), z1, z2) -> if(false, s(z08), z1, 0) [0]
   quot(0, z1, 0) -> if(true, 0, z1, s(0)) [0]
   quot(0, z1, s(z010)) -> if(true, 0, z1, plus(z010, s(s(0)))) [0]
   quot(0, z1, z2) -> if(true, 0, z1, 0) [0]
   quot(z0, z1, 0) -> if(null_zero, z0, z1, s(0)) [0]
   quot(z0, z1, s(z011)) -> if(null_zero, z0, z1, plus(z011, s(s(0)))) [0]
   quot(z0, z1, z2) -> if(null_zero, z0, z1, 0) [0]
   if(true, z0, z1, z2) -> p(z2) [0]
   if(false, 0, s(z1), z2) -> quot(0, s(z1), z2) [0]
   if(false, s(z012), s(z1), z2) -> quot(minus(z012, z1), s(z1), z2) [0]
   if(false, z0, s(z1), z2) -> quot(0, s(z1), z2) [0]
   minus(v0, v1) -> 0 [0]
   plus(v0, v1) -> 0 [0]
   zero(v0) -> null_zero [0]
   p(v0) -> 0 [0]
   div(v0, v1) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]
   if(v0, v1, v2, v3) -> 0 [0]

The TRS has the following type information:
MINUS :: 0:s -> 0:s -> c:c1:c2
0 :: 0:s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0:s -> 0:s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0:s -> 0:s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0:s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0:s -> c7
c7 :: c7
DIV :: 0:s -> 0:s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0:s -> 0:s -> 0:s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false:null_zero -> 0:s -> 0:s -> 0:s -> c11:c12
zero :: 0:s -> true:false:null_zero
plus :: 0:s -> 0:s -> 0:s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false:null_zero
c11 :: c7 -> c11:c12
false :: true:false:null_zero
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s -> 0:s
if :: true:false:null_zero -> 0:s -> 0:s -> 0:s -> 0:s
null_zero :: true:false:null_zero
const :: c8
const1 :: c9:c10
const2 :: c11:c12

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
   c5 => 0
   c6 => 1
   c7 => 0
   true => 2
   false => 1
   null_zero => 0
   const => 0
   const1 => 0
   const2 => 0

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + P(z2) :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(minus(z07, z1), 1 + z1, z2) + MINUS(1 + z07, 1 + z1) :|: z1 >= 0, z07 >= 0, z = 1, z' = 1 + z07, z'' = 1 + z1, z3 = z2, z2 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + z1, z2) + MINUS(z0, 1 + z1) :|: z1 >= 0, z = 1, z0 >= 0, z'' = 1 + z1, z3 = z2, z' = z0, z2 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + z1, z2) + MINUS(0, 1 + z1) :|: z1 >= 0, z = 1, z'' = 1 + z1, z3 = z2, z2 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z = 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
   P(z) -{ 1 }-> 0 :|: z = 1 + z0, z0 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z1, plus(z01, 1 + (1 + 0))) + ZERO(0) :|: z1 >= 0, z01 >= 0, z' = z1, z = 0, z'' = 1 + z01
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z1, plus(z05, 1 + (1 + 0))) + PLUS(1 + z05, 1 + 0) :|: z1 >= 0, z'' = 1 + z05, z' = z1, z05 >= 0, z = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z1, 0) + ZERO(0) :|: z'' = z2, z1 >= 0, z' = z1, z = 0, z2 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z1, 0) + PLUS(z2, 1 + 0) :|: z'' = z2, z1 >= 0, z' = z1, z = 0, z2 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z1, 1 + 0) + ZERO(0) :|: z'' = 0, z1 >= 0, z' = z1, z = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z1, 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z1 >= 0, z' = z1, z = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + z0', z1, plus(z0'', 1 + (1 + 0))) + ZERO(1 + z0') :|: z1 >= 0, z'' = 1 + z0'', z0' >= 0, z' = z1, z0'' >= 0, z = 1 + z0'
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + z0', z1, 0) + ZERO(1 + z0') :|: z'' = z2, z1 >= 0, z0' >= 0, z' = z1, z = 1 + z0', z2 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + z0', z1, 1 + 0) + ZERO(1 + z0') :|: z'' = 0, z1 >= 0, z0' >= 0, z' = z1, z = 1 + z0'
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + z03, z1, plus(z04, 1 + (1 + 0))) + PLUS(1 + z04, 1 + 0) :|: z'' = 1 + z04, z04 >= 0, z1 >= 0, z' = z1, z = 1 + z03, z03 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + z03, z1, 0) + PLUS(z2, 1 + 0) :|: z'' = z2, z1 >= 0, z' = z1, z = 1 + z03, z03 >= 0, z2 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + z03, z1, 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z1 >= 0, z' = z1, z = 1 + z03, z03 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, plus(z02, 1 + (1 + 0))) + ZERO(z0) :|: z = z0, z1 >= 0, z02 >= 0, z' = z1, z0 >= 0, z'' = 1 + z02
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, plus(z06, 1 + (1 + 0))) + PLUS(1 + z06, 1 + 0) :|: z'' = 1 + z06, z = z0, z1 >= 0, z06 >= 0, z' = z1, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 0) + ZERO(z0) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 0) + PLUS(z2, 1 + 0) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + 0) + ZERO(z0) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z = 1 + z0, z0 >= 0
   div(z, z') -{ 0 }-> quot(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if(z, z', z'', z3) -{ 0 }-> quot(minus(z012, z1), 1 + z1, z2) :|: z1 >= 0, z = 1, z' = 1 + z012, z012 >= 0, z'' = 1 + z1, z3 = z2, z2 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + z1, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1, z3 = z2, z2 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + z1, z2) :|: z1 >= 0, z = 1, z0 >= 0, z'' = 1 + z1, z3 = z2, z' = z0, z2 >= 0
   if(z, z', z'', z3) -{ 0 }-> p(z2) :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   if(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   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
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   plus(z, z') -{ 0 }-> plus(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z', z'') -{ 0 }-> if(2, 0, z1, plus(z010, 1 + (1 + 0))) :|: z1 >= 0, z'' = 1 + z010, z' = z1, z = 0, z010 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z1, 0) :|: z'' = z2, z1 >= 0, z' = z1, z = 0, z2 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z1, 1 + 0) :|: z'' = 0, z1 >= 0, z' = z1, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + z08, z1, plus(z09, 1 + (1 + 0))) :|: z08 >= 0, z1 >= 0, z'' = 1 + z09, z = 1 + z08, z' = z1, z09 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + z08, z1, 0) :|: z08 >= 0, z'' = z2, z1 >= 0, z = 1 + z08, z' = z1, z2 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + z08, z1, 1 + 0) :|: z'' = 0, z08 >= 0, z1 >= 0, z = 1 + z08, z' = z1
   quot(z, z', z'') -{ 0 }-> if(0, z0, z1, plus(z011, 1 + (1 + 0))) :|: z = z0, z1 >= 0, z011 >= 0, z'' = 1 + z011, z' = z1, z0 >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z0, z1, 0) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z0, z1, 1 + 0) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0
   zero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z = 1 + z0, z0 >= 0
   P(z) -{ 1 }-> 0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1, z2 = 1 + z0', z0' >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(minus(z07, z1), 1 + z1, z2) + MINUS(1 + z07, 1 + z1) :|: z1 >= 0, z07 >= 0, z = 1, z' = 1 + z07, z'' = 1 + z1, z3 = z2, z2 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + z1, z2) + MINUS(z0, 1 + z1) :|: z1 >= 0, z = 1, z0 >= 0, z'' = 1 + z1, z3 = z2, z' = z0, z2 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + z1, z2) + MINUS(0, 1 + z1) :|: z1 >= 0, z = 1, z'' = 1 + z1, z3 = z2, z2 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z = 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
   P(z) -{ 1 }-> 0 :|: z = 1 + z0, z0 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z1, plus(z01, 1 + (1 + 0))) + 1 :|: z1 >= 0, z01 >= 0, z' = z1, z = 0, z'' = 1 + z01, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z1, plus(z05, 1 + (1 + 0))) + PLUS(1 + z05, 1 + 0) :|: z1 >= 0, z'' = 1 + z05, z' = z1, z05 >= 0, z = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z1, 0) + PLUS(z2, 1 + 0) :|: z'' = z2, z1 >= 0, z' = z1, z = 0, z2 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z1, 0) + 1 :|: z'' = z2, z1 >= 0, z' = z1, z = 0, z2 >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z1, 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z1 >= 0, z' = z1, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z1, 1 + 0) + 1 :|: z'' = 0, z1 >= 0, z' = z1, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + z0', z1, plus(z0'', 1 + (1 + 0))) + 0 :|: z1 >= 0, z'' = 1 + z0'', z0' >= 0, z' = z1, z0'' >= 0, z = 1 + z0', 1 + z0' = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + z0', z1, 0) + 0 :|: z'' = z2, z1 >= 0, z0' >= 0, z' = z1, z = 1 + z0', z2 >= 0, 1 + z0' = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + z0', z1, 1 + 0) + 0 :|: z'' = 0, z1 >= 0, z0' >= 0, z' = z1, z = 1 + z0', 1 + z0' = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + z03, z1, plus(z04, 1 + (1 + 0))) + PLUS(1 + z04, 1 + 0) :|: z'' = 1 + z04, z04 >= 0, z1 >= 0, z' = z1, z = 1 + z03, z03 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + z03, z1, 0) + PLUS(z2, 1 + 0) :|: z'' = z2, z1 >= 0, z' = z1, z = 1 + z03, z03 >= 0, z2 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + z03, z1, 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z1 >= 0, z' = z1, z = 1 + z03, z03 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z0, z1, plus(z02, 1 + (1 + 0))) + 1 :|: z = z0, z1 >= 0, z02 >= 0, z' = z1, z0 >= 0, z'' = 1 + z02, z0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z0, z1, plus(z02, 1 + (1 + 0))) + 0 :|: z = z0, z1 >= 0, z02 >= 0, z' = z1, z0 >= 0, z'' = 1 + z02, z0 = 1 + z0', z0' >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, plus(z06, 1 + (1 + 0))) + PLUS(1 + z06, 1 + 0) :|: z'' = 1 + z06, z = z0, z1 >= 0, z06 >= 0, z' = z1, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 0) + PLUS(z2, 1 + 0) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z0, z1, 0) + 1 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0, z0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z0, z1, 0) + 0 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0, z0 = 1 + z0', z0' >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z0, z1, 1 + 0) + 1 :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0, z0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z0, z1, 1 + 0) + 0 :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0, z0 = 1 + z0', z0' >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z = 1 + z0, z0 >= 0
   div(z, z') -{ 0 }-> quot(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if(z, z', z'', z3) -{ 0 }-> z0' :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1, z2 = 1 + z0', z0' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(minus(z012, z1), 1 + z1, z2) :|: z1 >= 0, z = 1, z' = 1 + z012, z012 >= 0, z'' = 1 + z1, z3 = z2, z2 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + z1, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1, z3 = z2, z2 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + z1, z2) :|: z1 >= 0, z = 1, z0 >= 0, z'' = 1 + z1, z3 = z2, z' = z0, z2 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1, v0 >= 0, z2 = v0
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   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
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   plus(z, z') -{ 0 }-> plus(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z', z'') -{ 0 }-> if(2, 0, z1, plus(z010, 1 + (1 + 0))) :|: z1 >= 0, z'' = 1 + z010, z' = z1, z = 0, z010 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z1, 0) :|: z'' = z2, z1 >= 0, z' = z1, z = 0, z2 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z1, 1 + 0) :|: z'' = 0, z1 >= 0, z' = z1, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + z08, z1, plus(z09, 1 + (1 + 0))) :|: z08 >= 0, z1 >= 0, z'' = 1 + z09, z = 1 + z08, z' = z1, z09 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + z08, z1, 0) :|: z08 >= 0, z'' = z2, z1 >= 0, z = 1 + z08, z' = z1, z2 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + z08, z1, 1 + 0) :|: z'' = 0, z08 >= 0, z1 >= 0, z = 1 + z08, z' = z1
   quot(z, z', z'') -{ 0 }-> if(0, z0, z1, plus(z011, 1 + (1 + 0))) :|: z = z0, z1 >= 0, z011 >= 0, z'' = 1 + z011, z' = z1, z0 >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z0, z1, 0) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z0, z1, 1 + 0) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0
   zero(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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: 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
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { P }
   { minus }
   { ZERO }
   { PLUS }
   { zero }
   { plus }
   { p }
   { MINUS }
   { if, quot }
   { QUOT, IF }
   { div }
   { DIV }

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: 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
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {P}, {minus}, {ZERO}, {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}

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(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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: 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
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {P}, {minus}, {ZERO}, {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: 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
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {P}, {minus}, {ZERO}, {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: ?, size: O(1) [0]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: 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
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {minus}, {ZERO}, {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]

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

(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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: 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
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {minus}, {ZERO}, {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]

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

(27) 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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: 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
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {minus}, {ZERO}, {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: ?, size: O(n^1) [z]

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

(29) 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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: 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
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ZERO}, {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]

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(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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ZERO}, {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]

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(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ZERO}, {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: ?, size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]

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

(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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {PLUS}, {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: ?, size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(2, 0, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + PLUS(1 + (z'' - 1), 1 + 0) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + PLUS(z'', 1 + 0) :|: z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + PLUS(0, 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + s6 :|: s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + s3 :|: s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + s9 :|: s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + s6 :|: s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + s3 :|: s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + s9 :|: s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {zero}, {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: ?, size: O(1) [2]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: zero
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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + s6 :|: s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + s3 :|: s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + s9 :|: s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]

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

(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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + s6 :|: s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + s3 :|: s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + s9 :|: s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + s6 :|: s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + s3 :|: s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + s9 :|: s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {plus}, {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: ?, size: O(n^1) [z + z']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + s6 :|: s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + s3 :|: s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + s9 :|: s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 1 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', plus(z'' - 1, 1 + (1 + 0))) + 0 :|: z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', plus(z'' - 1, 1 + (1 + 0))) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', plus(z'' - 1, 1 + (1 + 0))) :|: z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + 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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', s16) :|: s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', s15) :|: s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', s17) :|: s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', s16) :|: s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', s15) :|: s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', s17) :|: s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {p}, {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: ?, size: O(n^1) [z]

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', s16) :|: s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', s15) :|: s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', s17) :|: s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]

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

(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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', s16) :|: s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', s15) :|: s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', s17) :|: s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]

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

(63) 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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', s16) :|: s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', s15) :|: s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', s17) :|: s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {MINUS}, {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: ?, size: O(n^1) [1 + z]

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

(65) 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'

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + MINUS(0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 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
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', s16) :|: s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', s15) :|: s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', s17) :|: s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]

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

(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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + s24 :|: s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s23 :|: s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s25 :|: s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', s16) :|: s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', s15) :|: s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', s17) :|: s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]

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

(69) 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: 1 + z' + z3

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + s24 :|: s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s23 :|: s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s25 :|: s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', s16) :|: s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', s15) :|: s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', s17) :|: s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if,quot}, {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: ?, size: O(n^1) [1 + z' + z3]
  quot: runtime: ?, size: O(n^1) [2 + z + z'']

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

(71) 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

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

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + s24 :|: s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s23 :|: s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s25 :|: s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> quot(z, z', 0) :|: z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(s'', 1 + (z'' - 1), z3) :|: s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> quot(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', s16) :|: s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 0) :|: z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(2, 0, z', 1 + 0) :|: z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', s15) :|: s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 0) :|: z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(1, 1 + (z - 1), z', 1 + 0) :|: z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', s17) :|: s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3]
  quot: runtime: O(1) [0], size: O(n^1) [2 + z + z'']

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

(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:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + s24 :|: s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s23 :|: s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s25 :|: s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> s26 :|: s26 >= 0, s26 <= z + 0 + 2, z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s37 :|: s37 >= 0, s37 <= s'' + z3 + 2, s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s38 :|: s38 >= 0, s38 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s27 :|: s27 >= 0, s27 <= 1 + 0 + 1 + (1 + (z - 1)), z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s28 :|: s28 >= 0, s28 <= s15 + 1 + (1 + (z - 1)), s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 0 + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + 0 + 1 + 0, z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= s16 + 1 + 0, s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + 1 + 0, z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= s17 + 1 + z, s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3]
  quot: runtime: O(1) [0], size: O(n^1) [2 + z + z'']

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: QUOT
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 265 + 69*z + 6*z*z'' + 7*z^2 + 10*z''

Computed SIZE bound using KoAT for: IF
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 802 + 71*z' + 6*z'*z3 + 7*z'^2 + 30*z3

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + s24 :|: s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s23 :|: s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s25 :|: s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> s26 :|: s26 >= 0, s26 <= z + 0 + 2, z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s37 :|: s37 >= 0, s37 <= s'' + z3 + 2, s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s38 :|: s38 >= 0, s38 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s27 :|: s27 >= 0, s27 <= 1 + 0 + 1 + (1 + (z - 1)), z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s28 :|: s28 >= 0, s28 <= s15 + 1 + (1 + (z - 1)), s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 0 + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + 0 + 1 + 0, z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= s16 + 1 + 0, s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + 1 + 0, z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= s17 + 1 + z, s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {QUOT,IF}, {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3]
  quot: runtime: O(1) [0], size: O(n^1) [2 + z + z'']
  QUOT: runtime: ?, size: O(n^2) [265 + 69*z + 6*z*z'' + 7*z^2 + 10*z'']
  IF: runtime: ?, size: O(n^2) [802 + 71*z' + 6*z'*z3 + 7*z'^2 + 30*z3]

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

(77) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: QUOT
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2696 + 2787*z + 7*z*z' + 90*z*z'' + 90*z^2 + 6*z' + 90*z''

Computed RUNTIME bound using KoAT for: IF
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 5402 + 2607*z' + 7*z'*z'' + 90*z'*z3 + 90*z'^2 + 14*z'' + 180*z3

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

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

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', 0) :|: z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(s, 1 + (z'' - 1), z3) + s24 :|: s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s23 :|: s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 3 + z'' }-> 1 + QUOT(0, 1 + (z'' - 1), z3) + s25 :|: s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', s12) + s6 :|: s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', s19) + 1 :|: s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, z', 0) + s7 :|: s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 0) + 1 :|: z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + s5 :|: s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(2, 0, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', s11) + s3 :|: s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', s18) + 0 :|: s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, 1 + (z - 1), z', 0) + s4 :|: s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 0) + 0 :|: z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(1, 1 + (z - 1), z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', s13) + s9 :|: s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s20) + 1 :|: s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', s21) + 0 :|: s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s10 :|: s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 1 :|: z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 0) + 0 :|: z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 1 :|: z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + 0 :|: z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> s26 :|: s26 >= 0, s26 <= z + 0 + 2, z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s37 :|: s37 >= 0, s37 <= s'' + z3 + 2, s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s38 :|: s38 >= 0, s38 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s27 :|: s27 >= 0, s27 <= 1 + 0 + 1 + (1 + (z - 1)), z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s28 :|: s28 >= 0, s28 <= s15 + 1 + (1 + (z - 1)), s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 0 + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + 0 + 1 + 0, z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= s16 + 1 + 0, s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + 1 + 0, z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= s17 + 1 + z, s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3]
  quot: runtime: O(1) [0], size: O(n^1) [2 + z + z'']
  QUOT: runtime: O(n^2) [2696 + 2787*z + 7*z*z' + 90*z*z'' + 90*z^2 + 6*z' + 90*z''], size: O(n^2) [265 + 69*z + 6*z*z'' + 7*z^2 + 10*z'']
  IF: runtime: O(n^2) [5402 + 2607*z' + 7*z'*z'' + 90*z'*z3 + 90*z'^2 + 14*z'' + 180*z3], size: O(n^2) [802 + 71*z' + 6*z'*z3 + 7*z'^2 + 30*z3]

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

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

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 2697 + 2787*z + 7*z*z' + 90*z^2 + 6*z' }-> 1 + s39 :|: s39 >= 0, s39 <= 69 * z + 10 * 0 + 7 * (z * z) + 6 * (0 * z) + 265, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s49 + s23 :|: s49 >= 0, s49 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2699 + 2787*s + 7*s*z'' + 90*s*z3 + 90*s^2 + 7*z'' + 90*z3 }-> 1 + s50 + s24 :|: s50 >= 0, s50 <= 69 * s + 10 * z3 + 7 * (s * s) + 6 * (z3 * s) + 265, s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s51 + s25 :|: s51 >= 0, s51 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s11 + 90*s11*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s41 + s3 :|: s41 >= 0, s41 <= 71 * (1 + (z - 1)) + 30 * s11 + 6 * (s11 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s42 + s4 :|: s42 >= 0, s42 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s43 + s5 :|: s43 >= 0, s43 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s12 + 14*z' + z'' }-> 1 + s44 + s6 :|: s44 >= 0, s44 <= 71 * 0 + 30 * s12 + 6 * (s12 * 0) + 7 * (0 * 0) + 802, s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' + z'' }-> 1 + s45 + s7 :|: s45 >= 0, s45 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s46 + s8 :|: s46 >= 0, s46 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s13 + 90*s13*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s47 + s9 :|: s47 >= 0, s47 <= 71 * z + 30 * s13 + 6 * (s13 * z) + 7 * (z * z) + 802, s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s48 + s10 :|: s48 >= 0, s48 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s18 + 90*s18*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s53 + 0 :|: s53 >= 0, s53 <= 71 * (1 + (z - 1)) + 30 * s18 + 6 * (s18 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s54 + 0 :|: s54 >= 0, s54 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s55 + 1 :|: s55 >= 0, s55 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 180*s19 + 14*z' }-> 1 + s56 + 1 :|: s56 >= 0, s56 <= 71 * 0 + 30 * s19 + 6 * (s19 * 0) + 7 * (0 * 0) + 802, s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' }-> 1 + s57 + 1 :|: s57 >= 0, s57 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s58 + 1 :|: s58 >= 0, s58 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s59 + 0 :|: s59 >= 0, s59 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s20 + 90*s20*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s60 + 1 :|: s60 >= 0, s60 <= 71 * z + 30 * s20 + 6 * (s20 * z) + 7 * (z * z) + 802, s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s21 + 90*s21*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s61 + 0 :|: s61 >= 0, s61 <= 71 * z + 30 * s21 + 6 * (s21 * z) + 7 * (z * z) + 802, s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s62 + 1 :|: s62 >= 0, s62 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s63 + 0 :|: s63 >= 0, s63 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> s26 :|: s26 >= 0, s26 <= z + 0 + 2, z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s37 :|: s37 >= 0, s37 <= s'' + z3 + 2, s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s38 :|: s38 >= 0, s38 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s27 :|: s27 >= 0, s27 <= 1 + 0 + 1 + (1 + (z - 1)), z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s28 :|: s28 >= 0, s28 <= s15 + 1 + (1 + (z - 1)), s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 0 + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + 0 + 1 + 0, z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= s16 + 1 + 0, s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + 1 + 0, z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= s17 + 1 + z, s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3]
  quot: runtime: O(1) [0], size: O(n^1) [2 + z + z'']
  QUOT: runtime: O(n^2) [2696 + 2787*z + 7*z*z' + 90*z*z'' + 90*z^2 + 6*z' + 90*z''], size: O(n^2) [265 + 69*z + 6*z*z'' + 7*z^2 + 10*z'']
  IF: runtime: O(n^2) [5402 + 2607*z' + 7*z'*z'' + 90*z'*z3 + 90*z'^2 + 14*z'' + 180*z3], size: O(n^2) [802 + 71*z' + 6*z'*z3 + 7*z'^2 + 30*z3]

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

(81) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 2697 + 2787*z + 7*z*z' + 90*z^2 + 6*z' }-> 1 + s39 :|: s39 >= 0, s39 <= 69 * z + 10 * 0 + 7 * (z * z) + 6 * (0 * z) + 265, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s49 + s23 :|: s49 >= 0, s49 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2699 + 2787*s + 7*s*z'' + 90*s*z3 + 90*s^2 + 7*z'' + 90*z3 }-> 1 + s50 + s24 :|: s50 >= 0, s50 <= 69 * s + 10 * z3 + 7 * (s * s) + 6 * (z3 * s) + 265, s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s51 + s25 :|: s51 >= 0, s51 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s11 + 90*s11*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s41 + s3 :|: s41 >= 0, s41 <= 71 * (1 + (z - 1)) + 30 * s11 + 6 * (s11 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s42 + s4 :|: s42 >= 0, s42 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s43 + s5 :|: s43 >= 0, s43 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s12 + 14*z' + z'' }-> 1 + s44 + s6 :|: s44 >= 0, s44 <= 71 * 0 + 30 * s12 + 6 * (s12 * 0) + 7 * (0 * 0) + 802, s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' + z'' }-> 1 + s45 + s7 :|: s45 >= 0, s45 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s46 + s8 :|: s46 >= 0, s46 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s13 + 90*s13*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s47 + s9 :|: s47 >= 0, s47 <= 71 * z + 30 * s13 + 6 * (s13 * z) + 7 * (z * z) + 802, s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s48 + s10 :|: s48 >= 0, s48 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s18 + 90*s18*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s53 + 0 :|: s53 >= 0, s53 <= 71 * (1 + (z - 1)) + 30 * s18 + 6 * (s18 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s54 + 0 :|: s54 >= 0, s54 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s55 + 1 :|: s55 >= 0, s55 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 180*s19 + 14*z' }-> 1 + s56 + 1 :|: s56 >= 0, s56 <= 71 * 0 + 30 * s19 + 6 * (s19 * 0) + 7 * (0 * 0) + 802, s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' }-> 1 + s57 + 1 :|: s57 >= 0, s57 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s58 + 1 :|: s58 >= 0, s58 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s59 + 0 :|: s59 >= 0, s59 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s20 + 90*s20*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s60 + 1 :|: s60 >= 0, s60 <= 71 * z + 30 * s20 + 6 * (s20 * z) + 7 * (z * z) + 802, s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s21 + 90*s21*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s61 + 0 :|: s61 >= 0, s61 <= 71 * z + 30 * s21 + 6 * (s21 * z) + 7 * (z * z) + 802, s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s62 + 1 :|: s62 >= 0, s62 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s63 + 0 :|: s63 >= 0, s63 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> s26 :|: s26 >= 0, s26 <= z + 0 + 2, z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s37 :|: s37 >= 0, s37 <= s'' + z3 + 2, s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s38 :|: s38 >= 0, s38 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s27 :|: s27 >= 0, s27 <= 1 + 0 + 1 + (1 + (z - 1)), z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s28 :|: s28 >= 0, s28 <= s15 + 1 + (1 + (z - 1)), s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 0 + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + 0 + 1 + 0, z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= s16 + 1 + 0, s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + 1 + 0, z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= s17 + 1 + z, s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {div}, {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3]
  quot: runtime: O(1) [0], size: O(n^1) [2 + z + z'']
  QUOT: runtime: O(n^2) [2696 + 2787*z + 7*z*z' + 90*z*z'' + 90*z^2 + 6*z' + 90*z''], size: O(n^2) [265 + 69*z + 6*z*z'' + 7*z^2 + 10*z'']
  IF: runtime: O(n^2) [5402 + 2607*z' + 7*z'*z'' + 90*z'*z3 + 90*z'^2 + 14*z'' + 180*z3], size: O(n^2) [802 + 71*z' + 6*z'*z3 + 7*z'^2 + 30*z3]
  div: runtime: ?, size: O(n^1) [2 + z]

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

(83) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 2697 + 2787*z + 7*z*z' + 90*z^2 + 6*z' }-> 1 + s39 :|: s39 >= 0, s39 <= 69 * z + 10 * 0 + 7 * (z * z) + 6 * (0 * z) + 265, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s49 + s23 :|: s49 >= 0, s49 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2699 + 2787*s + 7*s*z'' + 90*s*z3 + 90*s^2 + 7*z'' + 90*z3 }-> 1 + s50 + s24 :|: s50 >= 0, s50 <= 69 * s + 10 * z3 + 7 * (s * s) + 6 * (z3 * s) + 265, s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s51 + s25 :|: s51 >= 0, s51 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s11 + 90*s11*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s41 + s3 :|: s41 >= 0, s41 <= 71 * (1 + (z - 1)) + 30 * s11 + 6 * (s11 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s42 + s4 :|: s42 >= 0, s42 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s43 + s5 :|: s43 >= 0, s43 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s12 + 14*z' + z'' }-> 1 + s44 + s6 :|: s44 >= 0, s44 <= 71 * 0 + 30 * s12 + 6 * (s12 * 0) + 7 * (0 * 0) + 802, s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' + z'' }-> 1 + s45 + s7 :|: s45 >= 0, s45 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s46 + s8 :|: s46 >= 0, s46 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s13 + 90*s13*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s47 + s9 :|: s47 >= 0, s47 <= 71 * z + 30 * s13 + 6 * (s13 * z) + 7 * (z * z) + 802, s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s48 + s10 :|: s48 >= 0, s48 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s18 + 90*s18*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s53 + 0 :|: s53 >= 0, s53 <= 71 * (1 + (z - 1)) + 30 * s18 + 6 * (s18 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s54 + 0 :|: s54 >= 0, s54 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s55 + 1 :|: s55 >= 0, s55 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 180*s19 + 14*z' }-> 1 + s56 + 1 :|: s56 >= 0, s56 <= 71 * 0 + 30 * s19 + 6 * (s19 * 0) + 7 * (0 * 0) + 802, s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' }-> 1 + s57 + 1 :|: s57 >= 0, s57 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s58 + 1 :|: s58 >= 0, s58 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s59 + 0 :|: s59 >= 0, s59 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s20 + 90*s20*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s60 + 1 :|: s60 >= 0, s60 <= 71 * z + 30 * s20 + 6 * (s20 * z) + 7 * (z * z) + 802, s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s21 + 90*s21*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s61 + 0 :|: s61 >= 0, s61 <= 71 * z + 30 * s21 + 6 * (s21 * z) + 7 * (z * z) + 802, s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s62 + 1 :|: s62 >= 0, s62 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s63 + 0 :|: s63 >= 0, s63 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> s26 :|: s26 >= 0, s26 <= z + 0 + 2, z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s37 :|: s37 >= 0, s37 <= s'' + z3 + 2, s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s38 :|: s38 >= 0, s38 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s27 :|: s27 >= 0, s27 <= 1 + 0 + 1 + (1 + (z - 1)), z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s28 :|: s28 >= 0, s28 <= s15 + 1 + (1 + (z - 1)), s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 0 + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + 0 + 1 + 0, z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= s16 + 1 + 0, s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + 1 + 0, z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= s17 + 1 + z, s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3]
  quot: runtime: O(1) [0], size: O(n^1) [2 + z + z'']
  QUOT: runtime: O(n^2) [2696 + 2787*z + 7*z*z' + 90*z*z'' + 90*z^2 + 6*z' + 90*z''], size: O(n^2) [265 + 69*z + 6*z*z'' + 7*z^2 + 10*z'']
  IF: runtime: O(n^2) [5402 + 2607*z' + 7*z'*z'' + 90*z'*z3 + 90*z'^2 + 14*z'' + 180*z3], size: O(n^2) [802 + 71*z' + 6*z'*z3 + 7*z'^2 + 30*z3]
  div: runtime: O(1) [0], size: O(n^1) [2 + z]

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

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

(86)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 2697 + 2787*z + 7*z*z' + 90*z^2 + 6*z' }-> 1 + s39 :|: s39 >= 0, s39 <= 69 * z + 10 * 0 + 7 * (z * z) + 6 * (0 * z) + 265, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s49 + s23 :|: s49 >= 0, s49 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2699 + 2787*s + 7*s*z'' + 90*s*z3 + 90*s^2 + 7*z'' + 90*z3 }-> 1 + s50 + s24 :|: s50 >= 0, s50 <= 69 * s + 10 * z3 + 7 * (s * s) + 6 * (z3 * s) + 265, s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s51 + s25 :|: s51 >= 0, s51 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s11 + 90*s11*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s41 + s3 :|: s41 >= 0, s41 <= 71 * (1 + (z - 1)) + 30 * s11 + 6 * (s11 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s42 + s4 :|: s42 >= 0, s42 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s43 + s5 :|: s43 >= 0, s43 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s12 + 14*z' + z'' }-> 1 + s44 + s6 :|: s44 >= 0, s44 <= 71 * 0 + 30 * s12 + 6 * (s12 * 0) + 7 * (0 * 0) + 802, s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' + z'' }-> 1 + s45 + s7 :|: s45 >= 0, s45 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s46 + s8 :|: s46 >= 0, s46 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s13 + 90*s13*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s47 + s9 :|: s47 >= 0, s47 <= 71 * z + 30 * s13 + 6 * (s13 * z) + 7 * (z * z) + 802, s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s48 + s10 :|: s48 >= 0, s48 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s18 + 90*s18*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s53 + 0 :|: s53 >= 0, s53 <= 71 * (1 + (z - 1)) + 30 * s18 + 6 * (s18 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s54 + 0 :|: s54 >= 0, s54 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s55 + 1 :|: s55 >= 0, s55 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 180*s19 + 14*z' }-> 1 + s56 + 1 :|: s56 >= 0, s56 <= 71 * 0 + 30 * s19 + 6 * (s19 * 0) + 7 * (0 * 0) + 802, s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' }-> 1 + s57 + 1 :|: s57 >= 0, s57 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s58 + 1 :|: s58 >= 0, s58 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s59 + 0 :|: s59 >= 0, s59 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s20 + 90*s20*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s60 + 1 :|: s60 >= 0, s60 <= 71 * z + 30 * s20 + 6 * (s20 * z) + 7 * (z * z) + 802, s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s21 + 90*s21*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s61 + 0 :|: s61 >= 0, s61 <= 71 * z + 30 * s21 + 6 * (s21 * z) + 7 * (z * z) + 802, s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s62 + 1 :|: s62 >= 0, s62 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s63 + 0 :|: s63 >= 0, s63 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> s26 :|: s26 >= 0, s26 <= z + 0 + 2, z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s37 :|: s37 >= 0, s37 <= s'' + z3 + 2, s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s38 :|: s38 >= 0, s38 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s27 :|: s27 >= 0, s27 <= 1 + 0 + 1 + (1 + (z - 1)), z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s28 :|: s28 >= 0, s28 <= s15 + 1 + (1 + (z - 1)), s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 0 + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + 0 + 1 + 0, z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= s16 + 1 + 0, s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + 1 + 0, z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= s17 + 1 + z, s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3]
  quot: runtime: O(1) [0], size: O(n^1) [2 + z + z'']
  QUOT: runtime: O(n^2) [2696 + 2787*z + 7*z*z' + 90*z*z'' + 90*z^2 + 6*z' + 90*z''], size: O(n^2) [265 + 69*z + 6*z*z'' + 7*z^2 + 10*z'']
  IF: runtime: O(n^2) [5402 + 2607*z' + 7*z'*z'' + 90*z'*z3 + 90*z'^2 + 14*z'' + 180*z3], size: O(n^2) [802 + 71*z' + 6*z'*z3 + 7*z'^2 + 30*z3]
  div: runtime: O(1) [0], size: O(n^1) [2 + z]

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

(87) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(88)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 2697 + 2787*z + 7*z*z' + 90*z^2 + 6*z' }-> 1 + s39 :|: s39 >= 0, s39 <= 69 * z + 10 * 0 + 7 * (z * z) + 6 * (0 * z) + 265, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s49 + s23 :|: s49 >= 0, s49 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2699 + 2787*s + 7*s*z'' + 90*s*z3 + 90*s^2 + 7*z'' + 90*z3 }-> 1 + s50 + s24 :|: s50 >= 0, s50 <= 69 * s + 10 * z3 + 7 * (s * s) + 6 * (z3 * s) + 265, s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s51 + s25 :|: s51 >= 0, s51 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s11 + 90*s11*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s41 + s3 :|: s41 >= 0, s41 <= 71 * (1 + (z - 1)) + 30 * s11 + 6 * (s11 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s42 + s4 :|: s42 >= 0, s42 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s43 + s5 :|: s43 >= 0, s43 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s12 + 14*z' + z'' }-> 1 + s44 + s6 :|: s44 >= 0, s44 <= 71 * 0 + 30 * s12 + 6 * (s12 * 0) + 7 * (0 * 0) + 802, s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' + z'' }-> 1 + s45 + s7 :|: s45 >= 0, s45 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s46 + s8 :|: s46 >= 0, s46 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s13 + 90*s13*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s47 + s9 :|: s47 >= 0, s47 <= 71 * z + 30 * s13 + 6 * (s13 * z) + 7 * (z * z) + 802, s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s48 + s10 :|: s48 >= 0, s48 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s18 + 90*s18*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s53 + 0 :|: s53 >= 0, s53 <= 71 * (1 + (z - 1)) + 30 * s18 + 6 * (s18 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s54 + 0 :|: s54 >= 0, s54 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s55 + 1 :|: s55 >= 0, s55 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 180*s19 + 14*z' }-> 1 + s56 + 1 :|: s56 >= 0, s56 <= 71 * 0 + 30 * s19 + 6 * (s19 * 0) + 7 * (0 * 0) + 802, s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' }-> 1 + s57 + 1 :|: s57 >= 0, s57 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s58 + 1 :|: s58 >= 0, s58 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s59 + 0 :|: s59 >= 0, s59 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s20 + 90*s20*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s60 + 1 :|: s60 >= 0, s60 <= 71 * z + 30 * s20 + 6 * (s20 * z) + 7 * (z * z) + 802, s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s21 + 90*s21*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s61 + 0 :|: s61 >= 0, s61 <= 71 * z + 30 * s21 + 6 * (s21 * z) + 7 * (z * z) + 802, s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s62 + 1 :|: s62 >= 0, s62 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s63 + 0 :|: s63 >= 0, s63 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> s26 :|: s26 >= 0, s26 <= z + 0 + 2, z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s37 :|: s37 >= 0, s37 <= s'' + z3 + 2, s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s38 :|: s38 >= 0, s38 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s27 :|: s27 >= 0, s27 <= 1 + 0 + 1 + (1 + (z - 1)), z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s28 :|: s28 >= 0, s28 <= s15 + 1 + (1 + (z - 1)), s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 0 + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + 0 + 1 + 0, z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= s16 + 1 + 0, s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + 1 + 0, z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= s17 + 1 + z, s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {DIV}
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3]
  quot: runtime: O(1) [0], size: O(n^1) [2 + z + z'']
  QUOT: runtime: O(n^2) [2696 + 2787*z + 7*z*z' + 90*z*z'' + 90*z^2 + 6*z' + 90*z''], size: O(n^2) [265 + 69*z + 6*z*z'' + 7*z^2 + 10*z'']
  IF: runtime: O(n^2) [5402 + 2607*z' + 7*z'*z'' + 90*z'*z3 + 90*z'^2 + 14*z'' + 180*z3], size: O(n^2) [802 + 71*z' + 6*z'*z3 + 7*z'^2 + 30*z3]
  div: runtime: O(1) [0], size: O(n^1) [2 + z]
  DIV: runtime: ?, size: INF

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

(89) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: DIV
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2697 + 2787*z + 7*z*z' + 90*z^2 + 6*z'

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

(90)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 2697 + 2787*z + 7*z*z' + 90*z^2 + 6*z' }-> 1 + s39 :|: s39 >= 0, s39 <= 69 * z + 10 * 0 + 7 * (z * z) + 6 * (0 * z) + 265, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s49 + s23 :|: s49 >= 0, s49 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s23 >= 0, s23 <= 0 + 1, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2699 + 2787*s + 7*s*z'' + 90*s*z3 + 90*s^2 + 7*z'' + 90*z3 }-> 1 + s50 + s24 :|: s50 >= 0, s50 <= 69 * s + 10 * z3 + 7 * (s * s) + 6 * (z3 * s) + 265, s24 >= 0, s24 <= 1 + (z' - 1) + 1, s >= 0, s <= z' - 1, z'' - 1 >= 0, z' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 2699 + 7*z'' + 90*z3 }-> 1 + s51 + s25 :|: s51 >= 0, s51 <= 69 * 0 + 10 * z3 + 7 * (0 * 0) + 6 * (z3 * 0) + 265, s25 >= 0, s25 <= z' + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   MINUS(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   MINUS(z, z') -{ 2 + z' }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   P(z) -{ 1 }-> 0 :|: z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   PLUS(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s2 >= 0, s2 <= 0, z'' = 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s11 + 90*s11*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s41 + s3 :|: s41 >= 0, s41 <= 71 * (1 + (z - 1)) + 30 * s11 + 6 * (s11 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s11 >= 0, s11 <= z'' - 1 + (1 + (1 + 0)), s3 >= 0, s3 <= 1 + (z'' - 1), z'' - 1 >= 0, z' >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s42 + s4 :|: s42 >= 0, s42 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s4 >= 0, s4 <= z'', z' >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s43 + s5 :|: s43 >= 0, s43 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, s5 >= 0, s5 <= 0, z'' = 0, z' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s12 + 14*z' + z'' }-> 1 + s44 + s6 :|: s44 >= 0, s44 <= 71 * 0 + 30 * s12 + 6 * (s12 * 0) + 7 * (0 * 0) + 802, s12 >= 0, s12 <= z'' - 1 + (1 + (1 + 0)), s6 >= 0, s6 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' + z'' }-> 1 + s45 + s7 :|: s45 >= 0, s45 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, s7 >= 0, s7 <= z'', z' >= 0, z = 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s46 + s8 :|: s46 >= 0, s46 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, s8 >= 0, s8 <= 0, z'' = 0, z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s13 + 90*s13*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s47 + s9 :|: s47 >= 0, s47 <= 71 * z + 30 * s13 + 6 * (s13 * z) + 7 * (z * z) + 802, s13 >= 0, s13 <= z'' - 1 + (1 + (1 + 0)), s9 >= 0, s9 <= 1 + (z'' - 1), z' >= 0, z'' - 1 >= 0, z >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' + z'' }-> 1 + s48 + s10 :|: s48 >= 0, s48 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, s10 >= 0, s10 <= z'', z' >= 0, z >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 71 * (1 + (z - 1)) + 30 * (1 + 0) + 6 * ((1 + 0) * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z'' = 0, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s18 + 90*s18*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s53 + 0 :|: s53 >= 0, s53 <= 71 * (1 + (z - 1)) + 30 * s18 + 6 * (s18 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, s18 >= 0, s18 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s54 + 0 :|: s54 >= 0, s54 <= 71 * (1 + (z - 1)) + 30 * 0 + 6 * (0 * (1 + (z - 1))) + 7 * ((1 + (z - 1)) * (1 + (z - 1))) + 802, z' >= 0, z - 1 >= 0, z'' >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0
   QUOT(z, z', z'') -{ 5584 + 14*z' }-> 1 + s55 + 1 :|: s55 >= 0, s55 <= 71 * 0 + 30 * (1 + 0) + 6 * ((1 + 0) * 0) + 7 * (0 * 0) + 802, z'' = 0, z' >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 180*s19 + 14*z' }-> 1 + s56 + 1 :|: s56 >= 0, s56 <= 71 * 0 + 30 * s19 + 6 * (s19 * 0) + 7 * (0 * 0) + 802, s19 >= 0, s19 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z = 0, 0 = 0
   QUOT(z, z', z'') -{ 5404 + 14*z' }-> 1 + s57 + 1 :|: s57 >= 0, s57 <= 71 * 0 + 30 * 0 + 6 * (0 * 0) + 7 * (0 * 0) + 802, z' >= 0, z = 0, z'' >= 0, 0 = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s58 + 1 :|: s58 >= 0, s58 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5584 + 2697*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s59 + 0 :|: s59 >= 0, s59 <= 71 * z + 30 * (1 + 0) + 6 * ((1 + 0) * z) + 7 * (z * z) + 802, z'' = 0, z' >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 180*s20 + 90*s20*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s60 + 1 :|: s60 >= 0, s60 <= 71 * z + 30 * s20 + 6 * (s20 * z) + 7 * (z * z) + 802, s20 >= 0, s20 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 180*s21 + 90*s21*z + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s61 + 0 :|: s61 >= 0, s61 <= 71 * z + 30 * s21 + 6 * (s21 * z) + 7 * (z * z) + 802, s21 >= 0, s21 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0, z - 1 >= 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s62 + 1 :|: s62 >= 0, s62 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z = 0
   QUOT(z, z', z'') -{ 5404 + 2607*z + 7*z*z' + 90*z^2 + 14*z' }-> 1 + s63 + 0 :|: s63 >= 0, s63 <= 71 * z + 30 * 0 + 6 * (0 * z) + 7 * (z * z) + 802, z' >= 0, z >= 0, z'' >= 0, z - 1 >= 0
   ZERO(z) -{ 1 }-> 1 :|: z = 0
   ZERO(z) -{ 1 }-> 0 :|: z - 1 >= 0
   div(z, z') -{ 0 }-> s26 :|: s26 >= 0, s26 <= z + 0 + 2, z' >= 0, z >= 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'', z3) -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s37 :|: s37 >= 0, s37 <= s'' + z3 + 2, s'' >= 0, s'' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s38 :|: s38 >= 0, s38 <= 0 + z3 + 2, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z3 - 1 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0, z3 - 1 >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= z - 1 + (1 + z'), z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s27 :|: s27 >= 0, s27 <= 1 + 0 + 1 + (1 + (z - 1)), z'' = 0, z - 1 >= 0, z' >= 0
   quot(z, z', z'') -{ 0 }-> s28 :|: s28 >= 0, s28 <= s15 + 1 + (1 + (z - 1)), s15 >= 0, s15 <= z'' - 1 + (1 + (1 + 0)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 0 + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + 0 + 1 + 0, z'' = 0, z' >= 0, z = 0
   quot(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= s16 + 1 + 0, s16 >= 0, s16 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z = 0, z'' - 1 >= 0
   quot(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + 1 + 0, z' >= 0, z = 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= s17 + 1 + z, s17 >= 0, s17 <= z'' - 1 + (1 + (1 + 0)), z' >= 0, z'' - 1 >= 0, z >= 0
   quot(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0
   quot(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   zero(z) -{ 0 }-> 2 :|: z = 0
   zero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   zero(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  P: runtime: O(1) [1], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  ZERO: runtime: O(1) [1], size: O(1) [1]
  PLUS: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MINUS: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z]
  if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3]
  quot: runtime: O(1) [0], size: O(n^1) [2 + z + z'']
  QUOT: runtime: O(n^2) [2696 + 2787*z + 7*z*z' + 90*z*z'' + 90*z^2 + 6*z' + 90*z''], size: O(n^2) [265 + 69*z + 6*z*z'' + 7*z^2 + 10*z'']
  IF: runtime: O(n^2) [5402 + 2607*z' + 7*z'*z'' + 90*z'*z3 + 90*z'^2 + 14*z'' + 180*z3], size: O(n^2) [802 + 71*z' + 6*z'*z3 + 7*z'^2 + 30*z3]
  div: runtime: O(1) [0], size: O(n^1) [2 + z]
  DIV: runtime: O(n^2) [2697 + 2787*z + 7*z*z' + 90*z^2 + 6*z'], size: INF

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

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

(92)
BOUNDS(1, n^2)

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

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

(94)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MINUS(0', z0) -> c
   MINUS(z0, 0') -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   PLUS(0', z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
   ZERO(s(z0)) -> c5
   ZERO(0') -> c6
   P(s(z0)) -> c7
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2, s(0')))
   IF(true, z0, z1, z2) -> c11(P(z2))
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))

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

   minus(0', z0) -> 0'
   minus(z0, 0') -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   plus(0', z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   zero(s(z0)) -> false
   zero(0') -> true
   p(s(z0)) -> z0
   div(z0, z1) -> quot(z0, z1, 0')
   quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
   if(true, z0, z1, z2) -> p(z2)
   if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

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

(95) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
PLUS/1

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

(96)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MINUS(0', z0) -> c
   MINUS(z0, 0') -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   PLUS(0') -> c3
   PLUS(s(z0)) -> c4(PLUS(z0))
   ZERO(s(z0)) -> c5
   ZERO(0') -> c6
   P(s(z0)) -> c7
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2))
   IF(true, z0, z1, z2) -> c11(P(z2))
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))

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

   minus(0', z0) -> 0'
   minus(z0, 0') -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   plus(0', z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   zero(s(z0)) -> false
   zero(0') -> true
   p(s(z0)) -> z0
   div(z0, z1) -> quot(z0, z1, 0')
   quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
   if(true, z0, z1, z2) -> p(z2)
   if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

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

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

(98)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0') -> c3
PLUS(s(z0)) -> c4(PLUS(z0))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4

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

(99) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, PLUS, QUOT, plus, minus, quot

They will be analysed ascendingly in the following order:
MINUS < QUOT
PLUS < QUOT
plus < QUOT
minus < QUOT
plus < quot
minus < quot

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

(100)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0') -> c3
PLUS(s(z0)) -> c4(PLUS(z0))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
MINUS, PLUS, QUOT, plus, minus, quot

They will be analysed ascendingly in the following order:
MINUS < QUOT
PLUS < QUOT
plus < QUOT
minus < QUOT
plus < quot
minus < quot

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

(101) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13)) -> gen_c:c1:c210_13(n14_13), rt in Omega(1 + n14_13)

Induction Base:
MINUS(gen_0':s11_13(0), gen_0':s11_13(0)) ->_R^Omega(1)
c

Induction Step:
MINUS(gen_0':s11_13(+(n14_13, 1)), gen_0':s11_13(+(n14_13, 1))) ->_R^Omega(1)
c2(MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13))) ->_IH
c2(gen_c:c1:c210_13(c15_13))

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

(102)
Complex Obligation (BEST)

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

(103)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0') -> c3
PLUS(s(z0)) -> c4(PLUS(z0))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
MINUS, PLUS, QUOT, plus, minus, quot

They will be analysed ascendingly in the following order:
MINUS < QUOT
PLUS < QUOT
plus < QUOT
minus < QUOT
plus < quot
minus < quot

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

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

(105)
BOUNDS(n^1, INF)

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

(106)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0') -> c3
PLUS(s(z0)) -> c4(PLUS(z0))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Lemmas:
MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13)) -> gen_c:c1:c210_13(n14_13), rt in Omega(1 + n14_13)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
PLUS, QUOT, plus, minus, quot

They will be analysed ascendingly in the following order:
PLUS < QUOT
plus < QUOT
minus < QUOT
plus < quot
minus < quot

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

(107) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s11_13(n491_13)) -> gen_c3:c412_13(n491_13), rt in Omega(1 + n491_13)

Induction Base:
PLUS(gen_0':s11_13(0)) ->_R^Omega(1)
c3

Induction Step:
PLUS(gen_0':s11_13(+(n491_13, 1))) ->_R^Omega(1)
c4(PLUS(gen_0':s11_13(n491_13))) ->_IH
c4(gen_c3:c412_13(c492_13))

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

(108)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0') -> c3
PLUS(s(z0)) -> c4(PLUS(z0))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Lemmas:
MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13)) -> gen_c:c1:c210_13(n14_13), rt in Omega(1 + n14_13)
PLUS(gen_0':s11_13(n491_13)) -> gen_c3:c412_13(n491_13), rt in Omega(1 + n491_13)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
plus, QUOT, minus, quot

They will be analysed ascendingly in the following order:
plus < QUOT
minus < QUOT
plus < quot
minus < quot

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

(109) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s11_13(n857_13), gen_0':s11_13(b)) -> gen_0':s11_13(+(n857_13, b)), rt in Omega(0)

Induction Base:
plus(gen_0':s11_13(0), gen_0':s11_13(b)) ->_R^Omega(0)
gen_0':s11_13(b)

Induction Step:
plus(gen_0':s11_13(+(n857_13, 1)), gen_0':s11_13(b)) ->_R^Omega(0)
plus(gen_0':s11_13(n857_13), s(gen_0':s11_13(b))) ->_IH
gen_0':s11_13(+(+(b, 1), c858_13))

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

(110)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0') -> c3
PLUS(s(z0)) -> c4(PLUS(z0))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Lemmas:
MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13)) -> gen_c:c1:c210_13(n14_13), rt in Omega(1 + n14_13)
PLUS(gen_0':s11_13(n491_13)) -> gen_c3:c412_13(n491_13), rt in Omega(1 + n491_13)
plus(gen_0':s11_13(n857_13), gen_0':s11_13(b)) -> gen_0':s11_13(+(n857_13, b)), rt in Omega(0)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
minus, QUOT, quot

They will be analysed ascendingly in the following order:
minus < QUOT
minus < quot

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

(111) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s11_13(n2079_13), gen_0':s11_13(n2079_13)) -> gen_0':s11_13(0), rt in Omega(0)

Induction Base:
minus(gen_0':s11_13(0), gen_0':s11_13(0)) ->_R^Omega(0)
0'

Induction Step:
minus(gen_0':s11_13(+(n2079_13, 1)), gen_0':s11_13(+(n2079_13, 1))) ->_R^Omega(0)
minus(gen_0':s11_13(n2079_13), gen_0':s11_13(n2079_13)) ->_IH
gen_0':s11_13(0)

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

(112)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0') -> c3
PLUS(s(z0)) -> c4(PLUS(z0))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Lemmas:
MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13)) -> gen_c:c1:c210_13(n14_13), rt in Omega(1 + n14_13)
PLUS(gen_0':s11_13(n491_13)) -> gen_c3:c412_13(n491_13), rt in Omega(1 + n491_13)
plus(gen_0':s11_13(n857_13), gen_0':s11_13(b)) -> gen_0':s11_13(+(n857_13, b)), rt in Omega(0)
minus(gen_0':s11_13(n2079_13), gen_0':s11_13(n2079_13)) -> gen_0':s11_13(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
QUOT, quot