WORST_CASE(Omega(n^1),O(n^2))
proof of input_LIUeIBo1SG.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRNTS
    (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 262 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 64 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 281 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 180 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 76 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 368 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 23 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 198 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
    (58) CpxRNTS
    (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 5941 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 4261 ms]
    (64) CpxRNTS
    (65) FinalProof [FINISHED, 0 ms]
    (66) BOUNDS(1, n^2)
(67) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(68) CdtProblem
    (69) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CpxRelTRS
    (71) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CpxRelTRS
    (73) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) typed CpxTrs
    (75) OrderProof [LOWER BOUND(ID), 12 ms]
    (76) typed CpxTrs
    (77) RewriteLemmaProof [LOWER BOUND(ID), 308 ms]
    (78) typed CpxTrs
    (79) RewriteLemmaProof [LOWER BOUND(ID), 46 ms]
    (80) typed CpxTrs
    (81) RewriteLemmaProof [LOWER BOUND(ID), 106 ms]
    (82) BEST
        (83) proven lower bound
            (84) LowerBoundPropagationProof [FINISHED, 0 ms]
            (85) BOUNDS(n^1, INF)
        (86) typed CpxTrs
            (87) RewriteLemmaProof [LOWER BOUND(ID), 62 ms]
            (88) typed CpxTrs
            (89) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
            (90) typed CpxTrs
            (91) RewriteLemmaProof [LOWER BOUND(ID), 56 ms]
            (92) typed CpxTrs


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

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


The TRS R consists of the following rules:

   division(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(lt(x, y), x, y, inc(z))
   if(true, x, y, z) -> z
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(x))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   division(z0, z1) -> div(z0, z1, 0)
   div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
   if(true, z0, z1, z2) -> z2
   if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   inc(0) -> s(0)
   inc(s(z0)) -> s(inc(z0))
Tuples:
   DIVISION(z0, z1) -> c(DIV(z0, z1, 0))
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(z0, 0) -> c5
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(z0, 0) -> c7
   LT(0, s(z0)) -> c8
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
S tuples:
   DIVISION(z0, z1) -> c(DIV(z0, z1, 0))
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(z0, 0) -> c5
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(z0, 0) -> c7
   LT(0, s(z0)) -> c8
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
K tuples:none
Defined Rule Symbols:   division_2, div_3, if_4, minus_2, lt_2, inc_1

Defined Pair Symbols:   DIVISION_2, DIV_3, IF_4, MINUS_2, LT_2, INC_1

Compound Symbols:   c_1, c1_2, c2_2, c3, c4_2, c5, c6_1, c7, c8, c9_1, c10, c11_1


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   DIVISION(z0, z1) -> c(DIV(z0, z1, 0))
Removed 5 trailing nodes:
   LT(z0, 0) -> c7
   MINUS(z0, 0) -> c5
   LT(0, s(z0)) -> c8
   INC(0) -> c10
   IF(true, z0, z1, z2) -> c3

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   division(z0, z1) -> div(z0, z1, 0)
   div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
   if(true, z0, z1, z2) -> z2
   if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   inc(0) -> s(0)
   inc(s(z0)) -> s(inc(z0))
Tuples:
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(s(z0)) -> c11(INC(z0))
S tuples:
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(s(z0)) -> c11(INC(z0))
K tuples:none
Defined Rule Symbols:   division_2, div_3, if_4, minus_2, lt_2, inc_1

Defined Pair Symbols:   DIV_3, IF_4, MINUS_2, LT_2, INC_1

Compound Symbols:   c1_2, c2_2, c4_2, c6_1, c9_1, c11_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   division(z0, z1) -> div(z0, z1, 0)
   div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
   if(true, z0, z1, z2) -> z2
   if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   inc(0) -> s(0)
   inc(s(z0)) -> s(inc(z0))
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(z0, 0) -> z0
Tuples:
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(s(z0)) -> c11(INC(z0))
S tuples:
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(s(z0)) -> c11(INC(z0))
K tuples:none
Defined Rule Symbols:   lt_2, inc_1, minus_2

Defined Pair Symbols:   DIV_3, IF_4, MINUS_2, LT_2, INC_1

Compound Symbols:   c1_2, c2_2, c4_2, c6_1, c9_1, c11_1


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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(s(z0)) -> c11(INC(z0))

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

   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   inc(0) -> s(0)
   inc(s(z0)) -> s(inc(z0))
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(z0, 0) -> z0

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

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

(10)
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:

   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) [1]
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) [1]
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) [1]
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) [1]
   LT(s(z0), s(z1)) -> c9(LT(z0, z1)) [1]
   INC(s(z0)) -> c11(INC(z0)) [1]
   lt(z0, 0) -> false [0]
   lt(0, s(z0)) -> true [0]
   lt(s(z0), s(z1)) -> lt(z0, z1) [0]
   inc(0) -> s(0) [0]
   inc(s(z0)) -> s(inc(z0)) [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) [1]
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) [1]
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) [1]
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) [1]
   LT(s(z0), s(z1)) -> c9(LT(z0, z1)) [1]
   INC(s(z0)) -> c11(INC(z0)) [1]
   lt(z0, 0) -> false [0]
   lt(0, s(z0)) -> true [0]
   lt(s(z0), s(z1)) -> lt(z0, z1) [0]
   inc(0) -> s(0) [0]
   inc(s(z0)) -> s(inc(z0)) [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]

The TRS has the following type information:
DIV :: s:0 -> s:0 -> s:0 -> c1:c2
c1 :: c4 -> c9 -> c1:c2
IF :: false:true -> s:0 -> s:0 -> s:0 -> c4
lt :: s:0 -> s:0 -> false:true
inc :: s:0 -> s:0
LT :: s:0 -> s:0 -> c9
c2 :: c4 -> c11 -> c1:c2
INC :: s:0 -> c11
false :: false:true
s :: s:0 -> s:0
c4 :: c1:c2 -> c6 -> c4
minus :: s:0 -> s:0 -> s:0
MINUS :: s:0 -> s:0 -> c6
c6 :: c6 -> c6
c9 :: c9 -> c9
c11 :: c11 -> c11
0 :: s:0
true :: false:true

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

(13) 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:

   DIV_3
   IF_4
   MINUS_2
   LT_2
   INC_1

(c) The following functions are completely defined:

   lt_2
   inc_1
   minus_2

Due to the following rules being added:

   lt(v0, v1) -> null_lt [0]
   inc(v0) -> 0 [0]
   minus(v0, v1) -> 0 [0]

And the following fresh constants:    null_lt, const, const1, const2, const3, const4

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

(14)
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:

   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) [1]
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) [1]
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) [1]
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) [1]
   LT(s(z0), s(z1)) -> c9(LT(z0, z1)) [1]
   INC(s(z0)) -> c11(INC(z0)) [1]
   lt(z0, 0) -> false [0]
   lt(0, s(z0)) -> true [0]
   lt(s(z0), s(z1)) -> lt(z0, z1) [0]
   inc(0) -> s(0) [0]
   inc(s(z0)) -> s(inc(z0)) [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]
   lt(v0, v1) -> null_lt [0]
   inc(v0) -> 0 [0]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
DIV :: s:0 -> s:0 -> s:0 -> c1:c2
c1 :: c4 -> c9 -> c1:c2
IF :: false:true:null_lt -> s:0 -> s:0 -> s:0 -> c4
lt :: s:0 -> s:0 -> false:true:null_lt
inc :: s:0 -> s:0
LT :: s:0 -> s:0 -> c9
c2 :: c4 -> c11 -> c1:c2
INC :: s:0 -> c11
false :: false:true:null_lt
s :: s:0 -> s:0
c4 :: c1:c2 -> c6 -> c4
minus :: s:0 -> s:0 -> s:0
MINUS :: s:0 -> s:0 -> c6
c6 :: c6 -> c6
c9 :: c9 -> c9
c11 :: c11 -> c11
0 :: s:0
true :: false:true:null_lt
null_lt :: false:true:null_lt
const :: c1:c2
const1 :: c4
const2 :: c9
const3 :: c11
const4 :: c6

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

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

(16)
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:

   DIV(z0, 0, 0) -> c1(IF(false, z0, 0, s(0)), LT(z0, 0)) [1]
   DIV(z0, 0, s(z01)) -> c1(IF(false, z0, 0, s(inc(z01))), LT(z0, 0)) [1]
   DIV(z0, 0, z2) -> c1(IF(false, z0, 0, 0), LT(z0, 0)) [1]
   DIV(0, s(z0'), 0) -> c1(IF(true, 0, s(z0'), s(0)), LT(0, s(z0'))) [1]
   DIV(0, s(z0'), s(z02)) -> c1(IF(true, 0, s(z0'), s(inc(z02))), LT(0, s(z0'))) [1]
   DIV(0, s(z0'), z2) -> c1(IF(true, 0, s(z0'), 0), LT(0, s(z0'))) [1]
   DIV(s(z0''), s(z1'), 0) -> c1(IF(lt(z0'', z1'), s(z0''), s(z1'), s(0)), LT(s(z0''), s(z1'))) [1]
   DIV(s(z0''), s(z1'), s(z03)) -> c1(IF(lt(z0'', z1'), s(z0''), s(z1'), s(inc(z03))), LT(s(z0''), s(z1'))) [1]
   DIV(s(z0''), s(z1'), z2) -> c1(IF(lt(z0'', z1'), s(z0''), s(z1'), 0), LT(s(z0''), s(z1'))) [1]
   DIV(z0, z1, 0) -> c1(IF(null_lt, z0, z1, s(0)), LT(z0, z1)) [1]
   DIV(z0, z1, s(z04)) -> c1(IF(null_lt, z0, z1, s(inc(z04))), LT(z0, z1)) [1]
   DIV(z0, z1, z2) -> c1(IF(null_lt, z0, z1, 0), LT(z0, z1)) [1]
   DIV(z0, 0, 0) -> c2(IF(false, z0, 0, s(0)), INC(0)) [1]
   DIV(z0, 0, s(z07)) -> c2(IF(false, z0, 0, s(inc(z07))), INC(s(z07))) [1]
   DIV(z0, 0, z2) -> c2(IF(false, z0, 0, 0), INC(z2)) [1]
   DIV(0, s(z05), 0) -> c2(IF(true, 0, s(z05), s(0)), INC(0)) [1]
   DIV(0, s(z05), s(z08)) -> c2(IF(true, 0, s(z05), s(inc(z08))), INC(s(z08))) [1]
   DIV(0, s(z05), z2) -> c2(IF(true, 0, s(z05), 0), INC(z2)) [1]
   DIV(s(z06), s(z1''), 0) -> c2(IF(lt(z06, z1''), s(z06), s(z1''), s(0)), INC(0)) [1]
   DIV(s(z06), s(z1''), s(z09)) -> c2(IF(lt(z06, z1''), s(z06), s(z1''), s(inc(z09))), INC(s(z09))) [1]
   DIV(s(z06), s(z1''), z2) -> c2(IF(lt(z06, z1''), s(z06), s(z1''), 0), INC(z2)) [1]
   DIV(z0, z1, 0) -> c2(IF(null_lt, z0, z1, s(0)), INC(0)) [1]
   DIV(z0, z1, s(z010)) -> c2(IF(null_lt, z0, z1, s(inc(z010))), INC(s(z010))) [1]
   DIV(z0, z1, z2) -> c2(IF(null_lt, z0, z1, 0), INC(z2)) [1]
   IF(false, s(z011), s(z1), z2) -> c4(DIV(minus(z011, z1), s(z1), z2), MINUS(s(z011), s(z1))) [1]
   IF(false, z0, s(z1), z2) -> c4(DIV(0, s(z1), z2), MINUS(z0, s(z1))) [1]
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) [1]
   LT(s(z0), s(z1)) -> c9(LT(z0, z1)) [1]
   INC(s(z0)) -> c11(INC(z0)) [1]
   lt(z0, 0) -> false [0]
   lt(0, s(z0)) -> true [0]
   lt(s(z0), s(z1)) -> lt(z0, z1) [0]
   inc(0) -> s(0) [0]
   inc(s(z0)) -> s(inc(z0)) [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]
   lt(v0, v1) -> null_lt [0]
   inc(v0) -> 0 [0]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
DIV :: s:0 -> s:0 -> s:0 -> c1:c2
c1 :: c4 -> c9 -> c1:c2
IF :: false:true:null_lt -> s:0 -> s:0 -> s:0 -> c4
lt :: s:0 -> s:0 -> false:true:null_lt
inc :: s:0 -> s:0
LT :: s:0 -> s:0 -> c9
c2 :: c4 -> c11 -> c1:c2
INC :: s:0 -> c11
false :: false:true:null_lt
s :: s:0 -> s:0
c4 :: c1:c2 -> c6 -> c4
minus :: s:0 -> s:0 -> s:0
MINUS :: s:0 -> s:0 -> c6
c6 :: c6 -> c6
c9 :: c9 -> c9
c11 :: c11 -> c11
0 :: s:0
true :: false:true:null_lt
null_lt :: false:true:null_lt
const :: c1:c2
const1 :: c4
const2 :: c9
const3 :: c11
const4 :: c6

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

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

   false => 1
   0 => 0
   true => 2
   null_lt => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z0'', z1'), 1 + z0'', 1 + z1', 0) + LT(1 + z0'', 1 + z1') :|: z'' = z2, z' = 1 + z1', z1' >= 0, z0'' >= 0, z2 >= 0, z = 1 + z0''
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z0'', z1'), 1 + z0'', 1 + z1', 1 + inc(z03)) + LT(1 + z0'', 1 + z1') :|: z'' = 1 + z03, z' = 1 + z1', z1' >= 0, z0'' >= 0, z03 >= 0, z = 1 + z0''
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z0'', z1'), 1 + z0'', 1 + z1', 1 + 0) + LT(1 + z0'', 1 + z1') :|: z'' = 0, z' = 1 + z1', z1' >= 0, z0'' >= 0, z = 1 + z0''
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z06, z1''), 1 + z06, 1 + z1'', 0) + INC(z2) :|: z'' = z2, z' = 1 + z1'', z06 >= 0, z2 >= 0, z = 1 + z06, z1'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z06, z1''), 1 + z06, 1 + z1'', 1 + inc(z09)) + INC(1 + z09) :|: z' = 1 + z1'', z'' = 1 + z09, z06 >= 0, z09 >= 0, z = 1 + z06, z1'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z06, z1''), 1 + z06, 1 + z1'', 1 + 0) + INC(0) :|: z'' = 0, z' = 1 + z1'', z06 >= 0, z = 1 + z06, z1'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z0', 0) + LT(0, 1 + z0') :|: z'' = z2, z0' >= 0, z = 0, z' = 1 + z0', z2 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z0', 1 + inc(z02)) + LT(0, 1 + z0') :|: z0' >= 0, z02 >= 0, z = 0, z' = 1 + z0', z'' = 1 + z02
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z0', 1 + 0) + LT(0, 1 + z0') :|: z'' = 0, z0' >= 0, z = 0, z' = 1 + z0'
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z05, 0) + INC(z2) :|: z'' = z2, z05 >= 0, z = 0, z' = 1 + z05, z2 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z05, 1 + inc(z08)) + INC(1 + z08) :|: z08 >= 0, z'' = 1 + z08, z05 >= 0, z = 0, z' = 1 + z05
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z05, 1 + 0) + INC(0) :|: z'' = 0, z05 >= 0, z = 0, z' = 1 + z05
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 0) + LT(z0, 0) :|: z'' = z2, z = z0, z0 >= 0, z2 >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 0) + INC(z2) :|: z'' = z2, z = z0, z0 >= 0, z2 >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 1 + inc(z01)) + LT(z0, 0) :|: z = z0, z01 >= 0, z0 >= 0, z' = 0, z'' = 1 + z01
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 1 + inc(z07)) + INC(1 + z07) :|: z = z0, z07 >= 0, z'' = 1 + z07, z0 >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 1 + 0) + LT(z0, 0) :|: z'' = 0, z = z0, z0 >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 1 + 0) + INC(0) :|: z'' = 0, z = z0, z0 >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 0) + LT(z0, z1) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 0) + INC(z2) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + inc(z010)) + INC(1 + z010) :|: z = z0, z1 >= 0, z'' = 1 + z010, z' = z1, z0 >= 0, z010 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + inc(z04)) + LT(z0, z1) :|: z'' = 1 + z04, z04 >= 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + 0) + LT(z0, z1) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + 0) + INC(0) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(minus(z011, z1), 1 + z1, z2) + MINUS(1 + z011, 1 + z1) :|: z011 >= 0, z1 >= 0, z = 1, z' = 1 + z011, z'' = 1 + z1, z3 = z2, z2 >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + z1, z2) + MINUS(z0, 1 + z1) :|: z1 >= 0, z = 1, z0 >= 0, z'' = 1 + z1, z3 = z2, z' = z0, z2 >= 0
   INC(z) -{ 1 }-> 1 + INC(z0) :|: z = 1 + z0, z0 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   inc(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   inc(z) -{ 0 }-> 1 + inc(z0) :|: z = 1 + z0, z0 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   lt(z, z') -{ 0 }-> 2 :|: z0 >= 0, z' = 1 + z0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z = z0, z0 >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   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 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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


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

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

   { minus }
   { inc }
   { LT }
   { INC }
   { lt }
   { MINUS }
   { IF, DIV }

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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

Function symbols to be analyzed: {minus}, {inc}, {LT}, {INC}, {lt}, {MINUS}, {IF,DIV}

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

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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

Function symbols to be analyzed: {minus}, {inc}, {LT}, {INC}, {lt}, {MINUS}, {IF,DIV}

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

(25) 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

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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

Function symbols to be analyzed: {minus}, {inc}, {LT}, {INC}, {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: ?, size: O(n^1) [z]

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

(27) 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

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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

Function symbols to be analyzed: {inc}, {LT}, {INC}, {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]

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

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {inc}, {LT}, {INC}, {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {inc}, {LT}, {INC}, {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: ?, size: O(n^1) [1 + z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {LT}, {INC}, {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]

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

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s2) + LT(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + LT(0, 1 + (z' - 1)) :|: s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + LT(z, 0) :|: s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s4) + INC(1 + (z'' - 1)) :|: s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s3) + LT(z, z') :|: s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {LT}, {INC}, {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s2) + LT(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + LT(0, 1 + (z' - 1)) :|: s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + LT(z, 0) :|: s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s4) + INC(1 + (z'' - 1)) :|: s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s3) + LT(z, z') :|: s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {LT}, {INC}, {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: ?, size: O(1) [0]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: LT
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', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s2) + LT(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + LT(0, 1 + (z' - 1)) :|: s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + LT(z, 0) :|: s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s4) + INC(1 + (z'' - 1)) :|: s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s3) + LT(z, z') :|: s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {INC}, {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]

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

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s4) + INC(1 + (z'' - 1)) :|: s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {INC}, {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s4) + INC(1 + (z'' - 1)) :|: s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {INC}, {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: ?, size: O(1) [0]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s4) + INC(1 + (z'' - 1)) :|: s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: O(n^1) [z], size: O(1) [0]

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

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s30 :|: s30 >= 0, s30 <= 0, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s6) + s29 :|: s29 >= 0, s29 <= 0, s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s27 :|: s27 >= 0, s27 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + s26 :|: s26 >= 0, s26 <= 0, s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s25 :|: s25 >= 0, s25 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 0) + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 1 + s4) + s23 :|: s23 >= 0, s23 <= 0, s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s22 :|: s22 >= 0, s22 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 0) + s33 :|: s33 >= 0, s33 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 1 + s7) + s32 :|: s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + s31 :|: s31 >= 0, s31 <= 0, z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ z }-> 1 + s34 :|: s34 >= 0, s34 <= 0, z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: O(n^1) [z], size: O(1) [0]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s30 :|: s30 >= 0, s30 <= 0, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s6) + s29 :|: s29 >= 0, s29 <= 0, s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s27 :|: s27 >= 0, s27 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + s26 :|: s26 >= 0, s26 <= 0, s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s25 :|: s25 >= 0, s25 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 0) + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 1 + s4) + s23 :|: s23 >= 0, s23 <= 0, s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s22 :|: s22 >= 0, s22 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 0) + s33 :|: s33 >= 0, s33 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 1 + s7) + s32 :|: s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + s31 :|: s31 >= 0, s31 <= 0, z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ z }-> 1 + s34 :|: s34 >= 0, s34 <= 0, z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: O(n^1) [z], size: O(1) [0]
  lt: runtime: ?, size: O(1) [2]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s30 :|: s30 >= 0, s30 <= 0, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s6) + s29 :|: s29 >= 0, s29 <= 0, s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s27 :|: s27 >= 0, s27 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + s26 :|: s26 >= 0, s26 <= 0, s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s25 :|: s25 >= 0, s25 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 0) + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 1 + s4) + s23 :|: s23 >= 0, s23 <= 0, s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s22 :|: s22 >= 0, s22 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 0) + s33 :|: s33 >= 0, s33 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 1 + s7) + s32 :|: s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + s31 :|: s31 >= 0, s31 <= 0, z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ z }-> 1 + s34 :|: s34 >= 0, s34 <= 0, z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: O(n^1) [z], size: O(1) [0]
  lt: runtime: O(1) [0], size: O(1) [2]

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

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s35, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s35 >= 0, s35 <= 2, s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s36, 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s36 >= 0, s36 <= 2, s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s37, 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s37 >= 0, s37 <= 2, s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(s38, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s28 :|: s38 >= 0, s38 <= 2, s28 >= 0, s28 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s39, 1 + (z - 1), 1 + (z' - 1), 1 + s6) + s29 :|: s39 >= 0, s39 <= 2, s29 >= 0, s29 <= 0, s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 0) + s30 :|: s40 >= 0, s40 <= 2, s30 >= 0, s30 <= 0, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s27 :|: s27 >= 0, s27 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + s26 :|: s26 >= 0, s26 <= 0, s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s25 :|: s25 >= 0, s25 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 0) + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 1 + s4) + s23 :|: s23 >= 0, s23 <= 0, s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s22 :|: s22 >= 0, s22 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 0) + s33 :|: s33 >= 0, s33 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 1 + s7) + s32 :|: s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + s31 :|: s31 >= 0, s31 <= 0, z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ z }-> 1 + s34 :|: s34 >= 0, s34 <= 0, z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 2, z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: O(n^1) [z], size: O(1) [0]
  lt: runtime: O(1) [0], size: O(1) [2]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s35, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s35 >= 0, s35 <= 2, s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s36, 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s36 >= 0, s36 <= 2, s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s37, 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s37 >= 0, s37 <= 2, s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(s38, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s28 :|: s38 >= 0, s38 <= 2, s28 >= 0, s28 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s39, 1 + (z - 1), 1 + (z' - 1), 1 + s6) + s29 :|: s39 >= 0, s39 <= 2, s29 >= 0, s29 <= 0, s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 0) + s30 :|: s40 >= 0, s40 <= 2, s30 >= 0, s30 <= 0, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s27 :|: s27 >= 0, s27 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + s26 :|: s26 >= 0, s26 <= 0, s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s25 :|: s25 >= 0, s25 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 0) + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 1 + s4) + s23 :|: s23 >= 0, s23 <= 0, s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s22 :|: s22 >= 0, s22 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 0) + s33 :|: s33 >= 0, s33 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 1 + s7) + s32 :|: s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + s31 :|: s31 >= 0, s31 <= 0, z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ z }-> 1 + s34 :|: s34 >= 0, s34 <= 0, z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 2, z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {MINUS}, {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: O(n^1) [z], size: O(1) [0]
  lt: runtime: O(1) [0], size: O(1) [2]
  MINUS: runtime: ?, size: O(1) [0]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: MINUS
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', z'') -{ 1 + z' }-> 1 + IF(s35, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s35 >= 0, s35 <= 2, s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s36, 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s36 >= 0, s36 <= 2, s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s37, 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s37 >= 0, s37 <= 2, s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(s38, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s28 :|: s38 >= 0, s38 <= 2, s28 >= 0, s28 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s39, 1 + (z - 1), 1 + (z' - 1), 1 + s6) + s29 :|: s39 >= 0, s39 <= 2, s29 >= 0, s29 <= 0, s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 0) + s30 :|: s40 >= 0, s40 <= 2, s30 >= 0, s30 <= 0, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s27 :|: s27 >= 0, s27 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + s26 :|: s26 >= 0, s26 <= 0, s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s25 :|: s25 >= 0, s25 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 0) + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 1 + s4) + s23 :|: s23 >= 0, s23 <= 0, s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s22 :|: s22 >= 0, s22 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 0) + s33 :|: s33 >= 0, s33 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 1 + s7) + s32 :|: s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + s31 :|: s31 >= 0, s31 <= 0, z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(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 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ z }-> 1 + s34 :|: s34 >= 0, s34 <= 0, z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 2, z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: O(n^1) [z], size: O(1) [0]
  lt: runtime: O(1) [0], size: O(1) [2]
  MINUS: runtime: O(n^1) [z'], size: O(1) [0]

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

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s35, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s35 >= 0, s35 <= 2, s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s36, 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s36 >= 0, s36 <= 2, s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s37, 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s37 >= 0, s37 <= 2, s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(s38, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s28 :|: s38 >= 0, s38 <= 2, s28 >= 0, s28 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s39, 1 + (z - 1), 1 + (z' - 1), 1 + s6) + s29 :|: s39 >= 0, s39 <= 2, s29 >= 0, s29 <= 0, s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 0) + s30 :|: s40 >= 0, s40 <= 2, s30 >= 0, s30 <= 0, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s27 :|: s27 >= 0, s27 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + s26 :|: s26 >= 0, s26 <= 0, s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s25 :|: s25 >= 0, s25 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 0) + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 1 + s4) + s23 :|: s23 >= 0, s23 <= 0, s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s22 :|: s22 >= 0, s22 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 0) + s33 :|: s33 >= 0, s33 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 1 + s7) + s32 :|: s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + s31 :|: s31 >= 0, s31 <= 0, z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 + z'' }-> 1 + DIV(s, 1 + (z'' - 1), z3) + s42 :|: s42 >= 0, s42 <= 0, s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 + z'' }-> 1 + DIV(0, 1 + (z'' - 1), z3) + s43 :|: s43 >= 0, s43 <= 0, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ z }-> 1 + s34 :|: s34 >= 0, s34 <= 0, z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ z' }-> 1 + s44 :|: s44 >= 0, s44 <= 0, z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 2, z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: O(n^1) [z], size: O(1) [0]
  lt: runtime: O(1) [0], size: O(1) [2]
  MINUS: runtime: O(n^1) [z'], size: O(1) [0]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s35, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s35 >= 0, s35 <= 2, s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s36, 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s36 >= 0, s36 <= 2, s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s37, 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s37 >= 0, s37 <= 2, s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(s38, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s28 :|: s38 >= 0, s38 <= 2, s28 >= 0, s28 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s39, 1 + (z - 1), 1 + (z' - 1), 1 + s6) + s29 :|: s39 >= 0, s39 <= 2, s29 >= 0, s29 <= 0, s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 0) + s30 :|: s40 >= 0, s40 <= 2, s30 >= 0, s30 <= 0, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s27 :|: s27 >= 0, s27 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + s26 :|: s26 >= 0, s26 <= 0, s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s25 :|: s25 >= 0, s25 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 0) + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 1 + s4) + s23 :|: s23 >= 0, s23 <= 0, s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s22 :|: s22 >= 0, s22 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 0) + s33 :|: s33 >= 0, s33 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 1 + s7) + s32 :|: s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + s31 :|: s31 >= 0, s31 <= 0, z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 + z'' }-> 1 + DIV(s, 1 + (z'' - 1), z3) + s42 :|: s42 >= 0, s42 <= 0, s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 + z'' }-> 1 + DIV(0, 1 + (z'' - 1), z3) + s43 :|: s43 >= 0, s43 <= 0, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ z }-> 1 + s34 :|: s34 >= 0, s34 <= 0, z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ z' }-> 1 + s44 :|: s44 >= 0, s44 <= 0, z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 2, z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {IF,DIV}
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: O(n^1) [z], size: O(1) [0]
  lt: runtime: O(1) [0], size: O(1) [2]
  MINUS: runtime: O(n^1) [z'], size: O(1) [0]
  IF: runtime: ?, size: O(1) [0]
  DIV: runtime: ?, size: O(1) [1]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: IF
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 133 + 967*z' + 52*z'*z'' + 90*z'*z3 + 90*z'^2 + 7*z'' + 12*z3

Computed RUNTIME bound using KoAT for: DIV
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 3408 + 18486*z + 624*z*z' + 540*z*z'' + 1620*z^2 + 135*z' + 104*z''

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

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

   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s35, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s15 :|: s35 >= 0, s35 <= 2, s15 >= 0, s15 <= 0, z'' = 0, z' - 1 >= 0, z - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s36, 1 + (z - 1), 1 + (z' - 1), 1 + s2) + s16 :|: s36 >= 0, s36 <= 2, s16 >= 0, s16 <= 0, s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(s37, 1 + (z - 1), 1 + (z' - 1), 0) + s17 :|: s37 >= 0, s37 <= 2, s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(s38, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s28 :|: s38 >= 0, s38 <= 2, s28 >= 0, s28 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s39, 1 + (z - 1), 1 + (z' - 1), 1 + s6) + s29 :|: s39 >= 0, s39 <= 2, s29 >= 0, s29 <= 0, s6 >= 0, s6 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 0) + s30 :|: s40 >= 0, s40 <= 2, s30 >= 0, s30 <= 0, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s27 :|: s27 >= 0, s27 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s1) + s13 :|: s13 >= 0, s13 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s5) + s26 :|: s26 >= 0, s26 <= 0, s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s25 :|: s25 >= 0, s25 <= 0, z'' = 0, z' - 1 >= 0, z = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + s11 :|: s11 >= 0, s11 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 0) + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s'') + s10 :|: s10 >= 0, s10 <= 0, s'' >= 0, s'' <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(1, z, 0, 1 + s4) + s23 :|: s23 >= 0, s23 <= 0, s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s22 :|: s22 >= 0, s22 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z'' = 0, z >= 0, z' = 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 0) + s20 :|: s20 >= 0, s20 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 0) + s33 :|: s33 >= 0, s33 <= 0, z' >= 0, z >= 0, z'' >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + s3) + s19 :|: s19 >= 0, s19 <= 0, s3 >= 0, s3 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 + z'' }-> 1 + IF(0, z, z', 1 + s7) + s32 :|: s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0
   DIV(z, z', z'') -{ 1 + z' }-> 1 + IF(0, z, z', 1 + 0) + s18 :|: s18 >= 0, s18 <= 0, z'' = 0, z' >= 0, z >= 0
   DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + s31 :|: s31 >= 0, s31 <= 0, z'' = 0, z' >= 0, z >= 0
   IF(z, z', z'', z3) -{ 1 + z'' }-> 1 + DIV(s, 1 + (z'' - 1), z3) + s42 :|: s42 >= 0, s42 <= 0, s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0
   IF(z, z', z'', z3) -{ 1 + z'' }-> 1 + DIV(0, 1 + (z'' - 1), z3) + s43 :|: s43 >= 0, s43 <= 0, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0
   INC(z) -{ z }-> 1 + s34 :|: s34 >= 0, s34 <= 0, z - 1 >= 0
   LT(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ z' }-> 1 + s44 :|: s44 >= 0, s44 <= 0, z' - 1 >= 0, z - 1 >= 0
   inc(z) -{ 0 }-> 0 :|: z >= 0
   inc(z) -{ 0 }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1 + 1, z - 1 >= 0
   inc(z) -{ 0 }-> 1 + 0 :|: z = 0
   lt(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 2, z' - 1 >= 0, z - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(1) [0], size: O(n^1) [z]
  inc: runtime: O(1) [0], size: O(n^1) [1 + z]
  LT: runtime: O(n^1) [z'], size: O(1) [0]
  INC: runtime: O(n^1) [z], size: O(1) [0]
  lt: runtime: O(1) [0], size: O(1) [2]
  MINUS: runtime: O(n^1) [z'], size: O(1) [0]
  IF: runtime: O(n^2) [133 + 967*z' + 52*z'*z'' + 90*z'*z3 + 90*z'^2 + 7*z'' + 12*z3], size: O(1) [0]
  DIV: runtime: O(n^2) [3408 + 18486*z + 624*z*z' + 540*z*z'' + 1620*z^2 + 135*z' + 104*z''], size: O(1) [1]

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

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

(66)
BOUNDS(1, n^2)

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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   division(z0, z1) -> div(z0, z1, 0)
   div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
   if(true, z0, z1, z2) -> z2
   if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   inc(0) -> s(0)
   inc(s(z0)) -> s(inc(z0))
Tuples:
   DIVISION(z0, z1) -> c(DIV(z0, z1, 0))
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(z0, 0) -> c5
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(z0, 0) -> c7
   LT(0, s(z0)) -> c8
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
S tuples:
   DIVISION(z0, z1) -> c(DIV(z0, z1, 0))
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(z0, 0) -> c5
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(z0, 0) -> c7
   LT(0, s(z0)) -> c8
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
K tuples:none
Defined Rule Symbols:   division_2, div_3, if_4, minus_2, lt_2, inc_1

Defined Pair Symbols:   DIVISION_2, DIV_3, IF_4, MINUS_2, LT_2, INC_1

Compound Symbols:   c_1, c1_2, c2_2, c3, c4_2, c5, c6_1, c7, c8, c9_1, c10, c11_1


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

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

(70)
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:

   DIVISION(z0, z1) -> c(DIV(z0, z1, 0))
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(z0, 0) -> c5
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(z0, 0) -> c7
   LT(0, s(z0)) -> c8
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))

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

   division(z0, z1) -> div(z0, z1, 0)
   div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
   if(true, z0, z1, z2) -> z2
   if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   inc(0) -> s(0)
   inc(s(z0)) -> s(inc(z0))

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

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

(72)
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:

   DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(z0, 0') -> c5
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(z0, 0') -> c7
   LT(0', s(z0)) -> c8
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(0') -> c10
   INC(s(z0)) -> c11(INC(z0))

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

   division(z0, z1) -> div(z0, z1, 0')
   div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
   if(true, z0, z1, z2) -> z2
   if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
   minus(z0, 0') -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   lt(z0, 0') -> false
   lt(0', s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   inc(0') -> s(0')
   inc(s(z0)) -> s(inc(z0))

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

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

(74)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6

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

(75) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
DIV, lt, inc, LT, INC, minus, MINUS, div

They will be analysed ascendingly in the following order:
lt < DIV
inc < DIV
LT < DIV
INC < DIV
minus < DIV
MINUS < DIV
lt < div
inc < div
minus < div

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

(76)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
lt, DIV, inc, LT, INC, minus, MINUS, div

They will be analysed ascendingly in the following order:
lt < DIV
inc < DIV
LT < DIV
INC < DIV
minus < DIV
MINUS < DIV
lt < div
inc < div
minus < div

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

(77) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)

Induction Base:
lt(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(0)
false

Induction Step:
lt(gen_0':s9_12(+(n14_12, 1)), gen_0':s9_12(+(n14_12, 1))) ->_R^Omega(0)
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) ->_IH
false

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

(78)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
inc, DIV, LT, INC, minus, MINUS, div

They will be analysed ascendingly in the following order:
inc < DIV
LT < DIV
INC < DIV
minus < DIV
MINUS < DIV
inc < div
minus < div

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

(79) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)

Induction Base:
inc(gen_0':s9_12(0)) ->_R^Omega(0)
s(0')

Induction Step:
inc(gen_0':s9_12(+(n398_12, 1))) ->_R^Omega(0)
s(inc(gen_0':s9_12(n398_12))) ->_IH
s(gen_0':s9_12(+(1, c399_12)))

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

(80)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
LT, DIV, INC, minus, MINUS, div

They will be analysed ascendingly in the following order:
LT < DIV
INC < DIV
minus < DIV
MINUS < DIV
minus < div

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

(81) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12)

Induction Base:
LT(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(1)
c7

Induction Step:
LT(gen_0':s9_12(+(n694_12, 1)), gen_0':s9_12(+(n694_12, 1))) ->_R^Omega(1)
c9(LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12))) ->_IH
c9(gen_c7:c8:c910_12(c695_12))

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

(82)
Complex Obligation (BEST)

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

(83)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
LT, DIV, INC, minus, MINUS, div

They will be analysed ascendingly in the following order:
LT < DIV
INC < DIV
minus < DIV
MINUS < DIV
minus < div

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

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

(85)
BOUNDS(n^1, INF)

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

(86)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)
LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
INC, DIV, minus, MINUS, div

They will be analysed ascendingly in the following order:
INC < DIV
minus < DIV
MINUS < DIV
minus < div

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

(87) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12)

Induction Base:
INC(gen_0':s9_12(0)) ->_R^Omega(1)
c10

Induction Step:
INC(gen_0':s9_12(+(n1356_12, 1))) ->_R^Omega(1)
c11(INC(gen_0':s9_12(n1356_12))) ->_IH
c11(gen_c10:c1111_12(c1357_12))

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

(88)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)
LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12)
INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
minus, DIV, MINUS, div

They will be analysed ascendingly in the following order:
minus < DIV
MINUS < DIV
minus < div

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

(89) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) -> gen_0':s9_12(0), rt in Omega(0)

Induction Base:
minus(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(0)
gen_0':s9_12(0)

Induction Step:
minus(gen_0':s9_12(+(n1724_12, 1)), gen_0':s9_12(+(n1724_12, 1))) ->_R^Omega(0)
minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) ->_IH
gen_0':s9_12(0)

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

(90)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)
LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12)
INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12)
minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) -> gen_0':s9_12(0), rt in Omega(0)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
MINUS, DIV, div

They will be analysed ascendingly in the following order:
MINUS < DIV

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

(91) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s9_12(n2396_12), gen_0':s9_12(n2396_12)) -> gen_c5:c612_12(n2396_12), rt in Omega(1 + n2396_12)

Induction Base:
MINUS(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(1)
c5

Induction Step:
MINUS(gen_0':s9_12(+(n2396_12, 1)), gen_0':s9_12(+(n2396_12, 1))) ->_R^Omega(1)
c6(MINUS(gen_0':s9_12(n2396_12), gen_0':s9_12(n2396_12))) ->_IH
c6(gen_c5:c612_12(c2397_12))

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

(92)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)
LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12)
INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12)
minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) -> gen_0':s9_12(0), rt in Omega(0)
MINUS(gen_0':s9_12(n2396_12), gen_0':s9_12(n2396_12)) -> gen_c5:c612_12(n2396_12), rt in Omega(1 + n2396_12)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
DIV, div