WORST_CASE(Omega(n^1),O(n^3))
proof of input_hI2zAtxINH.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 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), 331 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 63 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 180 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 669 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 229 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 573 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 246 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 1049 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 20 ms]
    (52) CpxRNTS
    (53) FinalProof [FINISHED, 0 ms]
    (54) BOUNDS(1, n^3)
(55) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(56) CdtProblem
    (57) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CpxRelTRS
    (59) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CpxRelTRS
    (61) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) typed CpxTrs
    (63) OrderProof [LOWER BOUND(ID), 8 ms]
    (64) typed CpxTrs
    (65) RewriteLemmaProof [LOWER BOUND(ID), 372 ms]
    (66) BEST
        (67) proven lower bound
            (68) LowerBoundPropagationProof [FINISHED, 0 ms]
            (69) BOUNDS(n^1, INF)
        (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 78 ms]
            (72) typed CpxTrs


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

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


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0, y) -> 0
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(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:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(0, z0) -> 0
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
   if_minus(true, s(z0), z1) -> 0
   if_minus(false, s(z0), z1) -> s(minus(z0, z1))
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(0, z0) -> c3
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(true, s(z0), z1) -> c5
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c11
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(0, z0) -> c3
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(true, s(z0), z1) -> c5
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c11
K tuples:none
Defined Rule Symbols:   le_2, minus_2, if_minus_3, mod_2, if_mod_3

Defined Pair Symbols:   LE_2, MINUS_2, IF_MINUS_3, MOD_2, IF_MOD_3

Compound Symbols:   c, c1, c2_1, c3, c4_2, c5, c6_1, c7, c8, c9_2, c10_2, c11


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   MINUS(0, z0) -> c3
   LE(s(z0), 0) -> c1
   MOD(s(z0), 0) -> c8
   IF_MINUS(true, s(z0), z1) -> c5
   LE(0, z0) -> c
   IF_MOD(false, s(z0), s(z1)) -> c11
   MOD(0, z0) -> c7

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(0, z0) -> 0
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
   if_minus(true, s(z0), z1) -> 0
   if_minus(false, s(z0), z1) -> s(minus(z0, z1))
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   le_2, minus_2, if_minus_3, mod_2, if_mod_3

Defined Pair Symbols:   LE_2, MINUS_2, IF_MINUS_3, MOD_2, IF_MOD_3

Compound Symbols:   c2_1, c4_2, c6_1, c9_2, c10_2


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   le(0, z0) -> true
   minus(0, z0) -> 0
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
   if_minus(true, s(z0), z1) -> 0
   if_minus(false, s(z0), z1) -> s(minus(z0, z1))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   le_2, minus_2, if_minus_3

Defined Pair Symbols:   LE_2, MINUS_2, IF_MINUS_3, MOD_2, IF_MOD_3

Compound Symbols:   c2_1, c4_2, c6_1, c9_2, c10_2


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

(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^3).


The TRS R consists of the following rules:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))

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

   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   le(0, z0) -> true
   minus(0, z0) -> 0
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
   if_minus(true, s(z0), z1) -> 0
   if_minus(false, s(z0), z1) -> s(minus(z0, z1))

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^3).


The TRS R consists of the following rules:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1)) [1]
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) [1]
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1)) [1]
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) [1]
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) [1]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   le(0, z0) -> true [0]
   minus(0, z0) -> 0 [0]
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1) [0]
   if_minus(true, s(z0), z1) -> 0 [0]
   if_minus(false, s(z0), z1) -> s(minus(z0, z1)) [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:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1)) [1]
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) [1]
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1)) [1]
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) [1]
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) [1]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   le(0, z0) -> true [0]
   minus(0, z0) -> 0 [0]
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1) [0]
   if_minus(true, s(z0), z1) -> 0 [0]
   if_minus(false, s(z0), z1) -> s(minus(z0, z1)) [0]

The TRS has the following type information:
LE :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
MINUS :: s:0 -> s:0 -> c4
c4 :: c6 -> c2 -> c4
IF_MINUS :: false:true -> s:0 -> s:0 -> c6
le :: s:0 -> s:0 -> false:true
false :: false:true
c6 :: c4 -> c6
MOD :: s:0 -> s:0 -> c9
c9 :: c10 -> c2 -> c9
IF_MOD :: false:true -> s:0 -> s:0 -> c10
true :: false:true
c10 :: c9 -> c4 -> c10
minus :: s:0 -> s:0 -> s:0
0 :: s:0
if_minus :: false:true -> s:0 -> s:0 -> s:0

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:

   LE_2
   MINUS_2
   IF_MINUS_3
   MOD_2
   IF_MOD_3

(c) The following functions are completely defined:

   le_2
   minus_2
   if_minus_3

Due to the following rules being added:

   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> 0 [0]
   if_minus(v0, v1, v2) -> 0 [0]

And the following fresh constants:    null_le, 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:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1)) [1]
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) [1]
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1)) [1]
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) [1]
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) [1]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   le(0, z0) -> true [0]
   minus(0, z0) -> 0 [0]
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1) [0]
   if_minus(true, s(z0), z1) -> 0 [0]
   if_minus(false, s(z0), z1) -> s(minus(z0, z1)) [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> 0 [0]
   if_minus(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
LE :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
MINUS :: s:0 -> s:0 -> c4
c4 :: c6 -> c2 -> c4
IF_MINUS :: false:true:null_le -> s:0 -> s:0 -> c6
le :: s:0 -> s:0 -> false:true:null_le
false :: false:true:null_le
c6 :: c4 -> c6
MOD :: s:0 -> s:0 -> c9
c9 :: c10 -> c2 -> c9
IF_MOD :: false:true:null_le -> s:0 -> s:0 -> c10
true :: false:true:null_le
c10 :: c9 -> c4 -> c10
minus :: s:0 -> s:0 -> s:0
0 :: s:0
if_minus :: false:true:null_le -> s:0 -> s:0 -> s:0
null_le :: false:true:null_le
const :: c2
const1 :: c4
const2 :: c6
const3 :: c9
const4 :: c10

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:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1)) [1]
   MINUS(s(z0), 0) -> c4(IF_MINUS(false, s(z0), 0), LE(s(z0), 0)) [1]
   MINUS(s(z0), s(z1')) -> c4(IF_MINUS(le(z0, z1'), s(z0), s(z1')), LE(s(z0), s(z1'))) [1]
   MINUS(s(z0), z1) -> c4(IF_MINUS(null_le, s(z0), z1), LE(s(z0), z1)) [1]
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1)) [1]
   MOD(s(0), s(s(z0'))) -> c9(IF_MOD(false, s(0), s(s(z0'))), LE(s(z0'), 0)) [1]
   MOD(s(s(z1'')), s(s(z0''))) -> c9(IF_MOD(le(z0'', z1''), s(s(z1'')), s(s(z0''))), LE(s(z0''), s(z1''))) [1]
   MOD(s(z0), s(0)) -> c9(IF_MOD(true, s(z0), s(0)), LE(0, z0)) [1]
   MOD(s(z0), s(z1)) -> c9(IF_MOD(null_le, s(z0), s(z1)), LE(z1, z0)) [1]
   IF_MOD(true, s(0), s(z1)) -> c10(MOD(0, s(z1)), MINUS(0, z1)) [1]
   IF_MOD(true, s(s(z01)), s(z1)) -> c10(MOD(if_minus(le(s(z01), z1), s(z01), z1), s(z1)), MINUS(s(z01), z1)) [1]
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(0, s(z1)), MINUS(z0, z1)) [1]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   le(0, z0) -> true [0]
   minus(0, z0) -> 0 [0]
   minus(s(z0), 0) -> if_minus(false, s(z0), 0) [0]
   minus(s(z0), s(z11)) -> if_minus(le(z0, z11), s(z0), s(z11)) [0]
   minus(s(z0), z1) -> if_minus(null_le, s(z0), z1) [0]
   if_minus(true, s(z0), z1) -> 0 [0]
   if_minus(false, s(z0), z1) -> s(minus(z0, z1)) [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> 0 [0]
   if_minus(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
LE :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
MINUS :: s:0 -> s:0 -> c4
c4 :: c6 -> c2 -> c4
IF_MINUS :: false:true:null_le -> s:0 -> s:0 -> c6
le :: s:0 -> s:0 -> false:true:null_le
false :: false:true:null_le
c6 :: c4 -> c6
MOD :: s:0 -> s:0 -> c9
c9 :: c10 -> c2 -> c9
IF_MOD :: false:true:null_le -> s:0 -> s:0 -> c10
true :: false:true:null_le
c10 :: c9 -> c4 -> c10
minus :: s:0 -> s:0 -> s:0
0 :: s:0
if_minus :: false:true:null_le -> s:0 -> s:0 -> s:0
null_le :: false:true:null_le
const :: c2
const1 :: c4
const2 :: c6
const3 :: c9
const4 :: c10

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
   true => 2
   0 => 0
   null_le => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0

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

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

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z0, z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = 1 + z0, z'' = z1
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(le(1 + z01, z1), 1 + z01, z1), 1 + z1) + MINUS(1 + z01, z1) :|: z = 2, z1 >= 0, z01 >= 0, z' = 1 + (1 + z01), z'' = 1 + z1
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + z1) + MINUS(z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = 1 + z0, z'' = 1 + z1
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + z1) + MINUS(0, z1) :|: z = 2, z1 >= 0, z' = 1 + 0, z'' = 1 + z1
   LE(z, z') -{ 1 }-> 1 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(le(z0, z1'), 1 + z0, 1 + z1') + LE(1 + z0, 1 + z1') :|: z = 1 + z0, z' = 1 + z1', z1' >= 0, z0 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + z0, 0) + LE(1 + z0, 0) :|: z = 1 + z0, z0 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(0, 1 + z0, z1) + LE(1 + z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(le(z0'', z1''), 1 + (1 + z1''), 1 + (1 + z0'')) + LE(1 + z0'', 1 + z1'') :|: z' = 1 + (1 + z0''), z = 1 + (1 + z1''), z0'' >= 0, z1'' >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(2, 1 + z0, 1 + 0) + LE(0, z0) :|: z = 1 + z0, z' = 1 + 0, z0 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + z0')) + LE(1 + z0', 0) :|: z' = 1 + (1 + z0'), z0' >= 0, z = 1 + 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(0, 1 + z0, 1 + z1) + LE(z1, z0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z' = 1 + z0, z'' = z1
   if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z0, z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = 1 + z0, z'' = z1
   le(z, z') -{ 0 }-> le(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   le(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0
   le(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 0 }-> if_minus(le(z0, z11), 1 + z0, 1 + z11) :|: z' = 1 + z11, z11 >= 0, z = 1 + z0, z0 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + z0, 0) :|: z = 1 + z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   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:

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) + LE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + LE(1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + LE(1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 1 + (z - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + LE(0, z - 1) :|: z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 0) :|: z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + LE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { le }
   { LE }
   { minus, if_minus }
   { MINUS, IF_MINUS }
   { MOD, IF_MOD }

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

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

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) + LE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + LE(1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + LE(1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 1 + (z - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + LE(0, z - 1) :|: z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 0) :|: z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + LE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {le}, {LE}, {minus,if_minus}, {MINUS,IF_MINUS}, {MOD,IF_MOD}

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

(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:

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) + LE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + LE(1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + LE(1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 1 + (z - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + LE(0, z - 1) :|: z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 0) :|: z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + LE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {le}, {LE}, {minus,if_minus}, {MINUS,IF_MINUS}, {MOD,IF_MOD}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) + LE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + LE(1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + LE(1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 1 + (z - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + LE(0, z - 1) :|: z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 0) :|: z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + LE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {le}, {LE}, {minus,if_minus}, {MINUS,IF_MINUS}, {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: ?, size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) + LE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + LE(1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + LE(1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 1 + (z - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + LE(0, z - 1) :|: z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 0) :|: z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + LE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {LE}, {minus,if_minus}, {MINUS,IF_MINUS}, {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]

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

(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:

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(s'', 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(s, 1 + (z - 1), 1 + (z' - 1)) + LE(1 + (z - 1), 1 + (z' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + LE(1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + LE(1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 1 + (z - 2)) :|: s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + LE(0, z - 1) :|: z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 0) :|: z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + LE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(s2, 1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {LE}, {minus,if_minus}, {MINUS,IF_MINUS}, {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(s'', 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(s, 1 + (z - 1), 1 + (z' - 1)) + LE(1 + (z - 1), 1 + (z' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + LE(1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + LE(1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 1 + (z - 2)) :|: s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + LE(0, z - 1) :|: z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 0) :|: z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + LE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(s2, 1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {LE}, {minus,if_minus}, {MINUS,IF_MINUS}, {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: ?, size: O(1) [0]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(s'', 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(s, 1 + (z - 1), 1 + (z' - 1)) + LE(1 + (z - 1), 1 + (z' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + LE(1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + LE(1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 1 + (z - 2)) :|: s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + LE(0, z - 1) :|: z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + LE(1 + (z' - 2), 0) :|: z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + LE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(s2, 1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus,if_minus}, {MINUS,IF_MINUS}, {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]

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

(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:

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(s'', 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(s, 1 + (z - 1), 1 + (z' - 1)) + s5 :|: s5 >= 0, s5 <= 0, s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + s4 :|: s4 >= 0, s4 <= 0, z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + s6 :|: s6 >= 0, s6 <= 0, z' >= 0, z - 1 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + s7 :|: s7 >= 0, s7 <= 0, z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(s2, 1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus,if_minus}, {MINUS,IF_MINUS}, {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(s'', 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(s, 1 + (z - 1), 1 + (z' - 1)) + s5 :|: s5 >= 0, s5 <= 0, s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + s4 :|: s4 >= 0, s4 <= 0, z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + s6 :|: s6 >= 0, s6 <= 0, z' >= 0, z - 1 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + s7 :|: s7 >= 0, s7 <= 0, z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(s2, 1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus,if_minus}, {MINUS,IF_MINUS}, {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: ?, size: O(n^1) [z]
  if_minus: runtime: ?, size: O(n^1) [z']

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

(39) 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

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

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

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

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(if_minus(s'', 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(s, 1 + (z - 1), 1 + (z' - 1)) + s5 :|: s5 >= 0, s5 <= 0, s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + s4 :|: s4 >= 0, s4 <= 0, z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + s6 :|: s6 >= 0, s6 <= 0, z' >= 0, z - 1 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + s7 :|: s7 >= 0, s7 <= 0, z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> if_minus(s2, 1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> if_minus(0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {MINUS,IF_MINUS}, {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  if_minus: runtime: O(1) [0], size: O(n^1) [z']

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

(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:

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(s11, 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: s11 >= 0, s11 <= 1 + (z' - 2), s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(s, 1 + (z - 1), 1 + (z' - 1)) + s5 :|: s5 >= 0, s5 <= 0, s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + s4 :|: s4 >= 0, s4 <= 0, z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + s6 :|: s6 >= 0, s6 <= 0, z' >= 0, z - 1 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + s7 :|: s7 >= 0, s7 <= 0, z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + s15 :|: s15 >= 0, s15 <= z' - 1, z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (z - 1), s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 1 + (z - 1), z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {MINUS,IF_MINUS}, {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  if_minus: runtime: O(1) [0], size: O(n^1) [z']

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

(43) 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

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

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

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

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(s11, 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: s11 >= 0, s11 <= 1 + (z' - 2), s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(s, 1 + (z - 1), 1 + (z' - 1)) + s5 :|: s5 >= 0, s5 <= 0, s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + s4 :|: s4 >= 0, s4 <= 0, z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + s6 :|: s6 >= 0, s6 <= 0, z' >= 0, z - 1 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + s7 :|: s7 >= 0, s7 <= 0, z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + s15 :|: s15 >= 0, s15 <= z' - 1, z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (z - 1), s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 1 + (z - 1), z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {MINUS,IF_MINUS}, {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  if_minus: runtime: O(1) [0], size: O(n^1) [z']
  MINUS: runtime: ?, size: O(1) [0]
  IF_MINUS: runtime: ?, size: O(1) [1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   IF_MINUS(z, z', z'') -{ 1 }-> 1 + MINUS(z' - 1, z'') :|: z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(s11, 1 + (z'' - 1)) + MINUS(1 + (z' - 2), z'' - 1) :|: s11 >= 0, s11 <= 1 + (z' - 2), s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(0, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + MINUS(z' - 1, z'' - 1) :|: z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(s, 1 + (z - 1), 1 + (z' - 1)) + s5 :|: s5 >= 0, s5 <= 0, s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 }-> 1 + IF_MINUS(1, 1 + (z - 1), 0) + s4 :|: s4 >= 0, s4 <= 0, z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 + z' }-> 1 + IF_MINUS(0, 1 + (z - 1), z') + s6 :|: s6 >= 0, s6 <= 0, z' >= 0, z - 1 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + s7 :|: s7 >= 0, s7 <= 0, z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + s15 :|: s15 >= 0, s15 <= z' - 1, z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (z - 1), s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 1 + (z - 1), z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  if_minus: runtime: O(1) [0], size: O(n^1) [z']
  MINUS: runtime: O(n^2) [8*z + 4*z*z'], size: O(1) [0]
  IF_MINUS: runtime: O(n^2) [8*z' + 4*z'*z''], size: O(1) [1]

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

(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:

   IF_MINUS(z, z', z'') -{ -7 + 8*z' + 4*z'*z'' + -4*z'' }-> 1 + s19 :|: s19 >= 0, s19 <= 0, z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ -3 + 4*z' + 4*z'*z'' + -4*z'' }-> 1 + MOD(s11, 1 + (z'' - 1)) + s21 :|: s21 >= 0, s21 <= 0, s11 >= 0, s11 <= 1 + (z' - 2), s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + s20 :|: s20 >= 0, s20 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ -3 + 4*z' + 4*z'*z'' + -4*z'' }-> 1 + MOD(0, 1 + (z'' - 1)) + s22 :|: s22 >= 0, s22 <= 0, z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + 8*z }-> 1 + s16 + s4 :|: s16 >= 0, s16 <= 1, s4 >= 0, s4 <= 0, z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 + 8*z + 4*z*z' + z' }-> 1 + s17 + s5 :|: s17 >= 0, s17 <= 1, s5 >= 0, s5 <= 0, s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + 8*z + 4*z*z' + z' }-> 1 + s18 + s6 :|: s18 >= 0, s18 <= 1, s6 >= 0, s6 <= 0, z' >= 0, z - 1 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + s7 :|: s7 >= 0, s7 <= 0, z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + s15 :|: s15 >= 0, s15 <= z' - 1, z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (z - 1), s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 1 + (z - 1), z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  if_minus: runtime: O(1) [0], size: O(n^1) [z']
  MINUS: runtime: O(n^2) [8*z + 4*z*z'], size: O(1) [0]
  IF_MINUS: runtime: O(n^2) [8*z' + 4*z'*z''], size: O(1) [1]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   IF_MINUS(z, z', z'') -{ -7 + 8*z' + 4*z'*z'' + -4*z'' }-> 1 + s19 :|: s19 >= 0, s19 <= 0, z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ -3 + 4*z' + 4*z'*z'' + -4*z'' }-> 1 + MOD(s11, 1 + (z'' - 1)) + s21 :|: s21 >= 0, s21 <= 0, s11 >= 0, s11 <= 1 + (z' - 2), s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + s20 :|: s20 >= 0, s20 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ -3 + 4*z' + 4*z'*z'' + -4*z'' }-> 1 + MOD(0, 1 + (z'' - 1)) + s22 :|: s22 >= 0, s22 <= 0, z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + 8*z }-> 1 + s16 + s4 :|: s16 >= 0, s16 <= 1, s4 >= 0, s4 <= 0, z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 + 8*z + 4*z*z' + z' }-> 1 + s17 + s5 :|: s17 >= 0, s17 <= 1, s5 >= 0, s5 <= 0, s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + 8*z + 4*z*z' + z' }-> 1 + s18 + s6 :|: s18 >= 0, s18 <= 1, s6 >= 0, s6 <= 0, z' >= 0, z - 1 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + s7 :|: s7 >= 0, s7 <= 0, z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + s15 :|: s15 >= 0, s15 <= z' - 1, z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (z - 1), s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 1 + (z - 1), z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {MOD,IF_MOD}
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  if_minus: runtime: O(1) [0], size: O(n^1) [z']
  MINUS: runtime: O(n^2) [8*z + 4*z*z'], size: O(1) [0]
  IF_MINUS: runtime: O(n^2) [8*z' + 4*z'*z''], size: O(1) [1]
  MOD: runtime: ?, size: O(1) [0]
  IF_MOD: runtime: ?, size: O(1) [1]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: MOD
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 4*z + 38*z^2 + 16*z^2*z'

Computed RUNTIME bound using KoAT for: IF_MOD
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1 + 12*z' + 8*z'*z'' + 38*z'^2 + 16*z'^2*z''

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

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

   IF_MINUS(z, z', z'') -{ -7 + 8*z' + 4*z'*z'' + -4*z'' }-> 1 + s19 :|: s19 >= 0, s19 <= 0, z'' >= 0, z = 1, z' - 1 >= 0
   IF_MOD(z, z', z'') -{ -3 + 4*z' + 4*z'*z'' + -4*z'' }-> 1 + MOD(s11, 1 + (z'' - 1)) + s21 :|: s21 >= 0, s21 <= 0, s11 >= 0, s11 <= 1 + (z' - 2), s'' >= 0, s'' <= 2, z = 2, z'' - 1 >= 0, z' - 2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(0, 1 + (z'' - 1)) + s20 :|: s20 >= 0, s20 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   IF_MOD(z, z', z'') -{ -3 + 4*z' + 4*z'*z'' + -4*z'' }-> 1 + MOD(0, 1 + (z'' - 1)) + s22 :|: s22 >= 0, s22 <= 0, z = 2, z'' - 1 >= 0, z' - 1 >= 0
   LE(z, z') -{ z' }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + 8*z }-> 1 + s16 + s4 :|: s16 >= 0, s16 <= 1, s4 >= 0, s4 <= 0, z - 1 >= 0, z' = 0
   MINUS(z, z') -{ 1 + 8*z + 4*z*z' + z' }-> 1 + s17 + s5 :|: s17 >= 0, s17 <= 1, s5 >= 0, s5 <= 0, s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0
   MINUS(z, z') -{ 1 + 8*z + 4*z*z' + z' }-> 1 + s18 + s6 :|: s18 >= 0, s18 <= 1, s6 >= 0, s6 <= 0, z' >= 0, z - 1 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(s', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) + s8 :|: s8 >= 0, s8 <= 0, s' >= 0, s' <= 2, z' - 2 >= 0, z - 2 >= 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(2, 1 + (z - 1), 1 + 0) + s9 :|: s9 >= 0, s9 <= 0, z' = 1 + 0, z - 1 >= 0
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(1, 1 + 0, 1 + (1 + (z' - 2))) + s7 :|: s7 >= 0, s7 <= 0, z' - 2 >= 0, z = 1 + 0
   MOD(z, z') -{ z }-> 1 + IF_MOD(0, 1 + (z - 1), 1 + (z' - 1)) + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' - 1 >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 1 + s15 :|: s15 >= 0, s15 <= z' - 1, z'' >= 0, z = 1, z' - 1 >= 0
   le(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (z - 1), s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 1 + (z - 1), z' >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  if_minus: runtime: O(1) [0], size: O(n^1) [z']
  MINUS: runtime: O(n^2) [8*z + 4*z*z'], size: O(1) [0]
  IF_MINUS: runtime: O(n^2) [8*z' + 4*z'*z''], size: O(1) [1]
  MOD: runtime: O(n^3) [4*z + 38*z^2 + 16*z^2*z'], size: O(1) [0]
  IF_MOD: runtime: O(n^3) [1 + 12*z' + 8*z'*z'' + 38*z'^2 + 16*z'^2*z''], size: O(1) [1]

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

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

(54)
BOUNDS(1, n^3)

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

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(0, z0) -> 0
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
   if_minus(true, s(z0), z1) -> 0
   if_minus(false, s(z0), z1) -> s(minus(z0, z1))
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(0, z0) -> c3
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(true, s(z0), z1) -> c5
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c11
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(0, z0) -> c3
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(true, s(z0), z1) -> c5
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c11
K tuples:none
Defined Rule Symbols:   le_2, minus_2, if_minus_3, mod_2, if_mod_3

Defined Pair Symbols:   LE_2, MINUS_2, IF_MINUS_3, MOD_2, IF_MOD_3

Compound Symbols:   c, c1, c2_1, c3, c4_2, c5, c6_1, c7, c8, c9_2, c10_2, c11


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

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

(58)
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:

   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(0, z0) -> c3
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(true, s(z0), z1) -> c5
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c11

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

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(0, z0) -> 0
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
   if_minus(true, s(z0), z1) -> 0
   if_minus(false, s(z0), z1) -> s(minus(z0, z1))
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)

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

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

(60)
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:

   LE(0', z0) -> c
   LE(s(z0), 0') -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(0', z0) -> c3
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(true, s(z0), z1) -> c5
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   MOD(0', z0) -> c7
   MOD(s(z0), 0') -> c8
   MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c11

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

   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(0', z0) -> 0'
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
   if_minus(true, s(z0), z1) -> 0'
   if_minus(false, s(z0), z1) -> s(minus(z0, z1))
   mod(0', z0) -> 0'
   mod(s(z0), 0') -> 0'
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)

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

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

(62)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(0', z0) -> c3
MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(true, s(z0), z1) -> c5
IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c11
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(0', z0) -> 0'
minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) -> 0'
if_minus(false, s(z0), z1) -> s(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(z0)

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
MINUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c:c1:c2 -> c3:c4
IF_MINUS :: true:false -> 0':s -> 0':s -> c5:c6
le :: 0':s -> 0':s -> true:false
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c3:c4 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c:c1:c2 -> c7:c8:c9
IF_MOD :: true:false -> 0':s -> 0':s -> c10:c11
c10 :: c7:c8:c9 -> c3:c4 -> c10:c11
minus :: 0':s -> 0':s -> 0':s
c11 :: c10:c11
if_minus :: true:false -> 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_12 :: c:c1:c2
hole_0':s2_12 :: 0':s
hole_c3:c43_12 :: c3:c4
hole_c5:c64_12 :: c5:c6
hole_true:false5_12 :: true:false
hole_c7:c8:c96_12 :: c7:c8:c9
hole_c10:c117_12 :: c10:c11
gen_c:c1:c28_12 :: Nat -> c:c1:c2
gen_0':s9_12 :: Nat -> 0':s

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

(63) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LE, MINUS, le, MOD, minus, mod

They will be analysed ascendingly in the following order:
LE < MINUS
LE < MOD
le < MINUS
MINUS < MOD
le < MOD
le < minus
le < mod
minus < MOD
minus < mod

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

(64)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(0', z0) -> c3
MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(true, s(z0), z1) -> c5
IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c11
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(0', z0) -> 0'
minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) -> 0'
if_minus(false, s(z0), z1) -> s(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(z0)

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
MINUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c:c1:c2 -> c3:c4
IF_MINUS :: true:false -> 0':s -> 0':s -> c5:c6
le :: 0':s -> 0':s -> true:false
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c3:c4 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c:c1:c2 -> c7:c8:c9
IF_MOD :: true:false -> 0':s -> 0':s -> c10:c11
c10 :: c7:c8:c9 -> c3:c4 -> c10:c11
minus :: 0':s -> 0':s -> 0':s
c11 :: c10:c11
if_minus :: true:false -> 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_12 :: c:c1:c2
hole_0':s2_12 :: 0':s
hole_c3:c43_12 :: c3:c4
hole_c5:c64_12 :: c5:c6
hole_true:false5_12 :: true:false
hole_c7:c8:c96_12 :: c7:c8:c9
hole_c10:c117_12 :: c10:c11
gen_c:c1:c28_12 :: Nat -> c:c1:c2
gen_0':s9_12 :: Nat -> 0':s


Generator Equations:
gen_c:c1:c28_12(0) <=> c
gen_c:c1:c28_12(+(x, 1)) <=> c2(gen_c:c1:c28_12(x))
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))


The following defined symbols remain to be analysed:
LE, MINUS, le, MOD, minus, mod

They will be analysed ascendingly in the following order:
LE < MINUS
LE < MOD
le < MINUS
MINUS < MOD
le < MOD
le < minus
le < mod
minus < MOD
minus < mod

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s9_12(n11_12), gen_0':s9_12(n11_12)) -> gen_c:c1:c28_12(n11_12), rt in Omega(1 + n11_12)

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

Induction Step:
LE(gen_0':s9_12(+(n11_12, 1)), gen_0':s9_12(+(n11_12, 1))) ->_R^Omega(1)
c2(LE(gen_0':s9_12(n11_12), gen_0':s9_12(n11_12))) ->_IH
c2(gen_c:c1:c28_12(c12_12))

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

(66)
Complex Obligation (BEST)

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

(67)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(0', z0) -> c3
MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(true, s(z0), z1) -> c5
IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c11
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(0', z0) -> 0'
minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) -> 0'
if_minus(false, s(z0), z1) -> s(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(z0)

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
MINUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c:c1:c2 -> c3:c4
IF_MINUS :: true:false -> 0':s -> 0':s -> c5:c6
le :: 0':s -> 0':s -> true:false
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c3:c4 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c:c1:c2 -> c7:c8:c9
IF_MOD :: true:false -> 0':s -> 0':s -> c10:c11
c10 :: c7:c8:c9 -> c3:c4 -> c10:c11
minus :: 0':s -> 0':s -> 0':s
c11 :: c10:c11
if_minus :: true:false -> 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_12 :: c:c1:c2
hole_0':s2_12 :: 0':s
hole_c3:c43_12 :: c3:c4
hole_c5:c64_12 :: c5:c6
hole_true:false5_12 :: true:false
hole_c7:c8:c96_12 :: c7:c8:c9
hole_c10:c117_12 :: c10:c11
gen_c:c1:c28_12 :: Nat -> c:c1:c2
gen_0':s9_12 :: Nat -> 0':s


Generator Equations:
gen_c:c1:c28_12(0) <=> c
gen_c:c1:c28_12(+(x, 1)) <=> c2(gen_c:c1:c28_12(x))
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))


The following defined symbols remain to be analysed:
LE, MINUS, le, MOD, minus, mod

They will be analysed ascendingly in the following order:
LE < MINUS
LE < MOD
le < MINUS
MINUS < MOD
le < MOD
le < minus
le < mod
minus < MOD
minus < mod

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

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

(69)
BOUNDS(n^1, INF)

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

(70)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(0', z0) -> c3
MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(true, s(z0), z1) -> c5
IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c11
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(0', z0) -> 0'
minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) -> 0'
if_minus(false, s(z0), z1) -> s(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(z0)

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
MINUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c:c1:c2 -> c3:c4
IF_MINUS :: true:false -> 0':s -> 0':s -> c5:c6
le :: 0':s -> 0':s -> true:false
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c3:c4 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c:c1:c2 -> c7:c8:c9
IF_MOD :: true:false -> 0':s -> 0':s -> c10:c11
c10 :: c7:c8:c9 -> c3:c4 -> c10:c11
minus :: 0':s -> 0':s -> 0':s
c11 :: c10:c11
if_minus :: true:false -> 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_12 :: c:c1:c2
hole_0':s2_12 :: 0':s
hole_c3:c43_12 :: c3:c4
hole_c5:c64_12 :: c5:c6
hole_true:false5_12 :: true:false
hole_c7:c8:c96_12 :: c7:c8:c9
hole_c10:c117_12 :: c10:c11
gen_c:c1:c28_12 :: Nat -> c:c1:c2
gen_0':s9_12 :: Nat -> 0':s


Lemmas:
LE(gen_0':s9_12(n11_12), gen_0':s9_12(n11_12)) -> gen_c:c1:c28_12(n11_12), rt in Omega(1 + n11_12)


Generator Equations:
gen_c:c1:c28_12(0) <=> c
gen_c:c1:c28_12(+(x, 1)) <=> c2(gen_c:c1:c28_12(x))
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))


The following defined symbols remain to be analysed:
le, MINUS, MOD, minus, mod

They will be analysed ascendingly in the following order:
le < MINUS
MINUS < MOD
le < MOD
le < minus
le < mod
minus < MOD
minus < mod

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s9_12(n619_12), gen_0':s9_12(n619_12)) -> true, rt in Omega(0)

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

Induction Step:
le(gen_0':s9_12(+(n619_12, 1)), gen_0':s9_12(+(n619_12, 1))) ->_R^Omega(0)
le(gen_0':s9_12(n619_12), gen_0':s9_12(n619_12)) ->_IH
true

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

(72)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(0', z0) -> c3
MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(true, s(z0), z1) -> c5
IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c11
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(0', z0) -> 0'
minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) -> 0'
if_minus(false, s(z0), z1) -> s(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(z0)

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
MINUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c:c1:c2 -> c3:c4
IF_MINUS :: true:false -> 0':s -> 0':s -> c5:c6
le :: 0':s -> 0':s -> true:false
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c3:c4 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c:c1:c2 -> c7:c8:c9
IF_MOD :: true:false -> 0':s -> 0':s -> c10:c11
c10 :: c7:c8:c9 -> c3:c4 -> c10:c11
minus :: 0':s -> 0':s -> 0':s
c11 :: c10:c11
if_minus :: true:false -> 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_12 :: c:c1:c2
hole_0':s2_12 :: 0':s
hole_c3:c43_12 :: c3:c4
hole_c5:c64_12 :: c5:c6
hole_true:false5_12 :: true:false
hole_c7:c8:c96_12 :: c7:c8:c9
hole_c10:c117_12 :: c10:c11
gen_c:c1:c28_12 :: Nat -> c:c1:c2
gen_0':s9_12 :: Nat -> 0':s


Lemmas:
LE(gen_0':s9_12(n11_12), gen_0':s9_12(n11_12)) -> gen_c:c1:c28_12(n11_12), rt in Omega(1 + n11_12)
le(gen_0':s9_12(n619_12), gen_0':s9_12(n619_12)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c28_12(0) <=> c
gen_c:c1:c28_12(+(x, 1)) <=> c2(gen_c:c1:c28_12(x))
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))


The following defined symbols remain to be analysed:
MINUS, MOD, minus, mod

They will be analysed ascendingly in the following order:
MINUS < MOD
minus < MOD
minus < mod