/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), O(n^4))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^4).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 322 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 10.4 s]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 93 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 412 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 156 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 66 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 889 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 269 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 1236 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 162 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 463 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 347 ms]
    (60) CpxRNTS
    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 269 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 167 ms]
    (70) CpxRNTS
    (71) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (72) CpxRNTS
    (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 238 ms]
    (76) CpxRNTS
    (77) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (78) CpxRNTS
    (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (80) CpxRNTS
    (81) IntTrsBoundProof [UPPER BOUND(ID), 218 ms]
    (82) CpxRNTS
    (83) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (84) CpxRNTS
    (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (86) CpxRNTS
    (87) IntTrsBoundProof [UPPER BOUND(ID), 146 ms]
    (88) CpxRNTS
    (89) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (90) CpxRNTS
    (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (92) CpxRNTS
    (93) IntTrsBoundProof [UPPER BOUND(ID), 146 ms]
    (94) CpxRNTS
    (95) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (96) CpxRNTS
    (97) FinalProof [FINISHED, 0 ms]
    (98) BOUNDS(1, n^4)
(99) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(100) TRS for Loop Detection
    (101) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (102) BEST
        (103) proven lower bound
            (104) LowerBoundPropagationProof [FINISHED, 0 ms]
            (105) BOUNDS(n^1, INF)
        (106) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^4).


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: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_if_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^4).


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)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_if_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

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


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)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_if_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^4).


The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [1]
   encArg(0) -> 0 [0]
   encArg(true) -> true [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(false) -> false [0]
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_if_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_true -> true [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_false -> false [0]
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2)) [0]
   encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) [0]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [1]
   encArg(0) -> 0 [0]
   encArg(true) -> true [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(false) -> false [0]
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_if_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_true -> true [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_false -> false [0]
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2)) [0]
   encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) [0]

The TRS has the following type information:
le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
cons_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
cons_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
cons_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
cons_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
cons_if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
encode_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod
encode_if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod

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

(9) 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:
none

(c) The following functions are completely defined:

   le_2
   minus_2
   if_minus_3
   mod_2
   if_mod_3
   encArg_1
   encode_le_2
   encode_0
   encode_true
   encode_s_1
   encode_false
   encode_minus_2
   encode_if_minus_3
   encode_mod_2
   encode_if_mod_3

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_le(v0, v1) -> null_encode_le [0]
   encode_0 -> null_encode_0 [0]
   encode_true -> null_encode_true [0]
   encode_s(v0) -> null_encode_s [0]
   encode_false -> null_encode_false [0]
   encode_minus(v0, v1) -> null_encode_minus [0]
   encode_if_minus(v0, v1, v2) -> null_encode_if_minus [0]
   encode_mod(v0, v1) -> null_encode_mod [0]
   encode_if_mod(v0, v1, v2) -> null_encode_if_mod [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   if_minus(v0, v1, v2) -> null_if_minus [0]
   mod(v0, v1) -> null_mod [0]
   if_mod(v0, v1, v2) -> null_if_mod [0]

And the following fresh constants:    null_encArg, null_encode_le, null_encode_0, null_encode_true, null_encode_s, null_encode_false, null_encode_minus, null_encode_if_minus, null_encode_mod, null_encode_if_mod, null_le, null_minus, null_if_minus, null_mod, null_if_mod

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

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

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

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [1]
   encArg(0) -> 0 [0]
   encArg(true) -> true [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(false) -> false [0]
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_if_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_true -> true [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_false -> false [0]
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2)) [0]
   encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(v0) -> null_encArg [0]
   encode_le(v0, v1) -> null_encode_le [0]
   encode_0 -> null_encode_0 [0]
   encode_true -> null_encode_true [0]
   encode_s(v0) -> null_encode_s [0]
   encode_false -> null_encode_false [0]
   encode_minus(v0, v1) -> null_encode_minus [0]
   encode_if_minus(v0, v1, v2) -> null_encode_if_minus [0]
   encode_mod(v0, v1) -> null_encode_mod [0]
   encode_if_mod(v0, v1, v2) -> null_encode_if_mod [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   if_minus(v0, v1, v2) -> null_if_minus [0]
   mod(v0, v1) -> null_mod [0]
   if_mod(v0, v1, v2) -> null_if_mod [0]

The TRS has the following type information:
le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
cons_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
cons_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
cons_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
cons_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
cons_if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod

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

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

(12)
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(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), 0) -> if_minus(false, s(x), 0) [2]
   minus(s(x), s(y')) -> if_minus(le(x, y'), s(x), s(y')) [2]
   minus(s(x), y) -> if_minus(null_le, s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(0)) -> if_mod(true, s(x), s(0)) [2]
   mod(s(0), s(s(x'))) -> if_mod(false, s(0), s(s(x'))) [2]
   mod(s(s(y'')), s(s(x''))) -> if_mod(le(x'', y''), s(s(y'')), s(s(x''))) [2]
   mod(s(x), s(y)) -> if_mod(null_le, s(x), s(y)) [1]
   if_mod(true, s(0), s(y)) -> mod(0, s(y)) [2]
   if_mod(true, s(s(x1)), s(y)) -> mod(if_minus(le(s(x1), y), s(x1), y), s(y)) [2]
   if_mod(true, s(x), s(y)) -> mod(null_minus, s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [1]
   encArg(0) -> 0 [0]
   encArg(true) -> true [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(false) -> false [0]
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_if_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_true -> true [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_false -> false [0]
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2)) [0]
   encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(v0) -> null_encArg [0]
   encode_le(v0, v1) -> null_encode_le [0]
   encode_0 -> null_encode_0 [0]
   encode_true -> null_encode_true [0]
   encode_s(v0) -> null_encode_s [0]
   encode_false -> null_encode_false [0]
   encode_minus(v0, v1) -> null_encode_minus [0]
   encode_if_minus(v0, v1, v2) -> null_encode_if_minus [0]
   encode_mod(v0, v1) -> null_encode_mod [0]
   encode_if_mod(v0, v1, v2) -> null_encode_if_mod [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   if_minus(v0, v1, v2) -> null_if_minus [0]
   mod(v0, v1) -> null_mod [0]
   if_mod(v0, v1, v2) -> null_if_mod [0]

The TRS has the following type information:
le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
cons_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
cons_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
cons_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
cons_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
cons_if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
encode_if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_encode_if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod
null_if_mod :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_mod:cons_if_mod:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_mod:null_encode_if_mod:null_le:null_minus:null_if_minus:null_mod:null_if_mod

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

   0 => 0
   true => 2
   false => 1
   null_encArg => 0
   null_encode_le => 0
   null_encode_0 => 0
   null_encode_true => 0
   null_encode_s => 0
   null_encode_false => 0
   null_encode_minus => 0
   null_encode_if_minus => 0
   null_encode_mod => 0
   null_encode_if_mod => 0
   null_le => 0
   null_minus => 0
   null_if_minus => 0
   null_mod => 0
   null_if_mod => 0

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_minus(z, z') -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_mod(z, z') -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_mod(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = 1 + x, z'' = y, x >= 0, y >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z' = 1 + x, z'' = y, z = 1, x >= 0, y >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + x1, y), 1 + x1, y), 1 + y) :|: z = 2, x1 >= 0, z' = 1 + (1 + x1), y >= 0, z'' = 1 + y
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + y) :|: z = 2, y >= 0, z'' = 1 + y, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + y) :|: z = 2, z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   if_mod(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + x :|: z' = 1 + x, z = 1, x >= 0, y >= 0, z'' = 1 + y
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 2 }-> if_minus(le(x, y'), 1 + x, 1 + y') :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   minus(z, z') -{ 2 }-> if_minus(1, 1 + x, 0) :|: x >= 0, z = 1 + x, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   mod(z, z') -{ 2 }-> if_mod(le(x'', y''), 1 + (1 + y''), 1 + (1 + x'')) :|: z' = 1 + (1 + x''), z = 1 + (1 + y''), y'' >= 0, x'' >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + x')) :|: z' = 1 + (1 + x'), z = 1 + 0, x' >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   mod(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   mod(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { encode_0 }
   { le }
   { encode_false }
   { encode_true }
   { minus, if_minus }
   { mod, if_mod }
   { encArg }
   { encode_if_mod }
   { encode_mod }
   { encode_if_minus }
   { encode_minus }
   { encode_le }
   { encode_s }

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: ?, size: O(1) [0]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: ?, size: O(1) [2]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_false}, {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_false}, {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_false}, {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: ?, size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_true}, {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: ?, size: O(1) [2]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]

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

(45) 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'

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus,if_minus}, {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: ?, size: O(n^1) [z]
  if_minus: runtime: ?, size: O(n^1) [z']

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 4*z + z*z' + 2*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: 6 + 4*z' + z'*z'' + z''

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> mod(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> mod(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {mod,if_mod}, {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: ?, size: O(n^1) [z]
  if_mod: runtime: ?, size: O(n^1) [z']

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

(53) 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: 3 + 32*z + 2*z*z' + 10*z^2 + 2*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: 10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> mod(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> mod(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> if_mod(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: ?, size: O(n^1) [1 + z]

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]

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

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_if_mod}, {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: ?, size: O(n^1) [1 + z']

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_if_mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']

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

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_mod}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: ?, size: O(n^1) [1 + z]

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]

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

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: ?, size: O(n^1) [1 + z']

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

(77) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_if_minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: O(n^4) [363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']

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

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: O(n^4) [363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']

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

(81) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: O(n^4) [363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_minus: runtime: ?, size: O(n^1) [1 + z]

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

(83) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 249 + 476*z + z*z' + 396*z^2 + 112*z^3 + 12*z^4 + 474*z' + 396*z'^2 + 112*z'^3 + 12*z'^4

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: O(n^4) [363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_minus: runtime: O(n^4) [249 + 476*z + z*z' + 396*z^2 + 112*z^3 + 12*z^4 + 474*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: O(n^4) [363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_minus: runtime: O(n^4) [249 + 476*z + z*z' + 396*z^2 + 112*z^3 + 12*z^4 + 474*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]

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(87) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_le}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: O(n^4) [363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_minus: runtime: O(n^4) [249 + 476*z + z*z' + 396*z^2 + 112*z^3 + 12*z^4 + 474*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: ?, size: O(1) [2]

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

(89) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_le
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 237 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 472*z' + 396*z'^2 + 112*z'^3 + 12*z'^4

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

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

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: O(n^4) [363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_minus: runtime: O(n^4) [249 + 476*z + z*z' + 396*z^2 + 112*z^3 + 12*z^4 + 474*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [237 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 472*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(1) [2]

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

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

(92)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: O(n^4) [363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_minus: runtime: O(n^4) [249 + 476*z + z*z' + 396*z^2 + 112*z^3 + 12*z^4 + 474*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [237 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 472*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(1) [2]

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

(93) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(94)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: O(n^4) [363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_minus: runtime: O(n^4) [249 + 476*z + z*z' + 396*z^2 + 112*z^3 + 12*z^4 + 474*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [237 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 472*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(1) [2]
  encode_s: runtime: ?, size: O(n^1) [2 + z]

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

(95) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4

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

(96)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 236 + s16 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s17 :|: s15 >= 0, s15 <= x_1 + 1, s16 >= 0, s16 <= x_2 + 1, s17 >= 0, s17 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 242 + 4*s18 + s18*s19 + 2*s19 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s20 :|: s18 >= 0, s18 <= x_1 + 1, s19 >= 0, s19 <= x_2 + 1, s20 >= 0, s20 <= s18, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 357 + 4*s22 + s22*s23 + s23 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s24 :|: s21 >= 0, s21 <= x_1 + 1, s22 >= 0, s22 <= x_2 + 1, s23 >= 0, s23 <= x_3 + 1, s24 >= 0, s24 <= s22, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 237 + 32*s25 + 2*s25*s26 + 10*s25^2 + 2*s25^2*s26 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 }-> s27 :|: s25 >= 0, s25 <= x_1 + 1, s26 >= 0, s26 <= x_2 + 1, s27 >= 0, s27 <= s25, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 361 + 15*s29 + 10*s29^2 + 2*s29^2*s30 + s30 + 471*x_1 + 396*x_1^2 + 112*x_1^3 + 12*x_1^4 + 471*x_2 + 396*x_2^2 + 112*x_2^3 + 12*x_2^4 + 471*x_3 + 396*x_3^2 + 112*x_3^3 + 12*x_3^4 }-> s31 :|: s28 >= 0, s28 <= x_1 + 1, s29 >= 0, s29 <= x_2 + 1, s30 >= 0, s30 <= x_3 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -58 + -33*z + 132*z^2 + 64*z^3 + 12*z^4 }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 357 + 4*s40 + s40*s41 + s41 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s42 :|: s39 >= 0, s39 <= z + 1, s40 >= 0, s40 <= z' + 1, s41 >= 0, s41 <= z'' + 1, s42 >= 0, s42 <= s40, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_if_mod(z, z', z'') -{ 361 + 15*s47 + 10*s47^2 + 2*s47^2*s48 + s48 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 471*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4 }-> s49 :|: s46 >= 0, s46 <= z + 1, s47 >= 0, s47 <= z' + 1, s48 >= 0, s48 <= z'' + 1, s49 >= 0, s49 <= s47, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 236 + s33 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s34 :|: s32 >= 0, s32 <= z + 1, s33 >= 0, s33 <= z' + 1, s34 >= 0, s34 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 242 + 4*s36 + s36*s37 + 2*s37 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s38 :|: s36 >= 0, s36 <= z + 1, s37 >= 0, s37 <= z' + 1, s38 >= 0, s38 <= s36, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 237 + 32*s43 + 2*s43*s44 + 10*s43^2 + 2*s43^2*s44 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 471*z' + 396*z'^2 + 112*z'^3 + 12*z'^4 }-> s45 :|: s43 >= 0, s43 <= z + 1, s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= s43, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 }-> 1 + s35 :|: s35 >= 0, s35 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 5 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 8 + 32*s6 + 2*s6*z'' + 10*s6^2 + 2*s6^2*z'' + 3*z' + z'*z'' + z'' }-> s12 :|: s12 >= 0, s12 <= s6, s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 4 }-> s13 :|: s13 >= 0, s13 <= 0, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 11 + 15*z + 10*z^2 + 2*z^2*z' + z' }-> s10 :|: s10 >= 0, s10 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 13 + 15*z + 12*z^2 }-> s7 :|: s7 >= 0, s7 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 37 + 3*z' }-> s8 :|: s8 >= 0, s8 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 12 + 16*z + 10*z^2 + 2*z^2*z' + z' }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + (z - 2)), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  le: runtime: O(n^1) [2 + z'], size: O(1) [2]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 32*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [10 + 15*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']
  encArg: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^4) [389 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 510*z' + 4*z'*z'' + 408*z'^2 + 2*z'^2*z'' + 112*z'^3 + 12*z'^4 + 474*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^4) [283 + 529*z + 6*z*z' + 408*z^2 + 2*z^2*z' + 112*z^3 + 12*z^4 + 475*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_if_minus: runtime: O(n^4) [363 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 476*z' + z'*z'' + 396*z'^2 + 112*z'^3 + 12*z'^4 + 473*z'' + 396*z''^2 + 112*z''^3 + 12*z''^4], size: O(n^1) [1 + z']
  encode_minus: runtime: O(n^4) [249 + 476*z + z*z' + 396*z^2 + 112*z^3 + 12*z^4 + 474*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [237 + 471*z + 396*z^2 + 112*z^3 + 12*z^4 + 472*z' + 396*z'^2 + 112*z'^3 + 12*z'^4], size: O(1) [2]
  encode_s: runtime: O(n^4) [117 + 471*z + 396*z^2 + 112*z^3 + 12*z^4], size: O(n^1) [2 + z]

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

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

(98)
BOUNDS(1, n^4)

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

(100)
Obligation:
Analyzing the following TRS for decreasing loops:

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


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)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_if_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

The rewrite sequence

le(s(x), s(y)) ->^+ le(x, y)

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

The pumping substitution is [x / s(x), y / s(y)].

The result substitution is [ ].




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

(102)
Complex Obligation (BEST)

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

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

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


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)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_if_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

(105)
BOUNDS(n^1, INF)

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

(106)
Obligation:
Analyzing the following TRS for decreasing loops:

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


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)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_if_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST