/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 334 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), 2954 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 5 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 1 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 76 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 19 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 251 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 176 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 729 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 282 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 381 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 311 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
    (60) CpxRNTS
    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 901 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 662 ms]
    (66) CpxRNTS
    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 597 ms]
    (70) CpxRNTS
    (71) IntTrsBoundProof [UPPER BOUND(ID), 80 ms]
    (72) CpxRNTS
    (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 208 ms]
    (76) CpxRNTS
    (77) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (78) CpxRNTS
    (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (80) CpxRNTS
    (81) IntTrsBoundProof [UPPER BOUND(ID), 136 ms]
    (82) CpxRNTS
    (83) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (84) CpxRNTS
    (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (86) CpxRNTS
    (87) IntTrsBoundProof [UPPER BOUND(ID), 217 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), 67 ms]
    (94) CpxRNTS
    (95) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (96) CpxRNTS
    (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (98) CpxRNTS
    (99) IntTrsBoundProof [UPPER BOUND(ID), 188 ms]
    (100) CpxRNTS
    (101) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (102) CpxRNTS
    (103) FinalProof [FINISHED, 0 ms]
    (104) BOUNDS(1, n^4)
(105) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(106) TRS for Loop Detection
    (107) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (108) BEST
        (109) proven lower bound
            (110) LowerBoundPropagationProof [FINISHED, 0 ms]
            (111) BOUNDS(n^1, INF)
        (112) 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))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   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_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_log(x_1) -> log(encArg(x_1))


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

(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))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   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_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_log(x_1) -> log(encArg(x_1))

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))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   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_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_log(x_1) -> log(encArg(x_1))

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]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [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_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [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_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [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]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [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_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [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_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [0]

The TRS has the following type information:
le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
cons_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
cons_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
cons_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
cons_quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
cons_log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
encode_quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log
encode_log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log

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
   quot_2
   if_minus_3
   log_1
   encArg_1
   encode_le_2
   encode_0
   encode_true
   encode_s_1
   encode_false
   encode_minus_2
   encode_if_minus_3
   encode_quot_2
   encode_log_1

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_quot(v0, v1) -> null_encode_quot [0]
   encode_log(v0) -> null_encode_log [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   if_minus(v0, v1, v2) -> null_if_minus [0]
   log(v0) -> null_log [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_quot, null_encode_log, null_le, null_minus, null_quot, null_if_minus, null_log

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

(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]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [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_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [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_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [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_quot(v0, v1) -> null_encode_quot [0]
   encode_log(v0) -> null_encode_log [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   if_minus(v0, v1, v2) -> null_if_minus [0]
   log(v0) -> null_log [0]

The TRS has the following type information:
le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
cons_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
cons_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
cons_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
cons_quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
cons_log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log

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]
   quot(0, s(y)) -> 0 [1]
   quot(s(0), s(y)) -> s(quot(0, s(y))) [2]
   quot(s(s(x')), s(y)) -> s(quot(if_minus(le(s(x'), y), s(x'), y), s(y))) [2]
   quot(s(x), s(y)) -> s(quot(null_minus, s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(0))) -> s(log(s(0))) [2]
   log(s(s(s(x'')))) -> s(log(s(s(quot(minus(x'', s(0)), s(s(0))))))) [2]
   log(s(s(x))) -> s(log(s(null_quot))) [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_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_log(0)) -> log(0) [0]
   encArg(cons_log(true)) -> log(true) [0]
   encArg(cons_log(s(x_1863))) -> log(s(encArg(x_1863))) [0]
   encArg(cons_log(false)) -> log(false) [0]
   encArg(cons_log(cons_le(x_1864, x_2575))) -> log(le(encArg(x_1864), encArg(x_2575))) [0]
   encArg(cons_log(cons_minus(x_1865, x_2576))) -> log(minus(encArg(x_1865), encArg(x_2576))) [0]
   encArg(cons_log(cons_if_minus(x_1866, x_2577, x_3143))) -> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) [0]
   encArg(cons_log(cons_quot(x_1867, x_2578))) -> log(quot(encArg(x_1867), encArg(x_2578))) [0]
   encArg(cons_log(cons_log(x_1868))) -> log(log(encArg(x_1868))) [0]
   encArg(cons_log(x_1)) -> log(null_encArg) [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_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) [0]
   encode_log(0) -> log(0) [0]
   encode_log(true) -> log(true) [0]
   encode_log(s(x_11733)) -> log(s(encArg(x_11733))) [0]
   encode_log(false) -> log(false) [0]
   encode_log(cons_le(x_11734, x_21155)) -> log(le(encArg(x_11734), encArg(x_21155))) [0]
   encode_log(cons_minus(x_11735, x_21156)) -> log(minus(encArg(x_11735), encArg(x_21156))) [0]
   encode_log(cons_if_minus(x_11736, x_21157, x_3288)) -> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) [0]
   encode_log(cons_quot(x_11737, x_21158)) -> log(quot(encArg(x_11737), encArg(x_21158))) [0]
   encode_log(cons_log(x_11738)) -> log(log(encArg(x_11738))) [0]
   encode_log(x_1) -> log(null_encArg) [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_quot(v0, v1) -> null_encode_quot [0]
   encode_log(v0) -> null_encode_log [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   if_minus(v0, v1, v2) -> null_if_minus [0]
   log(v0) -> null_log [0]

The TRS has the following type information:
le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
cons_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
cons_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
cons_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
cons_quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
cons_log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
encode_log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_encode_log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_quot :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log
null_log :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_quot:cons_log: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_quot:null_encode_log:null_le:null_minus:null_quot:null_if_minus:null_log

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

(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_quot => 0
   null_encode_log => 0
   null_le => 0
   null_minus => 0
   null_quot => 0
   null_if_minus => 0
   null_log => 0

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

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

   encArg(z) -{ 0 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(x_1868))) :|: z = 1 + (1 + x_1868), x_1868 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(x_1863)) :|: z = 1 + (1 + x_1863), x_1863 >= 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_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_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_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(x_11738))) :|: z = 1 + x_11738, x_11738 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: x_1 >= 0, z = x_1
   encode_log(z) -{ 0 }-> log(1 + encArg(x_11733)) :|: x_11733 >= 0, z = 1 + x_11733
   encode_log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   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_quot(z, z') -{ 0 }-> quot(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: x >= 0, z = 1 + (1 + x)
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(x'', 1 + 0), 1 + (1 + 0)))) :|: x'' >= 0, z = 1 + (1 + (1 + x''))
   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
   quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + x', y), 1 + x', y), 1 + y) :|: z' = 1 + y, x' >= 0, y >= 0, z = 1 + (1 + x')
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + y) :|: z' = 1 + y, z = 1 + 0, y >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x


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

(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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 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 }
   { quot }
   { log }
   { encArg }
   { encode_log }
   { encode_minus }
   { encode_le }
   { encode_if_minus }
   { encode_quot }
   { encode_s }

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

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

   encArg(z) -{ 0 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {encode_s}

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

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

   encArg(z) -{ 0 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {encode_s}

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(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

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

   encArg(z) -{ 0 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_false}, {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_false}, {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_false}, {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_true}, {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {minus,if_minus}, {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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
   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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -3 + 5*z }-> 1 + log(1 + (1 + quot(s6, 1 + (1 + 0)))) :|: s6 >= 0, s6 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 5 + 3*z + z*z' + z' }-> 1 + quot(s5, 1 + (z' - 1)) :|: s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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: quot
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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -3 + 5*z }-> 1 + log(1 + (1 + quot(s6, 1 + (1 + 0)))) :|: s6 >= 0, s6 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 5 + 3*z + z*z' + z' }-> 1 + quot(s5, 1 + (z' - 1)) :|: s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {quot}, {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: ?, size: O(n^1) [z]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -3 + 5*z }-> 1 + log(1 + (1 + quot(s6, 1 + (1 + 0)))) :|: s6 >= 0, s6 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 5 + 3*z + z*z' + z' }-> 1 + quot(s5, 1 + (z' - 1)) :|: s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*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 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -2 + 10*s6 + 5*s6^2 + 5*z }-> 1 + log(1 + (1 + s10)) :|: s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -2 + 10*s6 + 5*s6^2 + 5*z }-> 1 + log(1 + (1 + s10)) :|: s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {log}, {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: ?, size: O(n^1) [z]

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: log
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1 + 3*z + 15*z^2 + 5*z^3

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

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

   encArg(z) -{ 0 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> log(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> log(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> log(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(2) :|: z = 2
   encode_log(z) -{ 0 }-> log(1) :|: z = 1
   encode_log(z) -{ 0 }-> log(0) :|: z = 0
   encode_log(z) -{ 0 }-> log(0) :|: z >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -2 + 10*s6 + 5*s6^2 + 5*z }-> 1 + log(1 + (1 + s10)) :|: s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]

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

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]

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

(63) 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

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encArg}, {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: ?, size: O(n^1) [1 + z]

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

(65) 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: 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> quot(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 }-> log(quot(encArg(x_1867), encArg(x_2578))) :|: z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ 0 }-> log(minus(encArg(x_1865), encArg(x_2576))) :|: x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 0 }-> log(log(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> log(le(encArg(x_1864), encArg(x_2575))) :|: z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 0 }-> log(if_minus(encArg(x_1866), encArg(x_2577), encArg(x_3143))) :|: x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 0 }-> log(1 + encArg(z - 2)) :|: z - 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_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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ 0 }-> log(quot(encArg(x_11737), encArg(x_21158))) :|: x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ 0 }-> log(minus(encArg(x_11735), encArg(x_21156))) :|: z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 0 }-> log(log(encArg(z - 1))) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> log(le(encArg(x_11734), encArg(x_21155))) :|: x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 0 }-> log(if_minus(encArg(x_11736), encArg(x_21157), encArg(x_3288))) :|: z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 0 }-> log(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_log(z) -{ 0 }-> 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_quot(z, z') -{ 0 }-> quot(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_quot(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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*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) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]

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(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_log
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) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_log}, {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: ?, size: O(n^1) [1 + z]

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_log
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*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) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]

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

(75) 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

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: ?, size: O(n^1) [1 + z]

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

(77) 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: 849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*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) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]

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

(81) 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

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_le}, {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: ?, size: O(1) [2]

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

(83) 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: 837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(1) [2]

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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) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(1) [2]

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

(87) 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'

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_if_minus}, {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(1) [2]
  encode_if_minus: runtime: ?, size: O(n^1) [1 + z']

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

(89) 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: 1263 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1473*z' + z'*z'' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1470*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(1) [2]
  encode_if_minus: runtime: O(n^4) [1263 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1473*z' + z'*z'' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1470*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4], size: O(n^1) [1 + z']

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(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) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(1) [2]
  encode_if_minus: runtime: O(n^4) [1263 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1473*z' + z'*z'' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1470*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4], size: O(n^1) [1 + z']

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

(93) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {encode_quot}, {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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(1) [2]
  encode_if_minus: runtime: O(n^4) [1263 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1473*z' + z'*z'' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1470*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4], size: O(n^1) [1 + z']
  encode_quot: runtime: ?, size: O(n^1) [1 + z]

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

(95) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 848 + 1485*z + 3*z*z' + 778*z^2 + z^2*z' + 401*z^3 + 96*z^4 + 1470*z' + 774*z'^2 + 401*z'^3 + 96*z'^4

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

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

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(1) [2]
  encode_if_minus: runtime: O(n^4) [1263 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1473*z' + z'*z'' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1470*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4], size: O(n^1) [1 + z']
  encode_quot: runtime: O(n^4) [848 + 1485*z + 3*z*z' + 778*z^2 + z^2*z' + 401*z^3 + 96*z^4 + 1470*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]

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

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

(98)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(1) [2]
  encode_if_minus: runtime: O(n^4) [1263 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1473*z' + z'*z'' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1470*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4], size: O(n^1) [1 + z']
  encode_quot: runtime: O(n^4) [848 + 1485*z + 3*z*z' + 778*z^2 + z^2*z' + 401*z^3 + 96*z^4 + 1470*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]

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

(99) 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

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

(100)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(1) [2]
  encode_if_minus: runtime: O(n^4) [1263 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1473*z' + z'*z'' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1470*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4], size: O(n^1) [1 + z']
  encode_quot: runtime: O(n^4) [848 + 1485*z + 3*z*z' + 778*z^2 + z^2*z' + 401*z^3 + 96*z^4 + 1470*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_s: runtime: ?, size: O(n^1) [2 + z]

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

(101) 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: 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4

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

(102)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 1 }-> s14 :|: s14 >= 0, s14 <= 0, z = 1 + 0
   encArg(z) -{ 107 }-> s15 :|: s15 >= 0, s15 <= 2, z = 1 + 2
   encArg(z) -{ 24 }-> s16 :|: s16 >= 0, s16 <= 1, z = 1 + 1
   encArg(z) -{ 1 }-> s17 :|: s17 >= 0, s17 <= 0, z - 1 >= 0
   encArg(z) -{ 836 + s24 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s25 :|: s23 >= 0, s23 <= x_1 + 1, s24 >= 0, s24 <= x_2 + 1, s25 >= 0, s25 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 842 + 4*s26 + s26*s27 + 2*s27 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s28 :|: s26 >= 0, s26 <= x_1 + 1, s27 >= 0, s27 <= x_2 + 1, s28 >= 0, s28 <= s26, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1257 + 4*s30 + s30*s31 + s31 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 + 1468*x_3 + 774*x_3^2 + 401*x_3^3 + 96*x_3^4 }-> s32 :|: s29 >= 0, s29 <= x_1 + 1, s30 >= 0, s30 <= x_2 + 1, s31 >= 0, s31 <= x_3 + 1, s32 >= 0, s32 <= s30, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 835 + 8*s33 + s33*s34 + 3*s33^2 + s33^2*s34 + 1468*x_1 + 774*x_1^2 + 401*x_1^3 + 96*x_1^4 + 1468*x_2 + 774*x_2^2 + 401*x_2^3 + 96*x_2^4 }-> s35 :|: s33 >= 0, s33 <= x_1 + 1, s34 >= 0, s34 <= x_2 + 1, s35 >= 0, s35 <= s33, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ -1071 + 48*s36 + 30*s36^2 + 5*s36^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s37 :|: s36 >= 0, s36 <= z - 2 + 1, s37 >= 0, s37 <= 1 + s36, z - 2 >= 0
   encArg(z) -{ 837 + s39 + 3*s40 + 15*s40^2 + 5*s40^3 + 1468*x_1864 + 774*x_1864^2 + 401*x_1864^3 + 96*x_1864^4 + 1468*x_2575 + 774*x_2575^2 + 401*x_2575^3 + 96*x_2575^4 }-> s41 :|: s38 >= 0, s38 <= x_1864 + 1, s39 >= 0, s39 <= x_2575 + 1, s40 >= 0, s40 <= 2, s41 >= 0, s41 <= s40, z = 1 + (1 + x_1864 + x_2575), x_2575 >= 0, x_1864 >= 0
   encArg(z) -{ 843 + 4*s42 + s42*s43 + 2*s43 + 3*s44 + 15*s44^2 + 5*s44^3 + 1468*x_1865 + 774*x_1865^2 + 401*x_1865^3 + 96*x_1865^4 + 1468*x_2576 + 774*x_2576^2 + 401*x_2576^3 + 96*x_2576^4 }-> s45 :|: s42 >= 0, s42 <= x_1865 + 1, s43 >= 0, s43 <= x_2576 + 1, s44 >= 0, s44 <= s42, s45 >= 0, s45 <= s44, x_1865 >= 0, x_2576 >= 0, z = 1 + (1 + x_1865 + x_2576)
   encArg(z) -{ 1258 + 4*s47 + s47*s48 + s48 + 3*s49 + 15*s49^2 + 5*s49^3 + 1468*x_1866 + 774*x_1866^2 + 401*x_1866^3 + 96*x_1866^4 + 1468*x_2577 + 774*x_2577^2 + 401*x_2577^3 + 96*x_2577^4 + 1468*x_3143 + 774*x_3143^2 + 401*x_3143^3 + 96*x_3143^4 }-> s50 :|: s46 >= 0, s46 <= x_1866 + 1, s47 >= 0, s47 <= x_2577 + 1, s48 >= 0, s48 <= x_3143 + 1, s49 >= 0, s49 <= s47, s50 >= 0, s50 <= s49, x_2577 >= 0, x_3143 >= 0, x_1866 >= 0, z = 1 + (1 + x_1866 + x_2577 + x_3143)
   encArg(z) -{ 836 + 8*s51 + s51*s52 + 3*s51^2 + s51^2*s52 + 3*s53 + 15*s53^2 + 5*s53^3 + 1468*x_1867 + 774*x_1867^2 + 401*x_1867^3 + 96*x_1867^4 + 1468*x_2578 + 774*x_2578^2 + 401*x_2578^3 + 96*x_2578^4 }-> s54 :|: s51 >= 0, s51 <= x_1867 + 1, s52 >= 0, s52 <= x_2578 + 1, s53 >= 0, s53 <= s51, s54 >= 0, s54 <= s53, z = 1 + (1 + x_1867 + x_2578), x_1867 >= 0, x_2578 >= 0
   encArg(z) -{ -1093 + 3*s55 + 15*s55^2 + 5*s55^3 + 3*s56 + 15*s56^2 + 5*s56^3 + 112*z + 672*z^2 + -367*z^3 + 96*z^4 }-> s57 :|: s55 >= 0, s55 <= z - 2 + 1, s56 >= 0, s56 <= s55, s57 >= 0, s57 <= s56, z - 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) -{ -582 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> 1 + s22 :|: s22 >= 0, s22 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_minus(z, z', z'') -{ 1257 + 4*s66 + s66*s67 + s67 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1468*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4 }-> s68 :|: s65 >= 0, s65 <= z + 1, s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, s68 >= 0, s68 <= s66, z >= 0, z'' >= 0, z' >= 0
   encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 836 + s59 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s60 :|: s58 >= 0, s58 <= z + 1, s59 >= 0, s59 <= z' + 1, s60 >= 0, s60 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_log(z) -{ 1 }-> s18 :|: s18 >= 0, s18 <= 0, z = 0
   encode_log(z) -{ 107 }-> s19 :|: s19 >= 0, s19 <= 2, z = 2
   encode_log(z) -{ 24 }-> s20 :|: s20 >= 0, s20 <= 1, z = 1
   encode_log(z) -{ 1 }-> s21 :|: s21 >= 0, s21 <= 0, z >= 0
   encode_log(z) -{ -558 + 48*s72 + 30*s72^2 + 5*s72^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s73 :|: s72 >= 0, s72 <= z - 1 + 1, s73 >= 0, s73 <= 1 + s72, z - 1 >= 0
   encode_log(z) -{ 837 + s75 + 3*s76 + 15*s76^2 + 5*s76^3 + 1468*x_11734 + 774*x_11734^2 + 401*x_11734^3 + 96*x_11734^4 + 1468*x_21155 + 774*x_21155^2 + 401*x_21155^3 + 96*x_21155^4 }-> s77 :|: s74 >= 0, s74 <= x_11734 + 1, s75 >= 0, s75 <= x_21155 + 1, s76 >= 0, s76 <= 2, s77 >= 0, s77 <= s76, x_11734 >= 0, x_21155 >= 0, z = 1 + x_11734 + x_21155
   encode_log(z) -{ 843 + 4*s78 + s78*s79 + 2*s79 + 3*s80 + 15*s80^2 + 5*s80^3 + 1468*x_11735 + 774*x_11735^2 + 401*x_11735^3 + 96*x_11735^4 + 1468*x_21156 + 774*x_21156^2 + 401*x_21156^3 + 96*x_21156^4 }-> s81 :|: s78 >= 0, s78 <= x_11735 + 1, s79 >= 0, s79 <= x_21156 + 1, s80 >= 0, s80 <= s78, s81 >= 0, s81 <= s80, z = 1 + x_11735 + x_21156, x_21156 >= 0, x_11735 >= 0
   encode_log(z) -{ 1258 + 4*s83 + s83*s84 + s84 + 3*s85 + 15*s85^2 + 5*s85^3 + 1468*x_11736 + 774*x_11736^2 + 401*x_11736^3 + 96*x_11736^4 + 1468*x_21157 + 774*x_21157^2 + 401*x_21157^3 + 96*x_21157^4 + 1468*x_3288 + 774*x_3288^2 + 401*x_3288^3 + 96*x_3288^4 }-> s86 :|: s82 >= 0, s82 <= x_11736 + 1, s83 >= 0, s83 <= x_21157 + 1, s84 >= 0, s84 <= x_3288 + 1, s85 >= 0, s85 <= s83, s86 >= 0, s86 <= s85, z = 1 + x_11736 + x_21157 + x_3288, x_21157 >= 0, x_11736 >= 0, x_3288 >= 0
   encode_log(z) -{ 836 + 8*s87 + s87*s88 + 3*s87^2 + s87^2*s88 + 3*s89 + 15*s89^2 + 5*s89^3 + 1468*x_11737 + 774*x_11737^2 + 401*x_11737^3 + 96*x_11737^4 + 1468*x_21158 + 774*x_21158^2 + 401*x_21158^3 + 96*x_21158^4 }-> s90 :|: s87 >= 0, s87 <= x_11737 + 1, s88 >= 0, s88 <= x_21158 + 1, s89 >= 0, s89 <= s87, s90 >= 0, s90 <= s89, x_11737 >= 0, z = 1 + x_11737 + x_21158, x_21158 >= 0
   encode_log(z) -{ -580 + 3*s91 + 15*s91^2 + 5*s91^3 + 3*s92 + 15*s92^2 + 5*s92^3 + 739*z + 147*z^2 + 17*z^3 + 96*z^4 }-> s93 :|: s91 >= 0, s91 <= z - 1 + 1, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z - 1 >= 0
   encode_log(z) -{ 0 }-> 0 :|: z >= 0
   encode_minus(z, z') -{ 842 + 4*s62 + s62*s63 + 2*s63 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s64 :|: s62 >= 0, s62 <= z + 1, s63 >= 0, s63 <= z' + 1, s64 >= 0, s64 <= s62, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_quot(z, z') -{ 835 + 8*s69 + s69*s70 + 3*s69^2 + s69^2*s70 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1468*z' + 774*z'^2 + 401*z'^3 + 96*z'^4 }-> s71 :|: s69 >= 0, s69 <= z + 1, s70 >= 0, s70 <= z' + 1, s71 >= 0, s71 <= s69, z >= 0, z' >= 0
   encode_quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 }-> 1 + s61 :|: s61 >= 0, s61 <= 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 + s4 :|: s4 >= 0, s4 <= z' - 1, z = 1, z' - 1 >= 0, z'' >= 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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: z >= 0
   log(z) -{ 26 }-> 1 + s11 :|: s11 >= 0, s11 <= 1 + 0, z = 1 + (1 + 0)
   log(z) -{ 105 + 123*s10 + 45*s10^2 + 5*s10^3 + 10*s6 + 5*s6^2 + 5*z }-> 1 + s12 :|: s12 >= 0, s12 <= 1 + (1 + s10), s10 >= 0, s10 <= s6, s6 >= 0, s6 <= z - 3, z - 3 >= 0
   log(z) -{ 25 }-> 1 + s13 :|: s13 >= 0, s13 <= 1 + 0, z - 2 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= 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
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 8*s5 + s5*z' + 3*s5^2 + s5^2*z' + 3*z + z*z' + z' }-> 1 + s8 :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= 1 + (z - 2), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' - 1 >= 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']
  quot: runtime: O(n^3) [1 + 8*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 15*z^2 + 5*z^3], size: O(n^1) [z]
  encArg: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [1 + z]
  encode_log: runtime: O(n^4) [4040 + 14788*z + 7374*z^2 + 3674*z^3 + 1056*z^4], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^4) [849 + 1473*z + z*z' + 774*z^2 + 401*z^3 + 96*z^4 + 1471*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^4) [837 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1469*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(1) [2]
  encode_if_minus: runtime: O(n^4) [1263 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4 + 1473*z' + z'*z'' + 774*z'^2 + 401*z'^3 + 96*z'^4 + 1470*z'' + 774*z''^2 + 401*z''^3 + 96*z''^4], size: O(n^1) [1 + z']
  encode_quot: runtime: O(n^4) [848 + 1485*z + 3*z*z' + 778*z^2 + z^2*z' + 401*z^3 + 96*z^4 + 1470*z' + 774*z'^2 + 401*z'^3 + 96*z'^4], size: O(n^1) [1 + z]
  encode_s: runtime: O(n^4) [417 + 1468*z + 774*z^2 + 401*z^3 + 96*z^4], size: O(n^1) [2 + z]

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(103) FinalProof (FINISHED)
Computed overall runtime complexity
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(104)
BOUNDS(1, n^4)

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(105) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(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))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   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_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_log(x_1) -> log(encArg(x_1))

Rewrite Strategy: INNERMOST
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(107) 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 [ ].




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(108)
Complex Obligation (BEST)

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(109)
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))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   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_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_log(x_1) -> log(encArg(x_1))

Rewrite Strategy: INNERMOST
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(110) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(111)
BOUNDS(n^1, INF)

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(112)
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))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   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_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_log(x_1) -> log(encArg(x_1))

Rewrite Strategy: INNERMOST