/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^3))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 256 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), 1693 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 2 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 411 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 98 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 56 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 251 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 135 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 21 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 77 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 1235 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 615 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 383 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 296 ms]
    (60) CpxRNTS
    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 259 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 13 ms]
    (66) CpxRNTS
    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 199 ms]
    (70) CpxRNTS
    (71) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (72) CpxRNTS
    (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 98 ms]
    (76) CpxRNTS
    (77) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (78) CpxRNTS
    (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (80) CpxRNTS
    (81) IntTrsBoundProof [UPPER BOUND(ID), 198 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), 166 ms]
    (88) CpxRNTS
    (89) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (90) CpxRNTS
    (91) FinalProof [FINISHED, 0 ms]
    (92) BOUNDS(1, n^3)
(93) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(94) CpxRelTRS
    (95) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) typed CpxTrs
    (97) OrderProof [LOWER BOUND(ID), 6 ms]
    (98) typed CpxTrs
    (99) RewriteLemmaProof [LOWER BOUND(ID), 296 ms]
    (100) BEST
        (101) proven lower bound
            (102) LowerBoundPropagationProof [FINISHED, 0 ms]
            (103) BOUNDS(n^1, INF)
        (104) typed CpxTrs
            (105) RewriteLemmaProof [LOWER BOUND(ID), 28 ms]
            (106) typed CpxTrs
            (107) RewriteLemmaProof [LOWER BOUND(ID), 144 ms]
            (108) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, x, y) -> mod(minus(x, y), y)
   if_mod(false, s(x), s(y)) -> s(x)

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

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

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


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, x, y) -> mod(minus(x, y), y)
   if_mod(false, s(x), s(y)) -> s(x)

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

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

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, x, y) -> mod(minus(x, y), y)
   if_mod(false, s(x), s(y)) -> s(x)

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

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

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


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

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

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

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

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

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

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

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

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

Due to the following rules being added:

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

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

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

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

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

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

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

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

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

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

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_mod => 0
   null_encode_if_mod => 0
   null_minus => 0
   null_le => 0
   null_mod => 0
   null_if_mod => 0

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

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

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


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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

(27) 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

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

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

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

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

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

(29) 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

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

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

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

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

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

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

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

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

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

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

(33) 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

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

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

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

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

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

(35) 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'

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

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

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

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

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

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

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

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

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

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

(39) 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

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

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

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

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

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

(41) 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

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

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

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

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

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

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

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

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

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

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

(45) 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

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

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

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

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

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

(47) 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

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

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

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

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

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

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

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

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

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

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 9 + 13*z + z*z' + z^2

Computed RUNTIME bound using KoAT for: if_mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 22 + 24*z' + z'*z'' + 2*z'^2

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

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

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

Function symbols to be analyzed: {encArg}, {encode_if_mod}, {encode_mod}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 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_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_if_mod}, {encode_mod}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 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_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_if_mod}, {encode_mod}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: ?, size: O(n^1) [1 + z]

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 78 + 283*z + 152*z^2 + 15*z^3

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

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

   encArg(z) -{ 0 }-> mod(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> if_mod(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 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_mod(z, z', z'') -{ 0 }-> if_mod(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> mod(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_if_mod}, {encode_mod}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]

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

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

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

   encArg(z) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_if_mod}, {encode_mod}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_if_mod}, {encode_mod}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: ?, size: O(n^1) [1 + z']

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_if_mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3

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

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

   encArg(z) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_mod}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], 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) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_mod}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_mod}, {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: ?, size: O(n^1) [1 + z]

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3

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

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

   encArg(z) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^3) [180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], 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) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^3) [180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], 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) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^3) [180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], 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^3) with polynomial bound: 158 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3

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

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

   encArg(z) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^3) [180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^3) [158 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], 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) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^3) [180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^3) [158 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], 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) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_le}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^3) [180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^3) [158 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], 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^3) with polynomial bound: 159 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3

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

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

   encArg(z) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^3) [180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^3) [158 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^3) [159 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(1) [2]

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

(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) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^3) [180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^3) [158 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^3) [159 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(1) [2]

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

(87) 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

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

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

   encArg(z) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^3) [180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^3) [158 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^3) [159 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(1) [2]
  encode_s: runtime: ?, size: O(n^1) [2 + z]

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

(89) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 78 + 283*z + 152*z^2 + 15*z^3

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

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

   encArg(z) -{ 158 + s11 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 157 + s14 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s15 :|: s13 >= 0, s13 <= x_1 + 1, s14 >= 0, s14 <= x_2 + 1, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 165 + 13*s16 + s16*s17 + s16^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 }-> s18 :|: s16 >= 0, s16 <= x_1 + 1, s17 >= 0, s17 <= x_2 + 1, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 256 + 24*s20 + s20*s21 + 2*s20^2 + 283*x_1 + 152*x_1^2 + 15*x_1^3 + 283*x_2 + 152*x_2^2 + 15*x_2^3 + 283*x_3 + 152*x_3^2 + 15*x_3^3 }-> s22 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= x_3 + 1, s22 >= 0, s22 <= s20, 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) -{ -68 + 24*z + 107*z^2 + 15*z^3 }-> 1 + s9 :|: s9 >= 0, s9 <= z - 1 + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_if_mod(z, z', z'') -{ 256 + 24*s34 + s34*s35 + 2*s34^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 + 283*z'' + 152*z''^2 + 15*z''^3 }-> s36 :|: s33 >= 0, s33 <= z + 1, s34 >= 0, s34 <= z' + 1, s35 >= 0, s35 <= z'' + 1, s36 >= 0, s36 <= s34, z >= 0, z'' >= 0, z' >= 0
   encode_if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_le(z, z') -{ 158 + s24 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s25 :|: s23 >= 0, s23 <= z + 1, s24 >= 0, s24 <= z' + 1, s25 >= 0, s25 <= 2, z >= 0, z' >= 0
   encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 157 + s28 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s29 :|: s27 >= 0, s27 <= z + 1, s28 >= 0, s28 <= z' + 1, s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_mod(z, z') -{ 165 + 13*s30 + s30*s31 + s30^2 + 283*z + 152*z^2 + 15*z^3 + 283*z' + 152*z'^2 + 15*z'^3 }-> s32 :|: s30 >= 0, s30 <= z + 1, s31 >= 0, s31 <= z' + 1, s32 >= 0, s32 <= s30, z >= 0, z' >= 0
   encode_mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 78 + 283*z + 152*z^2 + 15*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z + 1, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   if_mod(z, z', z'') -{ 11 + 13*z' + z'^2 }-> s6 :|: s6 >= 0, s6 <= z', z = 2, z'' = 0, z' >= 0
   if_mod(z, z', z'') -{ 11 + 13*s' + s'*z'' + s'^2 + z'' }-> s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z' - 1, z = 2, z' - 1 >= 0, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 10 }-> s8 :|: s8 >= 0, s8 <= 0, z = 2, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0
   le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 24 + 25*z + 2*z^2 }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 50 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0, z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 24 + 25*z + z*z' + 2*z^2 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + (z - 2)), s1 >= 0, s1 <= 2, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 23 + 24*z + z*z' + 2*z^2 }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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]
  mod: runtime: O(n^2) [9 + 13*z + z*z' + z^2], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [22 + 24*z' + z'*z'' + 2*z'^2], size: O(n^1) [z']
  encArg: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [1 + z]
  encode_if_mod: runtime: O(n^3) [283 + 283*z + 152*z^2 + 15*z^3 + 312*z' + z'*z'' + 154*z'^2 + 15*z'^3 + 284*z'' + 152*z''^2 + 15*z''^3], size: O(n^1) [1 + z']
  encode_mod: runtime: O(n^3) [180 + 299*z + z*z' + 153*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_minus: runtime: O(n^3) [158 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(n^1) [1 + z]
  encode_le: runtime: O(n^3) [159 + 283*z + 152*z^2 + 15*z^3 + 284*z' + 152*z'^2 + 15*z'^3], size: O(1) [2]
  encode_s: runtime: O(n^3) [78 + 283*z + 152*z^2 + 15*z^3], size: O(n^1) [2 + z]

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

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

(92)
BOUNDS(1, n^3)

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

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

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


The TRS R consists of the following rules:

   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(0', y) -> 0'
   mod(s(x), 0') -> 0'
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, x, y) -> mod(minus(x, y), y)
   if_mod(false, s(x), s(y)) -> s(x)

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

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

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

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

(96)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
if_mod(true, x, y) -> mod(minus(x, y), y)
if_mod(false, s(x), s(y)) -> s(x)
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_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
0' :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
true :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
s :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
false :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encArg :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_0 :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_true :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_s :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_false :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
hole_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod1_4 :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4 :: Nat -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod

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

(97) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, minus, mod, if_mod, encArg

They will be analysed ascendingly in the following order:
le < mod
le < encArg
minus < if_mod
minus < encArg
mod = if_mod
mod < encArg
if_mod < encArg

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

(98)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
if_mod(true, x, y) -> mod(minus(x, y), y)
if_mod(false, s(x), s(y)) -> s(x)
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_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
0' :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
true :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
s :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
false :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encArg :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_0 :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_true :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_s :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_false :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
hole_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod1_4 :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4 :: Nat -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod


Generator Equations:
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0) <=> 0'
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(x))


The following defined symbols remain to be analysed:
le, minus, mod, if_mod, encArg

They will be analysed ascendingly in the following order:
le < mod
le < encArg
minus < if_mod
minus < encArg
mod = if_mod
mod < encArg
if_mod < encArg

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

(99) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n4_4), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n4_4)) -> true, rt in Omega(1 + n4_4)

Induction Base:
le(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(+(n4_4, 1)), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(+(n4_4, 1))) ->_R^Omega(1)
le(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n4_4), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n4_4)) ->_IH
true

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

(100)
Complex Obligation (BEST)

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

(101)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
if_mod(true, x, y) -> mod(minus(x, y), y)
if_mod(false, s(x), s(y)) -> s(x)
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_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
0' :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
true :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
s :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
false :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encArg :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_0 :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_true :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_s :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_false :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
hole_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod1_4 :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4 :: Nat -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod


Generator Equations:
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0) <=> 0'
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(x))


The following defined symbols remain to be analysed:
le, minus, mod, if_mod, encArg

They will be analysed ascendingly in the following order:
le < mod
le < encArg
minus < if_mod
minus < encArg
mod = if_mod
mod < encArg
if_mod < encArg

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

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

(103)
BOUNDS(n^1, INF)

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

(104)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
if_mod(true, x, y) -> mod(minus(x, y), y)
if_mod(false, s(x), s(y)) -> s(x)
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_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
0' :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
true :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
s :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
false :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encArg :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_0 :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_true :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_s :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_false :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
hole_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod1_4 :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4 :: Nat -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod


Lemmas:
le(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n4_4), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n4_4)) -> true, rt in Omega(1 + n4_4)


Generator Equations:
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0) <=> 0'
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(x))


The following defined symbols remain to be analysed:
minus, mod, if_mod, encArg

They will be analysed ascendingly in the following order:
minus < if_mod
minus < encArg
mod = if_mod
mod < encArg
if_mod < encArg

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

(105) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n503_4), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n503_4)) -> gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0), rt in Omega(1 + n503_4)

Induction Base:
minus(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0)) ->_R^Omega(1)
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0)

Induction Step:
minus(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(+(n503_4, 1)), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(+(n503_4, 1))) ->_R^Omega(1)
minus(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n503_4), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n503_4)) ->_IH
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0)

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

(106)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
if_mod(true, x, y) -> mod(minus(x, y), y)
if_mod(false, s(x), s(y)) -> s(x)
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_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
encArg(cons_if_mod(x_1, x_2, x_3)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))
encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_true -> true
encode_s(x_1) -> s(encArg(x_1))
encode_false -> false
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
encode_if_mod(x_1, x_2, x_3) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
0' :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
true :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
s :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
false :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encArg :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
cons_if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_le :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_0 :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_true :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_s :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_false :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_minus :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
encode_if_mod :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
hole_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod1_4 :: 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4 :: Nat -> 0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod


Lemmas:
le(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n4_4), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n4_4)) -> true, rt in Omega(1 + n4_4)
minus(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n503_4), gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n503_4)) -> gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0), rt in Omega(1 + n503_4)


Generator Equations:
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0) <=> 0'
gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(x))


The following defined symbols remain to be analysed:
if_mod, mod, encArg

They will be analysed ascendingly in the following order:
mod = if_mod
mod < encArg
if_mod < encArg

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(107) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n1207_4)) -> gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n1207_4), rt in Omega(0)

Induction Base:
encArg(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(+(n1207_4, 1))) ->_R^Omega(0)
s(encArg(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(n1207_4))) ->_IH
s(gen_0':true:s:false:cons_le:cons_minus:cons_mod:cons_if_mod2_4(c1208_4))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(108)
BOUNDS(1, INF)