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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 227 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), 6 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 24 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 353 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 461 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 114 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 72 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 22 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 1169 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 630 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 2 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 99 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 168 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 638 ms]
    (70) CpxRNTS
    (71) IntTrsBoundProof [UPPER BOUND(ID), 227 ms]
    (72) CpxRNTS
    (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 209 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), 86 ms]
    (82) CpxRNTS
    (83) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (84) CpxRNTS
    (85) FinalProof [FINISHED, 0 ms]
    (86) BOUNDS(1, n^2)
(87) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(88) TRS for Loop Detection
    (89) DecreasingLoopProof [LOWER BOUND(ID), 2 ms]
    (90) BEST
        (91) proven lower bound
            (92) LowerBoundPropagationProof [FINISHED, 0 ms]
            (93) BOUNDS(n^1, INF)
        (94) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

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(nil) -> nil
   encArg(0) -> 0
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_norm(x_1)) -> norm(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_rem(x_1, x_2)) -> rem(encArg(x_1), encArg(x_2))
   encode_norm(x_1) -> norm(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_rem(x_1, x_2) -> rem(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

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

   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_norm(x_1)) -> norm(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_rem(x_1, x_2)) -> rem(encArg(x_1), encArg(x_2))
   encode_norm(x_1) -> norm(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_rem(x_1, x_2) -> rem(encArg(x_1), encArg(x_2))

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^2).


The TRS R consists of the following rules:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

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

   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_norm(x_1)) -> norm(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_rem(x_1, x_2)) -> rem(encArg(x_1), encArg(x_2))
   encode_norm(x_1) -> norm(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_rem(x_1, x_2) -> rem(encArg(x_1), encArg(x_2))

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^2).


The TRS R consists of the following rules:

   norm(nil) -> 0 [1]
   norm(g(x, y)) -> s(norm(x)) [1]
   f(x, nil) -> g(nil, x) [1]
   f(x, g(y, z)) -> g(f(x, y), z) [1]
   rem(nil, y) -> nil [1]
   rem(g(x, y), 0) -> g(x, y) [1]
   rem(g(x, y), s(z)) -> rem(x, z) [1]
   encArg(nil) -> nil [0]
   encArg(0) -> 0 [0]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_norm(x_1)) -> norm(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rem(x_1, x_2)) -> rem(encArg(x_1), encArg(x_2)) [0]
   encode_norm(x_1) -> norm(encArg(x_1)) [0]
   encode_nil -> nil [0]
   encode_0 -> 0 [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_rem(x_1, x_2) -> rem(encArg(x_1), encArg(x_2)) [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:

   norm(nil) -> 0 [1]
   norm(g(x, y)) -> s(norm(x)) [1]
   f(x, nil) -> g(nil, x) [1]
   f(x, g(y, z)) -> g(f(x, y), z) [1]
   rem(nil, y) -> nil [1]
   rem(g(x, y), 0) -> g(x, y) [1]
   rem(g(x, y), s(z)) -> rem(x, z) [1]
   encArg(nil) -> nil [0]
   encArg(0) -> 0 [0]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_norm(x_1)) -> norm(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rem(x_1, x_2)) -> rem(encArg(x_1), encArg(x_2)) [0]
   encode_norm(x_1) -> norm(encArg(x_1)) [0]
   encode_nil -> nil [0]
   encode_0 -> 0 [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_rem(x_1, x_2) -> rem(encArg(x_1), encArg(x_2)) [0]

The TRS has the following type information:
norm :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
nil :: nil:0:g:s:cons_norm:cons_f:cons_rem
0 :: nil:0:g:s:cons_norm:cons_f:cons_rem
g :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
s :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
f :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
rem :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
encArg :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
cons_norm :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
cons_f :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
cons_rem :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
encode_norm :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
encode_nil :: nil:0:g:s:cons_norm:cons_f:cons_rem
encode_0 :: nil:0:g:s:cons_norm:cons_f:cons_rem
encode_g :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
encode_s :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
encode_f :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem
encode_rem :: nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem -> nil:0:g:s:cons_norm:cons_f:cons_rem

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:

   rem_2
   norm_1
   f_2
   encArg_1
   encode_norm_1
   encode_nil
   encode_0
   encode_g_2
   encode_s_1
   encode_f_2
   encode_rem_2

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_norm(v0) -> null_encode_norm [0]
   encode_nil -> null_encode_nil [0]
   encode_0 -> null_encode_0 [0]
   encode_g(v0, v1) -> null_encode_g [0]
   encode_s(v0) -> null_encode_s [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_rem(v0, v1) -> null_encode_rem [0]
   rem(v0, v1) -> null_rem [0]
   norm(v0) -> null_norm [0]
   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_norm, null_encode_nil, null_encode_0, null_encode_g, null_encode_s, null_encode_f, null_encode_rem, null_rem, null_norm, null_f

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

(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:

   norm(nil) -> 0 [1]
   norm(g(x, y)) -> s(norm(x)) [1]
   f(x, nil) -> g(nil, x) [1]
   f(x, g(y, z)) -> g(f(x, y), z) [1]
   rem(nil, y) -> nil [1]
   rem(g(x, y), 0) -> g(x, y) [1]
   rem(g(x, y), s(z)) -> rem(x, z) [1]
   encArg(nil) -> nil [0]
   encArg(0) -> 0 [0]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_norm(x_1)) -> norm(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rem(x_1, x_2)) -> rem(encArg(x_1), encArg(x_2)) [0]
   encode_norm(x_1) -> norm(encArg(x_1)) [0]
   encode_nil -> nil [0]
   encode_0 -> 0 [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_rem(x_1, x_2) -> rem(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> null_encArg [0]
   encode_norm(v0) -> null_encode_norm [0]
   encode_nil -> null_encode_nil [0]
   encode_0 -> null_encode_0 [0]
   encode_g(v0, v1) -> null_encode_g [0]
   encode_s(v0) -> null_encode_s [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_rem(v0, v1) -> null_encode_rem [0]
   rem(v0, v1) -> null_rem [0]
   norm(v0) -> null_norm [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
norm :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
nil :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
0 :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
g :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
s :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
f :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
rem :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encArg :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
cons_norm :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
cons_f :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
cons_rem :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_norm :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_nil :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_0 :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_g :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_s :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_f :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_rem :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encArg :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_norm :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_nil :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_0 :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_g :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_s :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_f :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_rem :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_rem :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_norm :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_f :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f

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:

   norm(nil) -> 0 [1]
   norm(g(x, y)) -> s(norm(x)) [1]
   f(x, nil) -> g(nil, x) [1]
   f(x, g(y, z)) -> g(f(x, y), z) [1]
   rem(nil, y) -> nil [1]
   rem(g(x, y), 0) -> g(x, y) [1]
   rem(g(x, y), s(z)) -> rem(x, z) [1]
   encArg(nil) -> nil [0]
   encArg(0) -> 0 [0]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_norm(nil)) -> norm(nil) [0]
   encArg(cons_norm(0)) -> norm(0) [0]
   encArg(cons_norm(g(x_1', x_2'))) -> norm(g(encArg(x_1'), encArg(x_2'))) [0]
   encArg(cons_norm(s(x_1''))) -> norm(s(encArg(x_1''))) [0]
   encArg(cons_norm(cons_norm(x_11))) -> norm(norm(encArg(x_11))) [0]
   encArg(cons_norm(cons_f(x_12, x_2''))) -> norm(f(encArg(x_12), encArg(x_2''))) [0]
   encArg(cons_norm(cons_rem(x_13, x_21))) -> norm(rem(encArg(x_13), encArg(x_21))) [0]
   encArg(cons_norm(x_1)) -> norm(null_encArg) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rem(x_1, x_2)) -> rem(encArg(x_1), encArg(x_2)) [0]
   encode_norm(nil) -> norm(nil) [0]
   encode_norm(0) -> norm(0) [0]
   encode_norm(g(x_194, x_256)) -> norm(g(encArg(x_194), encArg(x_256))) [0]
   encode_norm(s(x_195)) -> norm(s(encArg(x_195))) [0]
   encode_norm(cons_norm(x_196)) -> norm(norm(encArg(x_196))) [0]
   encode_norm(cons_f(x_197, x_257)) -> norm(f(encArg(x_197), encArg(x_257))) [0]
   encode_norm(cons_rem(x_198, x_258)) -> norm(rem(encArg(x_198), encArg(x_258))) [0]
   encode_norm(x_1) -> norm(null_encArg) [0]
   encode_nil -> nil [0]
   encode_0 -> 0 [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_rem(x_1, x_2) -> rem(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> null_encArg [0]
   encode_norm(v0) -> null_encode_norm [0]
   encode_nil -> null_encode_nil [0]
   encode_0 -> null_encode_0 [0]
   encode_g(v0, v1) -> null_encode_g [0]
   encode_s(v0) -> null_encode_s [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_rem(v0, v1) -> null_encode_rem [0]
   rem(v0, v1) -> null_rem [0]
   norm(v0) -> null_norm [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
norm :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
nil :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
0 :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
g :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
s :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
f :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
rem :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encArg :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
cons_norm :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
cons_f :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
cons_rem :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_norm :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_nil :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_0 :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_g :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_s :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_f :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
encode_rem :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f -> nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encArg :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_norm :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_nil :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_0 :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_g :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_s :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_f :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_encode_rem :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_rem :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_norm :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f
null_f :: nil:0:g:s:cons_norm:cons_f:cons_rem:null_encArg:null_encode_norm:null_encode_nil:null_encode_0:null_encode_g:null_encode_s:null_encode_f:null_encode_rem:null_rem:null_norm:null_f

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:

   nil => 1
   0 => 0
   null_encArg => 0
   null_encode_norm => 0
   null_encode_nil => 0
   null_encode_0 => 0
   null_encode_g => 0
   null_encode_s => 0
   null_encode_f => 0
   null_encode_rem => 0
   null_rem => 0
   null_norm => 0
   null_f => 0

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(x_11))) :|: z' = 1 + (1 + x_11), x_11 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: x_1 >= 0, z' = 1 + x_1
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1'')) :|: x_1'' >= 0, z' = 1 + (1 + x_1'')
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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) :|: x_1 >= 0, z' = 1 + x_1
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_g(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(x_196))) :|: z' = 1 + x_196, x_196 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: x_1 >= 0, z' = x_1
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_195)) :|: z' = 1 + x_195, x_195 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_rem(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_s(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_s(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = x_1
   f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   f(z', z'') -{ 1 }-> 1 + f(x, y) + z :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + x :|: z' = x, x >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z) :|: z >= 0, z' = 1 + x + y, x >= 0, y >= 0, z'' = 1 + z
   rem(z', z'') -{ 1 }-> 1 :|: z'' = y, y >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0


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

(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 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0


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

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

   { encode_0 }
   { f }
   { rem }
   { norm }
   { encode_nil }
   { encArg }
   { encode_rem }
   { encode_g }
   { encode_norm }
   { encode_f }
   { encode_s }

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_0}, {f}, {rem}, {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {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 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_0}, {f}, {rem}, {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_0}, {f}, {rem}, {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: ?, size: O(1) [0]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}, {rem}, {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}, {rem}, {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}, {rem}, {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: ?, size: O(n^1) [1 + z' + z'']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {rem}, {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']

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

(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 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {rem}, {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {rem}, {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: ?, size: O(n^1) [z']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']

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

(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 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {norm}, {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: ?, size: O(n^1) [z']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> norm(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> norm(0) :|: z' - 1 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1) :|: z' = 1
   encode_norm(z') -{ 0 }-> norm(0) :|: z' = 0
   encode_norm(z') -{ 0 }-> norm(0) :|: z' >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 1 }-> 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']

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

(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') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_nil}, {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: ?, size: O(1) [1]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_nil
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') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]

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

(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') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 4*z'

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encArg}, {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: ?, size: O(n^1) [1 + 4*z']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2 + 26*z' + 16*z'^2

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 0 }-> rem(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> norm(rem(encArg(x_13), encArg(x_21))) :|: z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> norm(norm(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(f(encArg(x_12), encArg(x_2''))) :|: z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(z' - 2)) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> norm(1 + encArg(x_1') + encArg(x_2')) :|: x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_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
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> norm(rem(encArg(x_198), encArg(x_258))) :|: x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 0 }-> norm(norm(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(f(encArg(x_197), encArg(x_257))) :|: x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 0 }-> norm(1 + encArg(z' - 1)) :|: z' - 1 >= 0
   encode_norm(z') -{ 0 }-> norm(1 + encArg(x_194) + encArg(x_256)) :|: z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 0 }-> rem(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*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') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_rem
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 4*z'

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_rem}, {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: ?, size: O(n^1) [1 + 4*z']

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_rem
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*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') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 4*z' + 4*z''

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_g}, {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: ?, size: O(n^1) [3 + 4*z' + 4*z'']

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: O(n^2) [4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*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') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: O(n^2) [4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: encode_norm
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4*z'

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_norm}, {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: O(n^2) [4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']
  encode_norm: runtime: ?, size: O(n^1) [4*z']

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_norm
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 27 + 186*z' + 128*z'^2

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: O(n^2) [4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']
  encode_norm: runtime: O(n^2) [27 + 186*z' + 128*z'^2], size: O(n^1) [4*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') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: O(n^2) [4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']
  encode_norm: runtime: O(n^2) [27 + 186*z' + 128*z'^2], size: O(n^1) [4*z']

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 4*z' + 4*z''

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_f}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: O(n^2) [4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']
  encode_norm: runtime: O(n^2) [27 + 186*z' + 128*z'^2], size: O(n^1) [4*z']
  encode_f: runtime: ?, size: O(n^1) [3 + 4*z' + 4*z'']

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

(77) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: O(n^2) [4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']
  encode_norm: runtime: O(n^2) [27 + 186*z' + 128*z'^2], size: O(n^1) [4*z']
  encode_f: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*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') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: O(n^2) [4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']
  encode_norm: runtime: O(n^2) [27 + 186*z' + 128*z'^2], size: O(n^1) [4*z']
  encode_f: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']

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

(81) 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 + 4*z'

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: O(n^2) [4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']
  encode_norm: runtime: O(n^2) [27 + 186*z' + 128*z'^2], size: O(n^1) [4*z']
  encode_f: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']
  encode_s: runtime: ?, size: O(n^1) [2 + 4*z']

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

(83) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2 + 26*z' + 16*z'^2

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

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

   encArg(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= 1, z' = 1 + 1
   encArg(z') -{ 6 + s10 + s11 + 26*x_1' + 16*x_1'^2 + 26*x_2' + 16*x_2'^2 }-> s12 :|: s10 >= 0, s10 <= 4 * x_1' + 1, s11 >= 0, s11 <= 4 * x_2' + 1, s12 >= 0, s12 <= 1 + s10 + s11, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2')
   encArg(z') -{ 16 + s13 + -38*z' + 16*z'^2 }-> s14 :|: s13 >= 0, s13 <= 4 * (z' - 2) + 1, s14 >= 0, s14 <= 1 + s13, z' - 2 >= 0
   encArg(z') -{ 16 + s15 + s16 + -38*z' + 16*z'^2 }-> s17 :|: s15 >= 0, s15 <= 4 * (z' - 2) + 1, s16 >= 0, s16 <= s15, s17 >= 0, s17 <= s16, z' - 2 >= 0
   encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z' = 1 + 0
   encArg(z') -{ 6 + s19 + s20 + 26*x_12 + 16*x_12^2 + 26*x_2'' + 16*x_2''^2 }-> s21 :|: s18 >= 0, s18 <= 4 * x_12 + 1, s19 >= 0, s19 <= 4 * x_2'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= s20, z' = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z') -{ 6 + s23 + s24 + 26*x_13 + 16*x_13^2 + 26*x_21 + 16*x_21^2 }-> s25 :|: s22 >= 0, s22 <= 4 * x_13 + 1, s23 >= 0, s23 <= 4 * x_21 + 1, s24 >= 0, s24 <= s22, s25 >= 0, s25 <= s24, z' = 1 + (1 + x_13 + x_21), x_13 >= 0, x_21 >= 0
   encArg(z') -{ 5 + s27 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s28 :|: s26 >= 0, s26 <= 4 * x_1 + 1, s27 >= 0, s27 <= 4 * x_2 + 1, s28 >= 0, s28 <= s26 + s27 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0
   encArg(z') -{ 5 + s30 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> s31 :|: s29 >= 0, s29 <= 4 * x_1 + 1, s30 >= 0, s30 <= 4 * x_2 + 1, s31 >= 0, s31 <= s29, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ -8 + -6*z' + 16*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 4 * (z' - 1) + 1, z' - 1 >= 0
   encArg(z') -{ 4 + 26*x_1 + 16*x_1^2 + 26*x_2 + 16*x_2^2 }-> 1 + s7 + s8 :|: s7 >= 0, s7 <= 4 * x_1 + 1, s8 >= 0, s8 <= 4 * x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z', z'') -{ 5 + s52 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s53 :|: s51 >= 0, s51 <= 4 * z' + 1, s52 >= 0, s52 <= 4 * z'' + 1, s53 >= 0, s53 <= s51 + s52 + 1, z' >= 0, z'' >= 0
   encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> 1 + s48 + s49 :|: s48 >= 0, s48 <= 4 * z' + 1, s49 >= 0, s49 <= 4 * z'' + 1, z' >= 0, z'' >= 0
   encode_nil -{ 0 }-> 1 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_norm(z') -{ 6 + s32 + s33 + 26*x_194 + 16*x_194^2 + 26*x_256 + 16*x_256^2 }-> s34 :|: s32 >= 0, s32 <= 4 * x_194 + 1, s33 >= 0, s33 <= 4 * x_256 + 1, s34 >= 0, s34 <= 1 + s32 + s33, z' = 1 + x_194 + x_256, x_194 >= 0, x_256 >= 0
   encode_norm(z') -{ -6 + s35 + -6*z' + 16*z'^2 }-> s36 :|: s35 >= 0, s35 <= 4 * (z' - 1) + 1, s36 >= 0, s36 <= 1 + s35, z' - 1 >= 0
   encode_norm(z') -{ -6 + s37 + s38 + -6*z' + 16*z'^2 }-> s39 :|: s37 >= 0, s37 <= 4 * (z' - 1) + 1, s38 >= 0, s38 <= s37, s39 >= 0, s39 <= s38, z' - 1 >= 0
   encode_norm(z') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 1, z' = 1
   encode_norm(z') -{ 6 + s41 + s42 + 26*x_197 + 16*x_197^2 + 26*x_257 + 16*x_257^2 }-> s43 :|: s40 >= 0, s40 <= 4 * x_197 + 1, s41 >= 0, s41 <= 4 * x_257 + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= s42, x_197 >= 0, x_257 >= 0, z' = 1 + x_197 + x_257
   encode_norm(z') -{ 6 + s45 + s46 + 26*x_198 + 16*x_198^2 + 26*x_258 + 16*x_258^2 }-> s47 :|: s44 >= 0, s44 <= 4 * x_198 + 1, s45 >= 0, s45 <= 4 * x_258 + 1, s46 >= 0, s46 <= s44, s47 >= 0, s47 <= s46, x_258 >= 0, z' = 1 + x_198 + x_258, x_198 >= 0
   encode_norm(z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z' = 0
   encode_norm(z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0, z' >= 0
   encode_norm(z') -{ 0 }-> 0 :|: z' >= 0
   encode_rem(z', z'') -{ 5 + s55 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2 }-> s56 :|: s54 >= 0, s54 <= 4 * z' + 1, s55 >= 0, s55 <= 4 * z'' + 1, s56 >= 0, s56 <= s54, z' >= 0, z'' >= 0
   encode_rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2 + 26*z' + 16*z'^2 }-> 1 + s50 :|: s50 >= 0, s50 <= 4 * z' + 1, z' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 2 + y }-> 1 + s + z :|: s >= 0, s <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 1 + z' :|: z' >= 0, z'' = 1
   norm(z') -{ 1 }-> 0 :|: z' = 1
   norm(z') -{ 0 }-> 0 :|: z' >= 0
   norm(z') -{ 2 + x }-> 1 + s'' :|: s'' >= 0, s'' <= x, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 + z'' }-> s' :|: s' >= 0, s' <= x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
   rem(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   rem(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   rem(z', z'') -{ 1 }-> 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  rem: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  norm: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encode_nil: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [1 + 4*z']
  encode_rem: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [1 + 4*z']
  encode_g: runtime: O(n^2) [4 + 26*z' + 16*z'^2 + 26*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']
  encode_norm: runtime: O(n^2) [27 + 186*z' + 128*z'^2], size: O(n^1) [4*z']
  encode_f: runtime: O(n^2) [6 + 26*z' + 16*z'^2 + 30*z'' + 16*z''^2], size: O(n^1) [3 + 4*z' + 4*z'']
  encode_s: runtime: O(n^2) [2 + 26*z' + 16*z'^2], size: O(n^1) [2 + 4*z']

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

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

(86)
BOUNDS(1, n^2)

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

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

(88)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

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

   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_norm(x_1)) -> norm(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_rem(x_1, x_2)) -> rem(encArg(x_1), encArg(x_2))
   encode_norm(x_1) -> norm(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_rem(x_1, x_2) -> rem(encArg(x_1), encArg(x_2))

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

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

The rewrite sequence

rem(g(x, y), s(z)) ->^+ rem(x, z)

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

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

The result substitution is [ ].




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

(90)
Complex Obligation (BEST)

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

(91)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

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

   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_norm(x_1)) -> norm(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_rem(x_1, x_2)) -> rem(encArg(x_1), encArg(x_2))
   encode_norm(x_1) -> norm(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_rem(x_1, x_2) -> rem(encArg(x_1), encArg(x_2))

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

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

(93)
BOUNDS(n^1, INF)

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

(94)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

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

   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_norm(x_1)) -> norm(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_rem(x_1, x_2)) -> rem(encArg(x_1), encArg(x_2))
   encode_norm(x_1) -> norm(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_rem(x_1, x_2) -> rem(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST