WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 197 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 146 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) InliningProof [UPPER BOUND(ID), 2 ms]
    (16) CpxRNTS
    (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 152 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 34 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 8 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 352 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 77 ms]
    (44) CpxRNTS
    (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 413 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 241 ms]
    (50) CpxRNTS
    (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 200 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
    (56) CpxRNTS
    (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 167 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (62) CpxRNTS
    (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 147 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) FinalProof [FINISHED, 0 ms]
    (70) BOUNDS(1, n^2)
(71) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(72) TRS for Loop Detection
    (73) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (74) BEST
        (75) proven lower bound
            (76) LowerBoundPropagationProof [FINISHED, 0 ms]
            (77) BOUNDS(n^1, INF)
        (78) 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:

   g(x, y) -> x
   g(x, y) -> y
   f(0, 1, x) -> f(s(x), x, x)
   f(x, y, s(z)) -> s(f(0, 1, 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(0) -> 0
   encArg(1) -> 1
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_1 -> 1
   encode_s(x_1) -> s(encArg(x_1))


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

(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:

   g(x, y) -> x
   g(x, y) -> y
   f(0, 1, x) -> f(s(x), x, x)
   f(x, y, s(z)) -> s(f(0, 1, z))

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_1 -> 1
   encode_s(x_1) -> s(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   g(x, y) -> x
   g(x, y) -> y
   f(0, 1, x) -> f(s(x), x, x)
   f(x, y, s(z)) -> s(f(0, 1, z))

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_1 -> 1
   encode_s(x_1) -> s(encArg(x_1))

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   g(x, y) -> x [1]
   g(x, y) -> y [1]
   f(0, 1, x) -> f(s(x), x, x) [1]
   f(x, y, s(z)) -> s(f(0, 1, z)) [1]
   encArg(0) -> 0 [0]
   encArg(1) -> 1 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_1 -> 1 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

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

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

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

   g(x, y) -> x [1]
   g(x, y) -> y [1]
   f(0, 1, x) -> f(s(x), x, x) [1]
   f(x, y, s(z)) -> s(f(0, 1, z)) [1]
   encArg(0) -> 0 [0]
   encArg(1) -> 1 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_1 -> 1 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

The TRS has the following type information:
g :: 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f
f :: 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f
0 :: 0:1:s:cons_g:cons_f
1 :: 0:1:s:cons_g:cons_f
s :: 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f
encArg :: 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f
cons_g :: 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f
cons_f :: 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f
encode_g :: 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f
encode_f :: 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f
encode_0 :: 0:1:s:cons_g:cons_f
encode_1 :: 0:1:s:cons_g:cons_f
encode_s :: 0:1:s:cons_g:cons_f -> 0:1:s:cons_g:cons_f

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:

   f_3
   g_2
   encArg_1
   encode_g_2
   encode_f_3
   encode_0
   encode_1
   encode_s_1

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_g(v0, v1) -> null_encode_g [0]
   encode_f(v0, v1, v2) -> null_encode_f [0]
   encode_0 -> null_encode_0 [0]
   encode_1 -> null_encode_1 [0]
   encode_s(v0) -> null_encode_s [0]
   f(v0, v1, v2) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_g, null_encode_f, null_encode_0, null_encode_1, null_encode_s, 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:

   g(x, y) -> x [1]
   g(x, y) -> y [1]
   f(0, 1, x) -> f(s(x), x, x) [1]
   f(x, y, s(z)) -> s(f(0, 1, z)) [1]
   encArg(0) -> 0 [0]
   encArg(1) -> 1 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_1 -> 1 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_g(v0, v1) -> null_encode_g [0]
   encode_f(v0, v1, v2) -> null_encode_f [0]
   encode_0 -> null_encode_0 [0]
   encode_1 -> null_encode_1 [0]
   encode_s(v0) -> null_encode_s [0]
   f(v0, v1, v2) -> null_f [0]

The TRS has the following type information:
g :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
f :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
0 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
1 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
s :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encArg :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
cons_g :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
cons_f :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encode_g :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encode_f :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encode_0 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encode_1 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encode_s :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encArg :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encode_g :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encode_f :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encode_0 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encode_1 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encode_s :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_f :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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:

   g(x, y) -> x [1]
   g(x, y) -> y [1]
   f(0, 1, x) -> f(s(x), x, x) [1]
   f(x, y, s(z)) -> s(f(0, 1, z)) [1]
   encArg(0) -> 0 [0]
   encArg(1) -> 1 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_1 -> 1 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_g(v0, v1) -> null_encode_g [0]
   encode_f(v0, v1, v2) -> null_encode_f [0]
   encode_0 -> null_encode_0 [0]
   encode_1 -> null_encode_1 [0]
   encode_s(v0) -> null_encode_s [0]
   f(v0, v1, v2) -> null_f [0]

The TRS has the following type information:
g :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
f :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
0 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
1 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
s :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encArg :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
cons_g :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
cons_f :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encode_g :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encode_f :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encode_0 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encode_1 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
encode_s :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f -> 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encArg :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encode_g :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encode_f :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encode_0 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encode_1 :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_encode_s :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_f
null_f :: 0:1:s:cons_g:cons_f:null_encArg:null_encode_g:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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:

   0 => 0
   1 => 1
   null_encArg => 0
   null_encode_g => 0
   null_encode_f => 0
   null_encode_0 => 0
   null_encode_1 => 0
   null_encode_s => 0
   null_f => 0

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 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
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z1 = x_3, z' = x_1, x_3 >= 0, x_2 >= 0, z'' = x_2
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   encode_g(z', z'') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_g(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'', z1) -{ 1 }-> f(1 + x, x, x) :|: x >= 0, z'' = 1, z1 = x, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z) :|: z >= 0, z' = x, z'' = y, x >= 0, y >= 0, z1 = 1 + z
   g(z', z'') -{ 1 }-> x :|: z' = x, z'' = y, x >= 0, y >= 0
   g(z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0


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

(15) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   g(z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0
   g(z', z'') -{ 1 }-> x :|: z' = x, z'' = y, x >= 0, y >= 0

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 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
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z1 = x_3, z' = x_1, x_3 >= 0, x_2 >= 0, z'' = x_2
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   encode_g(z', z'') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_g(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'', z1) -{ 1 }-> f(1 + x, x, x) :|: x >= 0, z'' = 1, z1 = x, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z) :|: z >= 0, z' = x, z'' = y, x >= 0, y >= 0, z1 = 1 + z
   g(z', z'') -{ 1 }-> x :|: z' = x, z'' = y, x >= 0, y >= 0
   g(z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0


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

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0


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

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

   { g }
   { encode_0 }
   { encode_1 }
   { f }
   { encArg }
   { encode_f }
   { encode_g }
   { encode_s }

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {g}, {encode_0}, {encode_1}, {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}

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

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {g}, {encode_0}, {encode_1}, {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {g}, {encode_0}, {encode_1}, {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: ?, size: O(n^1) [z' + z'']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_0}, {encode_1}, {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']

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

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_0}, {encode_1}, {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']

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

(29) 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

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_0}, {encode_1}, {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: ?, size: O(1) [0]

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

(31) 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

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_1}, {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_1}, {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_1}, {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: ?, size: O(1) [1]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]

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

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z1

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: ?, size: O(n^1) [z1]

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

(43) 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 + 2*z1

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 1 }-> f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]

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

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]

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

(47) 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 + z'

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: ?, size: O(n^1) [1 + z']

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

(49) 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: z' + 2*z'^2

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

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

   encArg(z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 0 }-> g(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_g(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'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [1 + z']

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

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

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

   encArg(z') -{ 1 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= s11 + s10, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 + 2*s3 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 + x_3 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, s3 >= 0, s3 <= x_3 + 1, s4 >= 0, s4 <= s3, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -3*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 1 + 2*s7 + z' + 2*z'^2 + z'' + 2*z''^2 + z1 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z' + 1, s6 >= 0, s6 <= z'' + 1, s7 >= 0, s7 <= z1 + 1, s8 >= 0, s8 <= s7, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 1 + z' + 2*z'^2 + z'' + 2*z''^2 }-> s15 :|: s13 >= 0, s13 <= z' + 1, s14 >= 0, s14 <= z'' + 1, s15 >= 0, s15 <= s14 + s13, z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ z' + 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z' + 1, z' >= 0
   f(z', z'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [1 + z']

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

(53) 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: 1 + z1

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

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

   encArg(z') -{ 1 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= s11 + s10, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 + 2*s3 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 + x_3 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, s3 >= 0, s3 <= x_3 + 1, s4 >= 0, s4 <= s3, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -3*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 1 + 2*s7 + z' + 2*z'^2 + z'' + 2*z''^2 + z1 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z' + 1, s6 >= 0, s6 <= z'' + 1, s7 >= 0, s7 <= z1 + 1, s8 >= 0, s8 <= s7, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 1 + z' + 2*z'^2 + z'' + 2*z''^2 }-> s15 :|: s13 >= 0, s13 <= z' + 1, s14 >= 0, s14 <= z'' + 1, s15 >= 0, s15 <= s14 + s13, z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ z' + 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z' + 1, z' >= 0
   f(z', z'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [1 + z']
  encode_f: runtime: ?, size: O(n^1) [1 + z1]

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

(55) 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: 3 + z' + 2*z'^2 + z'' + 2*z''^2 + 3*z1 + 2*z1^2

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

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

   encArg(z') -{ 1 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= s11 + s10, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 + 2*s3 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 + x_3 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, s3 >= 0, s3 <= x_3 + 1, s4 >= 0, s4 <= s3, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -3*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 1 + 2*s7 + z' + 2*z'^2 + z'' + 2*z''^2 + z1 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z' + 1, s6 >= 0, s6 <= z'' + 1, s7 >= 0, s7 <= z1 + 1, s8 >= 0, s8 <= s7, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 1 + z' + 2*z'^2 + z'' + 2*z''^2 }-> s15 :|: s13 >= 0, s13 <= z' + 1, s14 >= 0, s14 <= z'' + 1, s15 >= 0, s15 <= s14 + s13, z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ z' + 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z' + 1, z' >= 0
   f(z', z'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [1 + z']
  encode_f: runtime: O(n^2) [3 + z' + 2*z'^2 + z'' + 2*z''^2 + 3*z1 + 2*z1^2], size: O(n^1) [1 + z1]

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

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

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

   encArg(z') -{ 1 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= s11 + s10, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 + 2*s3 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 + x_3 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, s3 >= 0, s3 <= x_3 + 1, s4 >= 0, s4 <= s3, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -3*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 1 + 2*s7 + z' + 2*z'^2 + z'' + 2*z''^2 + z1 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z' + 1, s6 >= 0, s6 <= z'' + 1, s7 >= 0, s7 <= z1 + 1, s8 >= 0, s8 <= s7, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 1 + z' + 2*z'^2 + z'' + 2*z''^2 }-> s15 :|: s13 >= 0, s13 <= z' + 1, s14 >= 0, s14 <= z'' + 1, s15 >= 0, s15 <= s14 + s13, z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ z' + 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z' + 1, z' >= 0
   f(z', z'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [1 + z']
  encode_f: runtime: O(n^2) [3 + z' + 2*z'^2 + z'' + 2*z''^2 + 3*z1 + 2*z1^2], size: O(n^1) [1 + z1]

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

(59) 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: 2 + z' + z''

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

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

   encArg(z') -{ 1 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= s11 + s10, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 + 2*s3 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 + x_3 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, s3 >= 0, s3 <= x_3 + 1, s4 >= 0, s4 <= s3, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -3*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 1 + 2*s7 + z' + 2*z'^2 + z'' + 2*z''^2 + z1 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z' + 1, s6 >= 0, s6 <= z'' + 1, s7 >= 0, s7 <= z1 + 1, s8 >= 0, s8 <= s7, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 1 + z' + 2*z'^2 + z'' + 2*z''^2 }-> s15 :|: s13 >= 0, s13 <= z' + 1, s14 >= 0, s14 <= z'' + 1, s15 >= 0, s15 <= s14 + s13, z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ z' + 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z' + 1, z' >= 0
   f(z', z'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_g}, {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [1 + z']
  encode_f: runtime: O(n^2) [3 + z' + 2*z'^2 + z'' + 2*z''^2 + 3*z1 + 2*z1^2], size: O(n^1) [1 + z1]
  encode_g: runtime: ?, size: O(n^1) [2 + z' + z'']

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

(61) 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: 1 + z' + 2*z'^2 + z'' + 2*z''^2

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

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

   encArg(z') -{ 1 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= s11 + s10, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 + 2*s3 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 + x_3 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, s3 >= 0, s3 <= x_3 + 1, s4 >= 0, s4 <= s3, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -3*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 1 + 2*s7 + z' + 2*z'^2 + z'' + 2*z''^2 + z1 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z' + 1, s6 >= 0, s6 <= z'' + 1, s7 >= 0, s7 <= z1 + 1, s8 >= 0, s8 <= s7, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 1 + z' + 2*z'^2 + z'' + 2*z''^2 }-> s15 :|: s13 >= 0, s13 <= z' + 1, s14 >= 0, s14 <= z'' + 1, s15 >= 0, s15 <= s14 + s13, z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ z' + 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z' + 1, z' >= 0
   f(z', z'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [1 + z']
  encode_f: runtime: O(n^2) [3 + z' + 2*z'^2 + z'' + 2*z''^2 + 3*z1 + 2*z1^2], size: O(n^1) [1 + z1]
  encode_g: runtime: O(n^2) [1 + z' + 2*z'^2 + z'' + 2*z''^2], size: O(n^1) [2 + z' + z'']

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

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

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

   encArg(z') -{ 1 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= s11 + s10, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 + 2*s3 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 + x_3 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, s3 >= 0, s3 <= x_3 + 1, s4 >= 0, s4 <= s3, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -3*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 1 + 2*s7 + z' + 2*z'^2 + z'' + 2*z''^2 + z1 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z' + 1, s6 >= 0, s6 <= z'' + 1, s7 >= 0, s7 <= z1 + 1, s8 >= 0, s8 <= s7, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 1 + z' + 2*z'^2 + z'' + 2*z''^2 }-> s15 :|: s13 >= 0, s13 <= z' + 1, s14 >= 0, s14 <= z'' + 1, s15 >= 0, s15 <= s14 + s13, z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ z' + 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z' + 1, z' >= 0
   f(z', z'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [1 + z']
  encode_f: runtime: O(n^2) [3 + z' + 2*z'^2 + z'' + 2*z''^2 + 3*z1 + 2*z1^2], size: O(n^1) [1 + z1]
  encode_g: runtime: O(n^2) [1 + z' + 2*z'^2 + z'' + 2*z''^2], size: O(n^1) [2 + z' + z'']

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 1 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= s11 + s10, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 + 2*s3 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 + x_3 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, s3 >= 0, s3 <= x_3 + 1, s4 >= 0, s4 <= s3, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -3*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 1 + 2*s7 + z' + 2*z'^2 + z'' + 2*z''^2 + z1 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z' + 1, s6 >= 0, s6 <= z'' + 1, s7 >= 0, s7 <= z1 + 1, s8 >= 0, s8 <= s7, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 1 + z' + 2*z'^2 + z'' + 2*z''^2 }-> s15 :|: s13 >= 0, s13 <= z' + 1, s14 >= 0, s14 <= z'' + 1, s15 >= 0, s15 <= s14 + s13, z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ z' + 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z' + 1, z' >= 0
   f(z', z'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [1 + z']
  encode_f: runtime: O(n^2) [3 + z' + 2*z'^2 + z'' + 2*z''^2 + 3*z1 + 2*z1^2], size: O(n^1) [1 + z1]
  encode_g: runtime: O(n^2) [1 + z' + 2*z'^2 + z'' + 2*z''^2], size: O(n^1) [2 + z' + z'']
  encode_s: runtime: ?, size: O(n^1) [2 + z']

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

(67) 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: z' + 2*z'^2

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

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

   encArg(z') -{ 1 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 }-> s12 :|: s10 >= 0, s10 <= x_1 + 1, s11 >= 0, s11 <= x_2 + 1, s12 >= 0, s12 <= s11 + s10, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 1 + 2*s3 + x_1 + 2*x_1^2 + x_2 + 2*x_2^2 + x_3 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, s3 >= 0, s3 <= x_3 + 1, s4 >= 0, s4 <= s3, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 1 + -3*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 1 + 2*s7 + z' + 2*z'^2 + z'' + 2*z''^2 + z1 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z' + 1, s6 >= 0, s6 <= z'' + 1, s7 >= 0, s7 <= z1 + 1, s8 >= 0, s8 <= s7, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'') -{ 1 + z' + 2*z'^2 + z'' + 2*z''^2 }-> s15 :|: s13 >= 0, s13 <= z' + 1, s14 >= 0, s14 <= z'' + 1, s15 >= 0, s15 <= s14 + s13, z' >= 0, z'' >= 0
   encode_g(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ z' + 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z' + 1, z' >= 0
   f(z', z'', z1) -{ 2 + 2*z1 }-> s :|: s >= 0, s <= z1, z1 >= 0, z'' = 1, z' = 0
   f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 2*z1 }-> 1 + s' :|: s' >= 0, s' <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0
   g(z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z' + z'']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(n^1) [1 + 2*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [1 + z']
  encode_f: runtime: O(n^2) [3 + z' + 2*z'^2 + z'' + 2*z''^2 + 3*z1 + 2*z1^2], size: O(n^1) [1 + z1]
  encode_g: runtime: O(n^2) [1 + z' + 2*z'^2 + z'' + 2*z''^2], size: O(n^1) [2 + z' + z'']
  encode_s: runtime: O(n^2) [z' + 2*z'^2], size: O(n^1) [2 + z']

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

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

(70)
BOUNDS(1, n^2)

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

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

(72)
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:

   g(x, y) -> x
   g(x, y) -> y
   f(0, 1, x) -> f(s(x), x, x)
   f(x, y, s(z)) -> s(f(0, 1, z))

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_1 -> 1
   encode_s(x_1) -> s(encArg(x_1))

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

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

The rewrite sequence

f(x, y, s(z)) ->^+ s(f(0, 1, z))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [z / s(z)].

The result substitution is [x / 0, y / 1].




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

(74)
Complex Obligation (BEST)

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

(75)
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:

   g(x, y) -> x
   g(x, y) -> y
   f(0, 1, x) -> f(s(x), x, x)
   f(x, y, s(z)) -> s(f(0, 1, z))

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_1 -> 1
   encode_s(x_1) -> s(encArg(x_1))

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

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

(77)
BOUNDS(n^1, INF)

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

(78)
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:

   g(x, y) -> x
   g(x, y) -> y
   f(0, 1, x) -> f(s(x), x, x)
   f(x, y, s(z)) -> s(f(0, 1, z))

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_1 -> 1
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST