WORST_CASE(?, O(n^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 166 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), 173 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), 1 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 17 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 409 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 159 ms]
(36) CpxRNTS
(37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 78 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 215 ms]
(42) CpxRNTS
(43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 154 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
(48) CpxRNTS
(49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(50) CpxRNTS
(51) IntTrsBoundProof [UPPER BOUND(ID), 67 ms]
(52) CpxRNTS
(53) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
(54) CpxRNTS
(55) FinalProof [FINISHED, 0 ms]
(56) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

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

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

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

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


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

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


The TRS R consists of the following rules:

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

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_1 -> 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(1, n^1).


The TRS R consists of the following rules:

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

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

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


The TRS R consists of the following rules:

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

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

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

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:

   g_3
   f_3
   encArg_1
   encode_f_3
   encode_g_3
   encode_0
   encode_1

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1, v2) -> null_encode_f [0]
   encode_g(v0, v1, v2) -> null_encode_g [0]
   encode_0 -> null_encode_0 [0]
   encode_1 -> null_encode_1 [0]
   g(v0, v1, v2) -> null_g [0]

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

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

(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:

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

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

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:

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

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

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_f => 0
   null_encode_g => 0
   null_encode_0 => 0
   null_encode_1 => 0
   null_g => 0

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

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

   encArg(z') -{ 0 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(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_g(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   f(z', z'', z1) -{ 1 }-> g(x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0
   g(z', z'', z1) -{ 1 }-> f(x, x, x) :|: x >= 0, z'' = 1, z1 = x, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0


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

(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 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0


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

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

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

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

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

   encArg(z') -{ 0 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

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

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

(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 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

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

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

(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 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encode_0}, {encode_1}, {f,g}, {encArg}, {encode_f}, {encode_g}
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 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encode_1}, {f,g}, {encArg}, {encode_f}, {encode_g}
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 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

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

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

(29) 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

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

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

   encArg(z') -{ 0 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

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

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

(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 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   encArg(z') -{ 0 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

Computed RUNTIME bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 4

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

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

   encArg(z') -{ 0 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 1 }-> g(z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 1 }-> f(z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encArg}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [3], size: O(1) [0]
  g: runtime: O(1) [4], size: O(1) [0]

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

(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 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 5 }-> s :|: s >= 0, s <= 0, z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= 0, z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encArg}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [3], size: O(1) [0]
  g: runtime: O(1) [4], size: O(1) [0]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 5 }-> s :|: s >= 0, s <= 0, z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= 0, z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encArg}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [3], size: O(1) [0]
  g: runtime: O(1) [4], size: O(1) [0]
  encArg: runtime: ?, size: O(1) [1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> g(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 }-> 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
   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'', z1) -{ 0 }-> g(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 5 }-> s :|: s >= 0, s <= 0, z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= 0, z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [3], size: O(1) [0]
  g: runtime: O(1) [4], size: O(1) [0]
  encArg: runtime: O(n^1) [4*z'], size: O(1) [1]

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

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

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

   encArg(z') -{ 3 + 4*x_1 + 4*x_2 + 4*x_3 }-> s3 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, s3 >= 0, s3 <= 0, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 4 + 4*x_1 + 4*x_2 + 4*x_3 }-> s7 :|: s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, s7 >= 0, s7 <= 0, 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
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 3 + 4*z' + 4*z'' + 4*z1 }-> s11 :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'', z1) -{ 4 + 4*z' + 4*z'' + 4*z1 }-> s15 :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 5 }-> s :|: s >= 0, s <= 0, z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= 0, z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [3], size: O(1) [0]
  g: runtime: O(1) [4], size: O(1) [0]
  encArg: runtime: O(n^1) [4*z'], size: O(1) [1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 3 + 4*x_1 + 4*x_2 + 4*x_3 }-> s3 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, s3 >= 0, s3 <= 0, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 4 + 4*x_1 + 4*x_2 + 4*x_3 }-> s7 :|: s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, s7 >= 0, s7 <= 0, 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
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 3 + 4*z' + 4*z'' + 4*z1 }-> s11 :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'', z1) -{ 4 + 4*z' + 4*z'' + 4*z1 }-> s15 :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 5 }-> s :|: s >= 0, s <= 0, z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= 0, z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [3], size: O(1) [0]
  g: runtime: O(1) [4], size: O(1) [0]
  encArg: runtime: O(n^1) [4*z'], size: O(1) [1]
  encode_f: runtime: ?, size: O(1) [0]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME 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'' + 4*z1

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

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

   encArg(z') -{ 3 + 4*x_1 + 4*x_2 + 4*x_3 }-> s3 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, s3 >= 0, s3 <= 0, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 4 + 4*x_1 + 4*x_2 + 4*x_3 }-> s7 :|: s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, s7 >= 0, s7 <= 0, 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
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 3 + 4*z' + 4*z'' + 4*z1 }-> s11 :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'', z1) -{ 4 + 4*z' + 4*z'' + 4*z1 }-> s15 :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 5 }-> s :|: s >= 0, s <= 0, z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= 0, z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [3], size: O(1) [0]
  g: runtime: O(1) [4], size: O(1) [0]
  encArg: runtime: O(n^1) [4*z'], size: O(1) [1]
  encode_f: runtime: O(n^1) [3 + 4*z' + 4*z'' + 4*z1], size: O(1) [0]

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

(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') -{ 3 + 4*x_1 + 4*x_2 + 4*x_3 }-> s3 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, s3 >= 0, s3 <= 0, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 4 + 4*x_1 + 4*x_2 + 4*x_3 }-> s7 :|: s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, s7 >= 0, s7 <= 0, 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
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 3 + 4*z' + 4*z'' + 4*z1 }-> s11 :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'', z1) -{ 4 + 4*z' + 4*z'' + 4*z1 }-> s15 :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 5 }-> s :|: s >= 0, s <= 0, z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= 0, z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [3], size: O(1) [0]
  g: runtime: O(1) [4], size: O(1) [0]
  encArg: runtime: O(n^1) [4*z'], size: O(1) [1]
  encode_f: runtime: O(n^1) [3 + 4*z' + 4*z'' + 4*z1], size: O(1) [0]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 3 + 4*x_1 + 4*x_2 + 4*x_3 }-> s3 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, s3 >= 0, s3 <= 0, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 4 + 4*x_1 + 4*x_2 + 4*x_3 }-> s7 :|: s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, s7 >= 0, s7 <= 0, 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
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 3 + 4*z' + 4*z'' + 4*z1 }-> s11 :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'', z1) -{ 4 + 4*z' + 4*z'' + 4*z1 }-> s15 :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 5 }-> s :|: s >= 0, s <= 0, z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= 0, z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [3], size: O(1) [0]
  g: runtime: O(1) [4], size: O(1) [0]
  encArg: runtime: O(n^1) [4*z'], size: O(1) [1]
  encode_f: runtime: O(n^1) [3 + 4*z' + 4*z'' + 4*z1], size: O(1) [0]
  encode_g: runtime: ?, size: O(1) [0]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 4*z' + 4*z'' + 4*z1

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

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

   encArg(z') -{ 3 + 4*x_1 + 4*x_2 + 4*x_3 }-> s3 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, s3 >= 0, s3 <= 0, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 4 + 4*x_1 + 4*x_2 + 4*x_3 }-> s7 :|: s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, s7 >= 0, s7 <= 0, 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
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 3 + 4*z' + 4*z'' + 4*z1 }-> s11 :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_g(z', z'', z1) -{ 4 + 4*z' + 4*z'' + 4*z1 }-> s15 :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 0, z' >= 0, z1 >= 0, z'' >= 0
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   f(z', z'', z1) -{ 5 }-> s :|: s >= 0, s <= 0, z1 >= 0, z' >= 0, z'' >= 0
   g(z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= 0, z1 >= 0, z'' = 1, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_1: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [3], size: O(1) [0]
  g: runtime: O(1) [4], size: O(1) [0]
  encArg: runtime: O(n^1) [4*z'], size: O(1) [1]
  encode_f: runtime: O(n^1) [3 + 4*z' + 4*z'' + 4*z1], size: O(1) [0]
  encode_g: runtime: O(n^1) [4 + 4*z' + 4*z'' + 4*z1], size: O(1) [0]

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(55) FinalProof (FINISHED)
Computed overall runtime complexity
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(56)
BOUNDS(1, n^1)