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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 175 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), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(14) CpxRNTS
(15) SimplificationProof [BOTH BOUNDS(ID, ID), 1 ms]
(16) CpxRNTS
(17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxRNTS
(19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
(24) CpxRNTS
(25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 118 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 192 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 72 ms]
(36) CpxRNTS
(37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 119 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
(42) CpxRNTS
(43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 139 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
(48) CpxRNTS
(49) FinalProof [FINISHED, 0 ms]
(50) 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(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))
   encode_f(x_1, x_2, x_3) -> f(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(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))
   encode_f(x_1, x_2, x_3) -> f(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(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))
   encode_f(x_1, x_2, x_3) -> f(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(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]
   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]

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(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]
   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]

The TRS has the following type information:
f :: 0:1:cons_f -> 0:1:cons_f -> 0:1:cons_f -> 0:1:cons_f
0 :: 0:1:cons_f
1 :: 0:1:cons_f
encArg :: 0:1:cons_f -> 0:1:cons_f
cons_f :: 0:1:cons_f -> 0:1:cons_f -> 0:1:cons_f -> 0:1:cons_f
encode_f :: 0:1:cons_f -> 0:1:cons_f -> 0:1:cons_f -> 0:1:cons_f
encode_0 :: 0:1:cons_f
encode_1 :: 0:1: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
   encArg_1
   encode_f_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_0 -> null_encode_0 [0]
   encode_1 -> null_encode_1 [0]
   f(v0, v1, v2) -> null_f [0]

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

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

The TRS has the following type information:
f :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
0 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
1 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
encArg :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
cons_f :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
encode_f :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
encode_0 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
encode_1 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
null_encArg :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
null_encode_f :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
null_encode_0 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
null_encode_1 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
null_f :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1: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:

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

The TRS has the following type information:
f :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
0 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
1 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
encArg :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
cons_f :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
encode_f :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f -> 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
encode_0 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
encode_1 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
null_encArg :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
null_encode_f :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
null_encode_0 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
null_encode_1 :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1:null_f
null_f :: 0:1:cons_f:null_encArg:null_encode_f:null_encode_0:null_encode_1: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_f => 0
   null_encode_0 => 0
   null_encode_1 => 0
   null_f => 0

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

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

   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', z'') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   f(z, z', z'') -{ 1 }-> f(x, x, x) :|: x >= 0, z'' = x, z' = 1, z = 0
   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0


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

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

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

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


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

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

   { encode_0 }
   { encode_1 }
   { f }
   { encArg }
   { encode_f }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Function symbols to be analyzed: {f}, {encArg}, {encode_f}
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

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

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

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

Function symbols to be analyzed: {f}, {encArg}, {encode_f}
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]

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

(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: 1

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

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

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

Function symbols to be analyzed: {encArg}, {encode_f}
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) [1], 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 }-> 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', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 2 }-> s :|: s >= 0, s <= 0, z'' >= 0, z' = 1, z = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_f}
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) [1], 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 }-> 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', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 2 }-> s :|: s >= 0, s <= 0, z'' >= 0, z' = 1, z = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_f}
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) [1], 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: z

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

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

   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', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 2 }-> s :|: s >= 0, s <= 0, z'' >= 0, z' = 1, z = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_f}
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) [1], size: O(1) [0]
  encArg: runtime: O(n^1) [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) -{ 1 + x_1 + x_2 + x_3 }-> s2 :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 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', z'') -{ 1 + z + z' + z'' }-> s6 :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 0, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 2 }-> s :|: s >= 0, s <= 0, z'' >= 0, z' = 1, z = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_f}
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) [1], size: O(1) [0]
  encArg: runtime: O(n^1) [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) -{ 1 + x_1 + x_2 + x_3 }-> s2 :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 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', z'') -{ 1 + z + z' + z'' }-> s6 :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 0, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 2 }-> s :|: s >= 0, s <= 0, z'' >= 0, z' = 1, z = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_f}
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) [1], size: O(1) [0]
  encArg: runtime: O(n^1) [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: 1 + z + z' + z''

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

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

   encArg(z) -{ 1 + x_1 + x_2 + x_3 }-> s2 :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 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', z'') -{ 1 + z + z' + z'' }-> s6 :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 0, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 2 }-> s :|: s >= 0, s <= 0, z'' >= 0, z' = 1, z = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 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) [1], size: O(1) [0]
  encArg: runtime: O(n^1) [z], size: O(1) [1]
  encode_f: runtime: O(n^1) [1 + z + z' + z''], size: O(1) [0]

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

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

(50)
BOUNDS(1, n^1)