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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 214 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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(12) CpxRNTS
(13) CompleteCoflocoProof [FINISHED, 145 ms]
(14) 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) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)
   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)
   h(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(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g1(x_1, x_2, x_3)) -> g1(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g2(x_1, x_2, x_3)) -> g2(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g1(x_1, x_2, x_3) -> g1(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g2(x_1, x_2, x_3) -> g2(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))


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

(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) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)
   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)
   h(x, x) -> x

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

   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g1(x_1, x_2, x_3)) -> g1(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g2(x_1, x_2, x_3)) -> g2(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g1(x_1, x_2, x_3) -> g1(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g2(x_1, x_2, x_3) -> g2(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))

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

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

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


The TRS R consists of the following rules:

   f(x, y) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)
   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)
   h(x, x) -> x

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

   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g1(x_1, x_2, x_3)) -> g1(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g2(x_1, x_2, x_3)) -> g2(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g1(x_1, x_2, x_3) -> g1(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g2(x_1, x_2, x_3) -> g2(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))

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

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

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


The TRS R consists of the following rules:

   f(x, y) -> g1(x, x, y) [1]
   f(x, y) -> g1(y, x, x) [1]
   f(x, y) -> g2(x, y, y) [1]
   f(x, y) -> g2(y, y, x) [1]
   g1(x, x, y) -> h(x, y) [1]
   g1(y, x, x) -> h(x, y) [1]
   g2(x, y, y) -> h(x, y) [1]
   g2(y, y, x) -> h(x, y) [1]
   h(x, x) -> x [1]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g1(x_1, x_2, x_3)) -> g1(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g2(x_1, x_2, x_3)) -> g2(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g1(x_1, x_2, x_3) -> g1(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g2(x_1, x_2, x_3) -> g2(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) [0]

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

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

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

   f(x, y) -> g1(x, x, y) [1]
   f(x, y) -> g1(y, x, x) [1]
   f(x, y) -> g2(x, y, y) [1]
   f(x, y) -> g2(y, y, x) [1]
   g1(x, x, y) -> h(x, y) [1]
   g1(y, x, x) -> h(x, y) [1]
   g2(x, y, y) -> h(x, y) [1]
   g2(y, y, x) -> h(x, y) [1]
   h(x, x) -> x [1]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g1(x_1, x_2, x_3)) -> g1(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g2(x_1, x_2, x_3)) -> g2(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g1(x_1, x_2, x_3) -> g1(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g2(x_1, x_2, x_3) -> g2(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) [0]

The TRS has the following type information:
f :: f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h
g1 :: f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h
g2 :: f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h
h :: f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h
encArg :: cons_f:cons_g1:cons_g2:cons_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h
cons_f :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h
cons_g1 :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h
cons_g2 :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h
cons_h :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h
encode_f :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h
encode_g1 :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h
encode_g2 :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h
encode_h :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> f:g1:g2:h:encArg:encode_f:encode_g1:encode_g2:encode_h

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

(9) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_g1(v0, v1, v2) -> null_encode_g1 [0]
   encode_g2(v0, v1, v2) -> null_encode_g2 [0]
   encode_h(v0, v1) -> null_encode_h [0]
   g1(v0, v1, v2) -> null_g1 [0]
   g2(v0, v1, v2) -> null_g2 [0]
   h(v0, v1) -> null_h [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_g1, null_encode_g2, null_encode_h, null_g1, null_g2, null_h, const

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

(10)
Obligation:
Runtime Complexity Weighted TRS where all 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) -> g1(x, x, y) [1]
   f(x, y) -> g1(y, x, x) [1]
   f(x, y) -> g2(x, y, y) [1]
   f(x, y) -> g2(y, y, x) [1]
   g1(x, x, y) -> h(x, y) [1]
   g1(y, x, x) -> h(x, y) [1]
   g2(x, y, y) -> h(x, y) [1]
   g2(y, y, x) -> h(x, y) [1]
   h(x, x) -> x [1]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g1(x_1, x_2, x_3)) -> g1(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g2(x_1, x_2, x_3)) -> g2(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g1(x_1, x_2, x_3) -> g1(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g2(x_1, x_2, x_3) -> g2(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_g1(v0, v1, v2) -> null_encode_g1 [0]
   encode_g2(v0, v1, v2) -> null_encode_g2 [0]
   encode_h(v0, v1) -> null_encode_h [0]
   g1(v0, v1, v2) -> null_g1 [0]
   g2(v0, v1, v2) -> null_g2 [0]
   h(v0, v1) -> null_h [0]

The TRS has the following type information:
f :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
g1 :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
g2 :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
h :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
encArg :: cons_f:cons_g1:cons_g2:cons_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
cons_f :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h
cons_g1 :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h
cons_g2 :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h
cons_h :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h
encode_f :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
encode_g1 :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
encode_g2 :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
encode_h :: cons_f:cons_g1:cons_g2:cons_h -> cons_f:cons_g1:cons_g2:cons_h -> null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
null_encArg :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
null_encode_f :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
null_encode_g1 :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
null_encode_g2 :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
null_encode_h :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
null_g1 :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
null_g2 :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
null_h :: null_encArg:null_encode_f:null_encode_g1:null_encode_g2:null_encode_h:null_g1:null_g2:null_h
const :: cons_f:cons_g1:cons_g2:cons_h

Rewrite Strategy: INNERMOST
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(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   null_encArg => 0
   null_encode_f => 0
   null_encode_g1 => 0
   null_encode_g2 => 0
   null_encode_h => 0
   null_g1 => 0
   null_g2 => 0
   null_h => 0
   const => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> h(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> g2(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 }-> g1(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)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g1(z, z', z'') -{ 0 }-> g1(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_g1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_g2(z, z', z'') -{ 0 }-> g2(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_g2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_h(z, z') -{ 0 }-> h(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_h(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 1 }-> g2(x, y, y) :|: x >= 0, y >= 0, z = x, z' = y
   f(z, z') -{ 1 }-> g2(y, y, x) :|: x >= 0, y >= 0, z = x, z' = y
   f(z, z') -{ 1 }-> g1(x, x, y) :|: x >= 0, y >= 0, z = x, z' = y
   f(z, z') -{ 1 }-> g1(y, x, x) :|: x >= 0, y >= 0, z = x, z' = y
   g1(z, z', z'') -{ 1 }-> h(x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = x
   g1(z, z', z'') -{ 1 }-> h(x, y) :|: z' = x, y >= 0, x >= 0, z'' = x, z = y
   g1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   g2(z, z', z'') -{ 1 }-> h(x, y) :|: z'' = y, x >= 0, y >= 0, z = x, z' = y
   g2(z, z', z'') -{ 1 }-> h(x, y) :|: y >= 0, x >= 0, z'' = x, z' = y, z = y
   g2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   h(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = x
   h(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

(13) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V12),0,[f(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[g1(V1, V, V12, Out)],[V1 >= 0,V >= 0,V12 >= 0]).
eq(start(V1, V, V12),0,[g2(V1, V, V12, Out)],[V1 >= 0,V >= 0,V12 >= 0]).
eq(start(V1, V, V12),0,[h(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V12),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[fun1(V1, V, V12, Out)],[V1 >= 0,V >= 0,V12 >= 0]).
eq(start(V1, V, V12),0,[fun2(V1, V, V12, Out)],[V1 >= 0,V >= 0,V12 >= 0]).
eq(start(V1, V, V12),0,[fun3(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(f(V1, V, Out),1,[g1(V3, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
eq(f(V1, V, Out),1,[g1(V5, V4, V4, Ret1)],[Out = Ret1,V4 >= 0,V5 >= 0,V1 = V4,V = V5]).
eq(f(V1, V, Out),1,[g2(V7, V6, V6, Ret2)],[Out = Ret2,V7 >= 0,V6 >= 0,V1 = V7,V = V6]).
eq(f(V1, V, Out),1,[g2(V9, V9, V8, Ret3)],[Out = Ret3,V8 >= 0,V9 >= 0,V1 = V8,V = V9]).
eq(g1(V1, V, V12, Out),1,[h(V11, V10, Ret4)],[Out = Ret4,V = V11,V12 = V10,V11 >= 0,V10 >= 0,V1 = V11]).
eq(g1(V1, V, V12, Out),1,[h(V14, V13, Ret5)],[Out = Ret5,V = V14,V13 >= 0,V14 >= 0,V12 = V14,V1 = V13]).
eq(g2(V1, V, V12, Out),1,[h(V16, V15, Ret6)],[Out = Ret6,V12 = V15,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
eq(g2(V1, V, V12, Out),1,[h(V17, V18, Ret7)],[Out = Ret7,V18 >= 0,V17 >= 0,V12 = V17,V = V18,V1 = V18]).
eq(h(V1, V, Out),1,[],[Out = V19,V = V19,V19 >= 0,V1 = V19]).
eq(encArg(V1, Out),0,[encArg(V21, Ret0),encArg(V20, Ret11),f(Ret0, Ret11, Ret8)],[Out = Ret8,V21 >= 0,V1 = 1 + V20 + V21,V20 >= 0]).
eq(encArg(V1, Out),0,[encArg(V23, Ret01),encArg(V24, Ret12),encArg(V22, Ret21),g1(Ret01, Ret12, Ret21, Ret9)],[Out = Ret9,V23 >= 0,V1 = 1 + V22 + V23 + V24,V22 >= 0,V24 >= 0]).
eq(encArg(V1, Out),0,[encArg(V26, Ret02),encArg(V25, Ret13),encArg(V27, Ret22),g2(Ret02, Ret13, Ret22, Ret10)],[Out = Ret10,V26 >= 0,V1 = 1 + V25 + V26 + V27,V27 >= 0,V25 >= 0]).
eq(encArg(V1, Out),0,[encArg(V29, Ret03),encArg(V28, Ret14),h(Ret03, Ret14, Ret15)],[Out = Ret15,V29 >= 0,V1 = 1 + V28 + V29,V28 >= 0]).
eq(fun(V1, V, Out),0,[encArg(V31, Ret04),encArg(V30, Ret16),f(Ret04, Ret16, Ret17)],[Out = Ret17,V31 >= 0,V30 >= 0,V1 = V31,V = V30]).
eq(fun1(V1, V, V12, Out),0,[encArg(V33, Ret05),encArg(V34, Ret18),encArg(V32, Ret23),g1(Ret05, Ret18, Ret23, Ret19)],[Out = Ret19,V33 >= 0,V32 >= 0,V34 >= 0,V1 = V33,V = V34,V12 = V32]).
eq(fun2(V1, V, V12, Out),0,[encArg(V37, Ret06),encArg(V36, Ret110),encArg(V35, Ret24),g2(Ret06, Ret110, Ret24, Ret20)],[Out = Ret20,V37 >= 0,V35 >= 0,V36 >= 0,V1 = V37,V = V36,V12 = V35]).
eq(fun3(V1, V, Out),0,[encArg(V39, Ret07),encArg(V38, Ret111),h(Ret07, Ret111, Ret25)],[Out = Ret25,V39 >= 0,V38 >= 0,V1 = V39,V = V38]).
eq(encArg(V1, Out),0,[],[Out = 0,V40 >= 0,V1 = V40]).
eq(fun(V1, V, Out),0,[],[Out = 0,V42 >= 0,V41 >= 0,V1 = V42,V = V41]).
eq(fun1(V1, V, V12, Out),0,[],[Out = 0,V44 >= 0,V12 = V45,V43 >= 0,V1 = V44,V = V43,V45 >= 0]).
eq(fun2(V1, V, V12, Out),0,[],[Out = 0,V46 >= 0,V12 = V47,V48 >= 0,V1 = V46,V = V48,V47 >= 0]).
eq(fun3(V1, V, Out),0,[],[Out = 0,V49 >= 0,V50 >= 0,V1 = V49,V = V50]).
eq(g1(V1, V, V12, Out),0,[],[Out = 0,V52 >= 0,V12 = V53,V51 >= 0,V1 = V52,V = V51,V53 >= 0]).
eq(g2(V1, V, V12, Out),0,[],[Out = 0,V55 >= 0,V12 = V56,V54 >= 0,V1 = V55,V = V54,V56 >= 0]).
eq(h(V1, V, Out),0,[],[Out = 0,V57 >= 0,V58 >= 0,V1 = V57,V = V58]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).
input_output_vars(g1(V1,V,V12,Out),[V1,V,V12],[Out]).
input_output_vars(g2(V1,V,V12,Out),[V1,V,V12],[Out]).
input_output_vars(h(V1,V,Out),[V1,V],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V12,Out),[V1,V,V12],[Out]).
input_output_vars(fun2(V1,V,V12,Out),[V1,V,V12],[Out]).
input_output_vars(fun3(V1,V,Out),[V1,V],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. non_recursive  : [h/3]
1. non_recursive  : [g1/4]
2. non_recursive  : [g2/4]
3. non_recursive  : [f/3]
4. recursive [non_tail,multiple] : [encArg/2]
5. non_recursive  : [fun/3]
6. non_recursive  : [fun1/4]
7. non_recursive  : [fun2/4]
8. non_recursive  : [fun3/3]
9. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into h/3
1. SCC is partially evaluated into g1/4
2. SCC is partially evaluated into g2/4
3. SCC is partially evaluated into f/3
4. SCC is partially evaluated into encArg/2
5. SCC is partially evaluated into fun/3
6. SCC is partially evaluated into fun1/4
7. SCC is partially evaluated into fun2/4
8. SCC is partially evaluated into fun3/3
9. SCC is partially evaluated into start/3

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations h/3 
* CE 20 is refined into CE [35] 
* CE 21 is refined into CE [36] 


### Cost equations --> "Loop" of h/3 
* CEs [35] --> Loop 18 
* CEs [36] --> Loop 19 

### Ranking functions of CR h(V1,V,Out) 

#### Partial ranking functions of CR h(V1,V,Out) 


### Specialization of cost equations g1/4 
* CE 15 is refined into CE [37,38] 
* CE 14 is refined into CE [39,40] 
* CE 16 is refined into CE [41] 


### Cost equations --> "Loop" of g1/4 
* CEs [38,40] --> Loop 20 
* CEs [37] --> Loop 21 
* CEs [39,41] --> Loop 22 

### Ranking functions of CR g1(V1,V,V12,Out) 

#### Partial ranking functions of CR g1(V1,V,V12,Out) 


### Specialization of cost equations g2/4 
* CE 17 is refined into CE [42,43] 
* CE 18 is refined into CE [44,45] 
* CE 19 is refined into CE [46] 


### Cost equations --> "Loop" of g2/4 
* CEs [43,45] --> Loop 23 
* CEs [42] --> Loop 24 
* CEs [44,46] --> Loop 25 

### Ranking functions of CR g2(V1,V,V12,Out) 

#### Partial ranking functions of CR g2(V1,V,V12,Out) 


### Specialization of cost equations f/3 
* CE 10 is refined into CE [47,48] 
* CE 11 is refined into CE [49,50] 
* CE 12 is refined into CE [51,52] 
* CE 13 is refined into CE [53,54] 


### Cost equations --> "Loop" of f/3 
* CEs [48,50,52,54] --> Loop 26 
* CEs [47,49,51,53] --> Loop 27 

### Ranking functions of CR f(V1,V,Out) 

#### Partial ranking functions of CR f(V1,V,Out) 


### Specialization of cost equations encArg/2 
* CE 26 is refined into CE [55] 
* CE 22 is refined into CE [56,57] 
* CE 25 is refined into CE [58,59] 
* CE 23 is refined into CE [60,61] 
* CE 24 is refined into CE [62,63] 


### Cost equations --> "Loop" of encArg/2 
* CEs [61,63] --> Loop 28 
* CEs [60,62] --> Loop 29 
* CEs [57,59] --> Loop 30 
* CEs [56,58] --> Loop 31 
* CEs [55] --> Loop 32 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [28,29,30,31]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [28,29,30,31]:
  - RF of loop [28:1,28:2,28:3,29:1,29:2,29:3,30:1,30:2,31:1,31:2]:
    V1


### Specialization of cost equations fun/3 
* CE 27 is refined into CE [64,65] 
* CE 28 is refined into CE [66] 


### Cost equations --> "Loop" of fun/3 
* CEs [64,65,66] --> Loop 33 

### Ranking functions of CR fun(V1,V,Out) 

#### Partial ranking functions of CR fun(V1,V,Out) 


### Specialization of cost equations fun1/4 
* CE 29 is refined into CE [67,68] 
* CE 30 is refined into CE [69] 


### Cost equations --> "Loop" of fun1/4 
* CEs [67,68,69] --> Loop 34 

### Ranking functions of CR fun1(V1,V,V12,Out) 

#### Partial ranking functions of CR fun1(V1,V,V12,Out) 


### Specialization of cost equations fun2/4 
* CE 31 is refined into CE [70,71] 
* CE 32 is refined into CE [72] 


### Cost equations --> "Loop" of fun2/4 
* CEs [70,71,72] --> Loop 35 

### Ranking functions of CR fun2(V1,V,V12,Out) 

#### Partial ranking functions of CR fun2(V1,V,V12,Out) 


### Specialization of cost equations fun3/3 
* CE 33 is refined into CE [73,74] 
* CE 34 is refined into CE [75] 


### Cost equations --> "Loop" of fun3/3 
* CEs [73,74,75] --> Loop 36 

### Ranking functions of CR fun3(V1,V,Out) 

#### Partial ranking functions of CR fun3(V1,V,Out) 


### Specialization of cost equations start/3 
* CE 1 is refined into CE [76,77] 
* CE 2 is refined into CE [78,79] 
* CE 3 is refined into CE [80,81] 
* CE 4 is refined into CE [82,83] 
* CE 5 is refined into CE [84] 
* CE 6 is refined into CE [85] 
* CE 7 is refined into CE [86] 
* CE 8 is refined into CE [87] 
* CE 9 is refined into CE [88] 


### Cost equations --> "Loop" of start/3 
* CEs [76,77,78,79,80,81,82,83,84,85,86,87,88] --> Loop 37 

### Ranking functions of CR start(V1,V,V12) 

#### Partial ranking functions of CR start(V1,V,V12) 


Computing Bounds
=====================================

#### Cost of chains of h(V1,V,Out):
* Chain [19]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [18]: 1
  with precondition: [V1=V,V1=Out,V1>=0] 


#### Cost of chains of g1(V1,V,V12,Out):
* Chain [22]: 1
  with precondition: [Out=0,V1>=0,V>=0,V12>=0] 

* Chain [21]: 1
  with precondition: [Out=0,V=V12,V1>=0,V>=0] 

* Chain [20]: 2
  with precondition: [V1=V,V1=V12,V1=Out,V1>=0] 


#### Cost of chains of g2(V1,V,V12,Out):
* Chain [25]: 1
  with precondition: [Out=0,V1>=0,V>=0,V12>=0] 

* Chain [24]: 1
  with precondition: [Out=0,V=V12,V1>=0,V>=0] 

* Chain [23]: 2
  with precondition: [V1=V,V1=V12,V1=Out,V1>=0] 


#### Cost of chains of f(V1,V,Out):
* Chain [27]: 2
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [26]: 3
  with precondition: [V1=V,V1=Out,V1>=0] 


#### Cost of chains of encArg(V1,Out):
* Chain [32]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([28,29,30,31],[[32]])]: 8*it(28)+0
  Such that:aux(1) =< V1
it(28) =< aux(1)

  with precondition: [Out=0,V1>=1] 


#### Cost of chains of fun(V1,V,Out):
* Chain [33]: 16*s(4)+16*s(6)+3
  Such that:aux(2) =< V1
aux(3) =< V
s(6) =< aux(3)
s(4) =< aux(2)

  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun1(V1,V,V12,Out):
* Chain [34]: 16*s(12)+16*s(14)+16*s(16)+2
  Such that:aux(4) =< V1
aux(5) =< V
aux(6) =< V12
s(16) =< aux(6)
s(14) =< aux(5)
s(12) =< aux(4)

  with precondition: [Out=0,V1>=0,V>=0,V12>=0] 


#### Cost of chains of fun2(V1,V,V12,Out):
* Chain [35]: 16*s(24)+16*s(26)+16*s(28)+2
  Such that:aux(7) =< V1
aux(8) =< V
aux(9) =< V12
s(28) =< aux(9)
s(26) =< aux(8)
s(24) =< aux(7)

  with precondition: [Out=0,V1>=0,V>=0,V12>=0] 


#### Cost of chains of fun3(V1,V,Out):
* Chain [36]: 16*s(36)+16*s(38)+1
  Such that:aux(10) =< V1
aux(11) =< V
s(38) =< aux(11)
s(36) =< aux(10)

  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of start(V1,V,V12):
* Chain [37]: 72*s(44)+64*s(47)+32*s(52)+3
  Such that:aux(12) =< V1
aux(13) =< V
aux(14) =< V12
s(44) =< aux(12)
s(47) =< aux(13)
s(52) =< aux(14)

  with precondition: [V1>=0] 


Closed-form bounds of start(V1,V,V12): 
-------------------------------------
* Chain [37] with precondition: [V1>=0] 
    - Upper bound: 72*V1+3+nat(V)*64+nat(V12)*32 
    - Complexity: n 

### Maximum cost of start(V1,V,V12): 72*V1+3+nat(V)*64+nat(V12)*32 
Asymptotic class: n 
* Total analysis performed in 377 ms.


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

(14)
BOUNDS(1, n^1)