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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 196 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxWeightedTrs
(9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedTrs
(11) CompletionProof [UPPER BOUND(ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(14) CpxRNTS
(15) CompleteCoflocoProof [FINISHED, 535 ms]
(16) 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:

   h(X) -> g(X, X)
   g(a, X) -> f(b, activate(X))
   f(X, X) -> h(a)
   a -> b
   activate(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(b) -> b
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_a) -> a
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_b -> b
   encode_activate(x_1) -> activate(encArg(x_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:

   h(X) -> g(X, X)
   g(a, X) -> f(b, activate(X))
   f(X, X) -> h(a)
   a -> b
   activate(X) -> X

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

   encArg(b) -> b
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_a) -> a
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_b -> b
   encode_activate(x_1) -> activate(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   h(X) -> g(X, X)
   g(a, X) -> f(b, activate(X))
   f(X, X) -> h(a)
   a -> b
   activate(X) -> X

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

   encArg(b) -> b
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_a) -> a
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_b -> b
   encode_activate(x_1) -> activate(encArg(x_1))

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

(5) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(6)
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:

   h(X) -> g(X, X)
   f(X, X) -> h(a)
   a -> b
   activate(X) -> X
   g(c_a, X) -> f(b, activate(X))

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

   encArg(b) -> b
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_a) -> a
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_b -> b
   encode_activate(x_1) -> activate(encArg(x_1))
   a -> c_a

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

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

(8)
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:

   h(X) -> g(X, X) [1]
   f(X, X) -> h(a) [1]
   a -> b [1]
   activate(X) -> X [1]
   g(c_a, X) -> f(b, activate(X)) [1]
   encArg(b) -> b [0]
   encArg(cons_h(x_1)) -> h(encArg(x_1)) [0]
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_a) -> a [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_h(x_1) -> h(encArg(x_1)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_b -> b [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   a -> c_a [0]

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   h(X) -> g(X, X) [1]
   f(X, X) -> h(a) [1]
   a -> b [1]
   activate(X) -> X [1]
   g(c_a, X) -> f(b, activate(X)) [1]
   encArg(b) -> b [0]
   encArg(cons_h(x_1)) -> h(encArg(x_1)) [0]
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_a) -> a [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_h(x_1) -> h(encArg(x_1)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_b -> b [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   a -> c_a [0]

The TRS has the following type information:
h :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
g :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
f :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
a :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
b :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
activate :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
c_a :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
encArg :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
cons_h :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
cons_g :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
cons_f :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
cons_a :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
cons_activate :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
encode_h :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
encode_g :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
encode_a :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
encode_f :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
encode_b :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate
encode_activate :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate

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

(11) 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_h(v0) -> null_encode_h [0]
   encode_g(v0, v1) -> null_encode_g [0]
   encode_a -> null_encode_a [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_b -> null_encode_b [0]
   encode_activate(v0) -> null_encode_activate [0]
   a -> null_a [0]
   f(v0, v1) -> null_f [0]
   g(v0, v1) -> null_g [0]

And the following fresh constants:    null_encArg, null_encode_h, null_encode_g, null_encode_a, null_encode_f, null_encode_b, null_encode_activate, null_a, null_f, null_g

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

(12)
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:

   h(X) -> g(X, X) [1]
   f(X, X) -> h(a) [1]
   a -> b [1]
   activate(X) -> X [1]
   g(c_a, X) -> f(b, activate(X)) [1]
   encArg(b) -> b [0]
   encArg(cons_h(x_1)) -> h(encArg(x_1)) [0]
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_a) -> a [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_h(x_1) -> h(encArg(x_1)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_b -> b [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   a -> c_a [0]
   encArg(v0) -> null_encArg [0]
   encode_h(v0) -> null_encode_h [0]
   encode_g(v0, v1) -> null_encode_g [0]
   encode_a -> null_encode_a [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_b -> null_encode_b [0]
   encode_activate(v0) -> null_encode_activate [0]
   a -> null_a [0]
   f(v0, v1) -> null_f [0]
   g(v0, v1) -> null_g [0]

The TRS has the following type information:
h :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
g :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
f :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
a :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
b :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
activate :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
c_a :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
encArg :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
cons_h :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
cons_g :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
cons_f :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
cons_a :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
cons_activate :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
encode_h :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
encode_g :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
encode_a :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
encode_f :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
encode_b :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
encode_activate :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g -> b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
null_encArg :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
null_encode_h :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
null_encode_g :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
null_encode_a :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
null_encode_f :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
null_encode_b :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
null_encode_activate :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
null_a :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
null_f :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f:null_g
null_g :: b:c_a:cons_h:cons_g:cons_f:cons_a:cons_activate:null_encArg:null_encode_h:null_encode_g:null_encode_a:null_encode_f:null_encode_b:null_encode_activate:null_a:null_f: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:

   b => 0
   c_a => 1
   cons_a => 2
   null_encArg => 0
   null_encode_h => 0
   null_encode_g => 0
   null_encode_a => 0
   null_encode_f => 0
   null_encode_b => 0
   null_encode_activate => 0
   null_a => 0
   null_f => 0
   null_g => 0

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   encArg(z) -{ 0 }-> h(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> activate(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> a :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a -{ 0 }-> a :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_activate(z) -{ 0 }-> activate(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_b -{ 0 }-> 0 :|: 
   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_g(z, z') -{ 0 }-> g(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_h(z) -{ 0 }-> h(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_h(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z, z') -{ 1 }-> h(a) :|: z' = X, X >= 0, z = X
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   g(z, z') -{ 1 }-> f(0, activate(X)) :|: z' = X, z = 1, X >= 0
   g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   h(z) -{ 1 }-> g(X, X) :|: X >= 0, z = X

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V1),0,[h(V, Out)],[V >= 0]).
eq(start(V, V1),0,[f(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[a(Out)],[]).
eq(start(V, V1),0,[activate(V, Out)],[V >= 0]).
eq(start(V, V1),0,[g(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[encArg(V, Out)],[V >= 0]).
eq(start(V, V1),0,[fun(V, Out)],[V >= 0]).
eq(start(V, V1),0,[fun1(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[fun2(Out)],[]).
eq(start(V, V1),0,[fun3(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[fun4(Out)],[]).
eq(start(V, V1),0,[fun5(V, Out)],[V >= 0]).
eq(h(V, Out),1,[g(X1, X1, Ret)],[Out = Ret,X1 >= 0,V = X1]).
eq(f(V, V1, Out),1,[a(Ret0),h(Ret0, Ret1)],[Out = Ret1,V1 = X2,X2 >= 0,V = X2]).
eq(a(Out),1,[],[Out = 0]).
eq(activate(V, Out),1,[],[Out = X3,X3 >= 0,V = X3]).
eq(g(V, V1, Out),1,[activate(X4, Ret11),f(0, Ret11, Ret2)],[Out = Ret2,V1 = X4,V = 1,X4 >= 0]).
eq(encArg(V, Out),0,[],[Out = 0,V = 0]).
eq(encArg(V, Out),0,[encArg(V2, Ret01),h(Ret01, Ret3)],[Out = Ret3,V = 1 + V2,V2 >= 0]).
eq(encArg(V, Out),0,[encArg(V3, Ret02),encArg(V4, Ret12),g(Ret02, Ret12, Ret4)],[Out = Ret4,V3 >= 0,V = 1 + V3 + V4,V4 >= 0]).
eq(encArg(V, Out),0,[encArg(V5, Ret03),encArg(V6, Ret13),f(Ret03, Ret13, Ret5)],[Out = Ret5,V5 >= 0,V = 1 + V5 + V6,V6 >= 0]).
eq(encArg(V, Out),0,[a(Ret6)],[Out = Ret6,V = 2]).
eq(encArg(V, Out),0,[encArg(V7, Ret04),activate(Ret04, Ret7)],[Out = Ret7,V = 1 + V7,V7 >= 0]).
eq(fun(V, Out),0,[encArg(V8, Ret05),h(Ret05, Ret8)],[Out = Ret8,V8 >= 0,V = V8]).
eq(fun1(V, V1, Out),0,[encArg(V10, Ret06),encArg(V9, Ret14),g(Ret06, Ret14, Ret9)],[Out = Ret9,V10 >= 0,V9 >= 0,V = V10,V1 = V9]).
eq(fun2(Out),0,[a(Ret10)],[Out = Ret10]).
eq(fun3(V, V1, Out),0,[encArg(V12, Ret07),encArg(V11, Ret15),f(Ret07, Ret15, Ret16)],[Out = Ret16,V12 >= 0,V11 >= 0,V = V12,V1 = V11]).
eq(fun4(Out),0,[],[Out = 0]).
eq(fun5(V, Out),0,[encArg(V13, Ret08),activate(Ret08, Ret17)],[Out = Ret17,V13 >= 0,V = V13]).
eq(a(Out),0,[],[Out = 1]).
eq(encArg(V, Out),0,[],[Out = 0,V14 >= 0,V = V14]).
eq(fun(V, Out),0,[],[Out = 0,V15 >= 0,V = V15]).
eq(fun1(V, V1, Out),0,[],[Out = 0,V17 >= 0,V16 >= 0,V = V17,V1 = V16]).
eq(fun2(Out),0,[],[Out = 0]).
eq(fun3(V, V1, Out),0,[],[Out = 0,V18 >= 0,V19 >= 0,V = V18,V1 = V19]).
eq(fun5(V, Out),0,[],[Out = 0,V20 >= 0,V = V20]).
eq(a(Out),0,[],[Out = 0]).
eq(f(V, V1, Out),0,[],[Out = 0,V21 >= 0,V22 >= 0,V = V21,V1 = V22]).
eq(g(V, V1, Out),0,[],[Out = 0,V23 >= 0,V24 >= 0,V = V23,V1 = V24]).
input_output_vars(h(V,Out),[V],[Out]).
input_output_vars(f(V,V1,Out),[V,V1],[Out]).
input_output_vars(a(Out),[],[Out]).
input_output_vars(activate(V,Out),[V],[Out]).
input_output_vars(g(V,V1,Out),[V,V1],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(fun1(V,V1,Out),[V,V1],[Out]).
input_output_vars(fun2(Out),[],[Out]).
input_output_vars(fun3(V,V1,Out),[V,V1],[Out]).
input_output_vars(fun4(Out),[],[Out]).
input_output_vars(fun5(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [a/1]
1. non_recursive  : [activate/2]
2. recursive  : [f/3,g/3,h/2]
3. recursive [non_tail,multiple] : [encArg/2]
4. non_recursive  : [fun/2]
5. non_recursive  : [fun1/3]
6. non_recursive  : [fun2/1]
7. non_recursive  : [fun3/3]
8. non_recursive  : [fun4/1]
9. non_recursive  : [fun5/2]
10. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into a/1
1. SCC is completely evaluated into other SCCs
2. SCC is partially evaluated into h/2
3. SCC is partially evaluated into encArg/2
4. SCC is partially evaluated into fun/2
5. SCC is partially evaluated into fun1/3
6. SCC is partially evaluated into fun2/1
7. SCC is partially evaluated into fun3/3
8. SCC is completely evaluated into other SCCs
9. SCC is partially evaluated into fun5/2
10. SCC is partially evaluated into start/2

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

### Specialization of cost equations a/1 
* CE 15 is refined into CE [40] 
* CE 14 is refined into CE [41] 
* CE 16 is refined into CE [42] 


### Cost equations --> "Loop" of a/1 
* CEs [40] --> Loop 16 
* CEs [41,42] --> Loop 17 

### Ranking functions of CR a(Out) 

#### Partial ranking functions of CR a(Out) 


### Specialization of cost equations h/2 
* CE 17 is refined into CE [43] 
* CE 18 is refined into CE [44] 


### Cost equations --> "Loop" of h/2 
* CEs [43,44] --> Loop 18 

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

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


### Specialization of cost equations encArg/2 
* CE 23 is refined into CE [45] 
* CE 25 is refined into CE [46,47] 
* CE 24 is refined into CE [48] 
* CE 26 is refined into CE [49] 
* CE 22 is refined into CE [50,51] 
* CE 21 is refined into CE [52,53] 
* CE 19 is refined into CE [54] 
* CE 20 is refined into CE [55] 


### Cost equations --> "Loop" of encArg/2 
* CEs [50,51] --> Loop 19 
* CEs [52,53,54,55] --> Loop 20 
* CEs [49] --> Loop 21 
* CEs [48] --> Loop 22 
* CEs [47] --> Loop 23 
* CEs [45,46] --> Loop 24 

### Ranking functions of CR encArg(V,Out) 
* RF of phase [19,20,21,22]: [V]

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [19,20,21,22]:
  - RF of loop [19:1,19:2,20:1,20:2,21:1,22:1]:
    V


### Specialization of cost equations fun/2 
* CE 27 is refined into CE [56,57,58] 
* CE 28 is refined into CE [59] 


### Cost equations --> "Loop" of fun/2 
* CEs [56,57,58,59] --> Loop 25 

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

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


### Specialization of cost equations fun1/3 
* CE 29 is refined into CE [60,61,62,63,64,65] 
* CE 30 is refined into CE [66,67,68,69,70,71,72,73] 
* CE 31 is refined into CE [74,75,76,77,78,79,80,81,82] 
* CE 32 is refined into CE [83] 


### Cost equations --> "Loop" of fun1/3 
* CEs [63,77,80] --> Loop 26 
* CEs [60,61,62,64,65,66,67,68,69,70,71,72,73,74,75,76,78,79,81,82,83] --> Loop 27 

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

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


### Specialization of cost equations fun2/1 
* CE 33 is refined into CE [84,85] 
* CE 34 is refined into CE [86] 


### Cost equations --> "Loop" of fun2/1 
* CEs [85] --> Loop 28 
* CEs [84,86] --> Loop 29 

### Ranking functions of CR fun2(Out) 

#### Partial ranking functions of CR fun2(Out) 


### Specialization of cost equations fun3/3 
* CE 35 is refined into CE [87,88,89,90,91,92,93,94,95] 
* CE 36 is refined into CE [96,97,98,99,100,101,102,103,104,105,106,107,108,109] 
* CE 37 is refined into CE [110] 


### Cost equations --> "Loop" of fun3/3 
* CEs [90,93,104,105] --> Loop 30 
* CEs [87,88,89,91,92,94,95,96,97,98,99,100,101,102,103,106,107,108,109,110] --> Loop 31 

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

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


### Specialization of cost equations fun5/2 
* CE 38 is refined into CE [111,112,113] 
* CE 39 is refined into CE [114] 


### Cost equations --> "Loop" of fun5/2 
* CEs [113] --> Loop 32 
* CEs [112,114] --> Loop 33 
* CEs [111] --> Loop 34 

### Ranking functions of CR fun5(V,Out) 

#### Partial ranking functions of CR fun5(V,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [115] 
* CE 2 is refined into CE [116] 
* CE 3 is refined into CE [117,118] 
* CE 4 is refined into CE [119,120] 
* CE 5 is refined into CE [121] 
* CE 6 is refined into CE [122,123] 
* CE 7 is refined into CE [124] 
* CE 8 is refined into CE [125,126,127] 
* CE 9 is refined into CE [128] 
* CE 10 is refined into CE [129] 
* CE 11 is refined into CE [130,131] 
* CE 12 is refined into CE [132] 
* CE 13 is refined into CE [133,134,135] 


### Cost equations --> "Loop" of start/2 
* CEs [115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135] --> Loop 35 

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

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


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

#### Cost of chains of a(Out):
* Chain [17]: 1
  with precondition: [Out=0] 

* Chain [16]: 0
  with precondition: [Out=1] 


#### Cost of chains of h(V,Out):
* Chain [18]: 3
  with precondition: [Out=0,V>=0] 


#### Cost of chains of encArg(V,Out):
* Chain [24]: 1
  with precondition: [Out=0,V>=0] 

* Chain [23]: 0
  with precondition: [V=2,Out=1] 

* Chain [multiple([19,20,21,22],[[24],[23]])]: 16*it(19)+1*it([24])+0
  Such that:it([24]) =< V+1
aux(1) =< V
it(19) =< aux(1)

  with precondition: [1>=Out,Out>=0,V>=2*Out+1] 


#### Cost of chains of fun(V,Out):
* Chain [25]: 1*s(1)+16*s(3)+4
  Such that:s(2) =< V
s(1) =< V+1
s(3) =< s(2)

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


#### Cost of chains of fun1(V,V1,Out):
* Chain [27]: 9*s(4)+144*s(6)+8*s(7)+128*s(9)+8
  Such that:aux(2) =< V
aux(3) =< V+1
aux(4) =< V1
aux(5) =< V1+1
s(7) =< aux(3)
s(4) =< aux(5)
s(9) =< aux(2)
s(6) =< aux(4)

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

* Chain [26]: 2*s(55)+32*s(57)+2
  Such that:aux(6) =< V
aux(7) =< V+1
s(55) =< aux(7)
s(57) =< aux(6)

  with precondition: [V1=2,Out=0,V>=0] 


#### Cost of chains of fun2(Out):
* Chain [29]: 1
  with precondition: [Out=0] 

* Chain [28]: 0
  with precondition: [Out=1] 


#### Cost of chains of fun3(V,V1,Out):
* Chain [31]: 9*s(73)+144*s(75)+6*s(79)+96*s(81)+7
  Such that:aux(10) =< V
aux(11) =< V+1
aux(12) =< V1
aux(13) =< V1+1
s(79) =< aux(11)
s(73) =< aux(13)
s(81) =< aux(10)
s(75) =< aux(12)

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

* Chain [30]: 3*s(118)+48*s(120)+5
  Such that:aux(14) =< V
aux(15) =< V+1
s(118) =< aux(15)
s(120) =< aux(14)

  with precondition: [V1=2,Out=0,V>=0] 


#### Cost of chains of fun5(V,Out):
* Chain [34]: 1
  with precondition: [V=2,Out=1] 

* Chain [33]: 2
  with precondition: [Out=0,V>=0] 

* Chain [32]: 1*s(139)+16*s(141)+1
  Such that:s(140) =< V
s(139) =< V+1
s(141) =< s(140)

  with precondition: [1>=Out,Out>=0,V>=2*Out+1] 


#### Cost of chains of start(V,V1):
* Chain [35]: 22*s(142)+352*s(144)+18*s(154)+288*s(155)+8
  Such that:aux(18) =< V
aux(19) =< V+1
aux(20) =< V1
aux(21) =< V1+1
s(142) =< aux(19)
s(144) =< aux(18)
s(154) =< aux(21)
s(155) =< aux(20)

  with precondition: [] 


Closed-form bounds of start(V,V1): 
-------------------------------------
* Chain [35] with precondition: [] 
    - Upper bound: nat(V)*352+8+nat(V1)*288+nat(V+1)*22+nat(V1+1)*18 
    - Complexity: n 

### Maximum cost of start(V,V1): nat(V)*352+8+nat(V1)*288+nat(V+1)*22+nat(V1+1)*18 
Asymptotic class: n 
* Total analysis performed in 441 ms.


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

(16)
BOUNDS(1, n^1)