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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 200 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, 404 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:

   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
   g(b) -> c
   b -> c
   g(X) -> n__g(X)
   activate(n__g(X)) -> g(X)
   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(n__g(x_1)) -> n__g(encArg(x_1))
   encArg(c) -> c
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_n__g(x_1) -> n__g(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b
   encode_c -> c


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

(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, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
   g(b) -> c
   b -> c
   g(X) -> n__g(X)
   activate(n__g(X)) -> g(X)
   activate(X) -> X

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

   encArg(n__g(x_1)) -> n__g(encArg(x_1))
   encArg(c) -> c
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_n__g(x_1) -> n__g(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b
   encode_c -> c

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, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
   g(b) -> c
   b -> c
   g(X) -> n__g(X)
   activate(n__g(X)) -> g(X)
   activate(X) -> X

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

   encArg(n__g(x_1)) -> n__g(encArg(x_1))
   encArg(c) -> c
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_n__g(x_1) -> n__g(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b
   encode_c -> c

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:

   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
   b -> c
   g(X) -> n__g(X)
   activate(n__g(X)) -> g(X)
   activate(X) -> X
   g(c_b) -> c

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

   encArg(n__g(x_1)) -> n__g(encArg(x_1))
   encArg(c) -> c
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_n__g(x_1) -> n__g(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b
   encode_c -> c
   b -> c_b

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:

   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y)) [1]
   b -> c [1]
   g(X) -> n__g(X) [1]
   activate(n__g(X)) -> g(X) [1]
   activate(X) -> X [1]
   g(c_b) -> c [1]
   encArg(n__g(x_1)) -> n__g(encArg(x_1)) [0]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_b) -> b [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_n__g(x_1) -> n__g(encArg(x_1)) [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_b -> b [0]
   encode_c -> c [0]
   b -> c_b [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:

   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y)) [1]
   b -> c [1]
   g(X) -> n__g(X) [1]
   activate(n__g(X)) -> g(X) [1]
   activate(X) -> X [1]
   g(c_b) -> c [1]
   encArg(n__g(x_1)) -> n__g(encArg(x_1)) [0]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_b) -> b [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_n__g(x_1) -> n__g(encArg(x_1)) [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_b -> b [0]
   encode_c -> c [0]
   b -> c_b [0]

The TRS has the following type information:
f :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
n__g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
activate :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
b :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
c :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
c_b :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
encArg :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
cons_f :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
cons_g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
cons_b :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
cons_activate :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
encode_f :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
encode_n__g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
encode_activate :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
encode_g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
encode_b :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate
encode_c :: n__g:c:c_b:cons_f:cons_g:cons_b: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_f(v0, v1, v2) -> null_encode_f [0]
   encode_n__g(v0) -> null_encode_n__g [0]
   encode_activate(v0) -> null_encode_activate [0]
   encode_g(v0) -> null_encode_g [0]
   encode_b -> null_encode_b [0]
   encode_c -> null_encode_c [0]
   b -> null_b [0]
   f(v0, v1, v2) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_n__g, null_encode_activate, null_encode_g, null_encode_b, null_encode_c, null_b, null_f

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

(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:

   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y)) [1]
   b -> c [1]
   g(X) -> n__g(X) [1]
   activate(n__g(X)) -> g(X) [1]
   activate(X) -> X [1]
   g(c_b) -> c [1]
   encArg(n__g(x_1)) -> n__g(encArg(x_1)) [0]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_b) -> b [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_n__g(x_1) -> n__g(encArg(x_1)) [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_b -> b [0]
   encode_c -> c [0]
   b -> c_b [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1, v2) -> null_encode_f [0]
   encode_n__g(v0) -> null_encode_n__g [0]
   encode_activate(v0) -> null_encode_activate [0]
   encode_g(v0) -> null_encode_g [0]
   encode_b -> null_encode_b [0]
   encode_c -> null_encode_c [0]
   b -> null_b [0]
   f(v0, v1, v2) -> null_f [0]

The TRS has the following type information:
f :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
n__g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
activate :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
b :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
c :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
c_b :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
encArg :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
cons_f :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
cons_g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
cons_b :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
cons_activate :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
encode_f :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
encode_n__g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
encode_activate :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
encode_g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f -> n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
encode_b :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
encode_c :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
null_encArg :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
null_encode_f :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
null_encode_n__g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
null_encode_activate :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
null_encode_g :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
null_encode_b :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
null_encode_c :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
null_b :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b:null_f
null_f :: n__g:c:c_b:cons_f:cons_g:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_n__g:null_encode_activate:null_encode_g:null_encode_b:null_encode_c:null_b: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:

   c => 0
   c_b => 1
   cons_b => 2
   null_encArg => 0
   null_encode_f => 0
   null_encode_n__g => 0
   null_encode_activate => 0
   null_encode_g => 0
   null_encode_b => 0
   null_encode_c => 0
   null_b => 0
   null_f => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> g(X) :|: z = 1 + X, X >= 0
   b -{ 0 }-> 1 :|: 
   b -{ 1 }-> 0 :|: 
   b -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> g(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> b :|: z = 2
   encArg(z) -{ 0 }-> activate(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 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 }-> b :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 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
   encode_g(z) -{ 0 }-> g(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_n__g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_n__g(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   f(z, z', z'') -{ 1 }-> f(activate(Y), activate(Y), activate(Y)) :|: Y >= 0, z'' = Y, z' = 1 + X, X >= 0, z = X
   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   g(z) -{ 1 }-> 0 :|: z = 1
   g(z) -{ 1 }-> 1 + 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(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[b(Out)],[]).
eq(start(V1, V, V2),0,[g(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[activate(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[fun(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[fun1(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[fun2(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[fun3(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[fun4(Out)],[]).
eq(start(V1, V, V2),0,[fun5(Out)],[]).
eq(f(V1, V, V2, Out),1,[activate(Y1, Ret0),activate(Y1, Ret1),activate(Y1, Ret2),f(Ret0, Ret1, Ret2, Ret)],[Out = Ret,Y1 >= 0,V2 = Y1,V = 1 + X1,X1 >= 0,V1 = X1]).
eq(b(Out),1,[],[Out = 0]).
eq(g(V1, Out),1,[],[Out = 1 + X2,X2 >= 0,V1 = X2]).
eq(activate(V1, Out),1,[g(X3, Ret3)],[Out = Ret3,V1 = 1 + X3,X3 >= 0]).
eq(activate(V1, Out),1,[],[Out = X4,X4 >= 0,V1 = X4]).
eq(g(V1, Out),1,[],[Out = 0,V1 = 1]).
eq(encArg(V1, Out),0,[encArg(V3, Ret11)],[Out = 1 + Ret11,V1 = 1 + V3,V3 >= 0]).
eq(encArg(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(encArg(V1, Out),0,[encArg(V5, Ret01),encArg(V6, Ret12),encArg(V4, Ret21),f(Ret01, Ret12, Ret21, Ret4)],[Out = Ret4,V5 >= 0,V1 = 1 + V4 + V5 + V6,V4 >= 0,V6 >= 0]).
eq(encArg(V1, Out),0,[encArg(V7, Ret02),g(Ret02, Ret5)],[Out = Ret5,V1 = 1 + V7,V7 >= 0]).
eq(encArg(V1, Out),0,[b(Ret6)],[Out = Ret6,V1 = 2]).
eq(encArg(V1, Out),0,[encArg(V8, Ret03),activate(Ret03, Ret7)],[Out = Ret7,V1 = 1 + V8,V8 >= 0]).
eq(fun(V1, V, V2, Out),0,[encArg(V11, Ret04),encArg(V10, Ret13),encArg(V9, Ret22),f(Ret04, Ret13, Ret22, Ret8)],[Out = Ret8,V11 >= 0,V9 >= 0,V10 >= 0,V1 = V11,V = V10,V2 = V9]).
eq(fun1(V1, Out),0,[encArg(V12, Ret14)],[Out = 1 + Ret14,V12 >= 0,V1 = V12]).
eq(fun2(V1, Out),0,[encArg(V13, Ret05),activate(Ret05, Ret9)],[Out = Ret9,V13 >= 0,V1 = V13]).
eq(fun3(V1, Out),0,[encArg(V14, Ret06),g(Ret06, Ret10)],[Out = Ret10,V14 >= 0,V1 = V14]).
eq(fun4(Out),0,[b(Ret15)],[Out = Ret15]).
eq(fun5(Out),0,[],[Out = 0]).
eq(b(Out),0,[],[Out = 1]).
eq(encArg(V1, Out),0,[],[Out = 0,V15 >= 0,V1 = V15]).
eq(fun(V1, V, V2, Out),0,[],[Out = 0,V17 >= 0,V2 = V18,V16 >= 0,V1 = V17,V = V16,V18 >= 0]).
eq(fun1(V1, Out),0,[],[Out = 0,V19 >= 0,V1 = V19]).
eq(fun2(V1, Out),0,[],[Out = 0,V20 >= 0,V1 = V20]).
eq(fun3(V1, Out),0,[],[Out = 0,V21 >= 0,V1 = V21]).
eq(fun4(Out),0,[],[Out = 0]).
eq(b(Out),0,[],[Out = 0]).
eq(f(V1, V, V2, Out),0,[],[Out = 0,V22 >= 0,V2 = V23,V24 >= 0,V1 = V22,V = V24,V23 >= 0]).
input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(b(Out),[],[Out]).
input_output_vars(g(V1,Out),[V1],[Out]).
input_output_vars(activate(V1,Out),[V1],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(fun1(V1,Out),[V1],[Out]).
input_output_vars(fun2(V1,Out),[V1],[Out]).
input_output_vars(fun3(V1,Out),[V1],[Out]).
input_output_vars(fun4(Out),[],[Out]).
input_output_vars(fun5(Out),[],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into g/2
1. SCC is partially evaluated into activate/2
2. SCC is partially evaluated into b/1
3. SCC is partially evaluated into f/4
4. SCC is partially evaluated into encArg/2
5. SCC is partially evaluated into fun/4
6. SCC is partially evaluated into fun1/2
7. SCC is partially evaluated into fun2/2
8. SCC is partially evaluated into fun3/2
9. SCC is partially evaluated into fun4/1
10. SCC is completely evaluated into other SCCs
11. SCC is partially evaluated into start/3

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

### Specialization of cost equations g/2 
* CE 17 is refined into CE [37] 
* CE 18 is refined into CE [38] 


### Cost equations --> "Loop" of g/2 
* CEs [37] --> Loop 19 
* CEs [38] --> Loop 20 

### Ranking functions of CR g(V1,Out) 

#### Partial ranking functions of CR g(V1,Out) 


### Specialization of cost equations activate/2 
* CE 19 is refined into CE [39,40] 
* CE 20 is refined into CE [41] 


### Cost equations --> "Loop" of activate/2 
* CEs [40,41] --> Loop 21 
* CEs [39] --> Loop 22 

### Ranking functions of CR activate(V1,Out) 

#### Partial ranking functions of CR activate(V1,Out) 


### Specialization of cost equations b/1 
* CE 15 is refined into CE [42] 
* CE 14 is refined into CE [43] 
* CE 16 is refined into CE [44] 


### Cost equations --> "Loop" of b/1 
* CEs [42] --> Loop 23 
* CEs [43,44] --> Loop 24 

### Ranking functions of CR b(Out) 

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


### Specialization of cost equations f/4 
* CE 13 is refined into CE [45] 
* CE 12 is refined into CE [46,47,48,49,50,51,52,53] 


### Cost equations --> "Loop" of f/4 
* CEs [53] --> Loop 25 
* CEs [52] --> Loop 26 
* CEs [51] --> Loop 27 
* CEs [50] --> Loop 28 
* CEs [49] --> Loop 29 
* CEs [48] --> Loop 30 
* CEs [47] --> Loop 31 
* CEs [46] --> Loop 32 
* CEs [45] --> Loop 33 

### Ranking functions of CR f(V1,V,V2,Out) 

#### Partial ranking functions of CR f(V1,V,V2,Out) 


### Specialization of cost equations encArg/2 
* CE 22 is refined into CE [54] 
* CE 25 is refined into CE [55,56] 
* CE 21 is refined into CE [57] 
* CE 24 is refined into CE [58,59] 
* CE 26 is refined into CE [60,61] 
* CE 23 is refined into CE [62] 


### Cost equations --> "Loop" of encArg/2 
* CEs [62] --> Loop 34 
* CEs [61] --> Loop 35 
* CEs [57,59] --> Loop 36 
* CEs [60] --> Loop 37 
* CEs [58] --> Loop 38 
* CEs [56] --> Loop 39 
* CEs [54,55] --> Loop 40 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [34,35,36,37,38]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [34,35,36,37,38]:
  - RF of loop [34:1,34:2,34:3,35:1,36:1,37:1,38:1]:
    V1


### Specialization of cost equations fun/4 
* CE 27 is refined into CE [63,64,65,66,67,68,69,70] 
* CE 28 is refined into CE [71] 


### Cost equations --> "Loop" of fun/4 
* CEs [63,64,65,66,67,68,69,70,71] --> Loop 41 

### Ranking functions of CR fun(V1,V,V2,Out) 

#### Partial ranking functions of CR fun(V1,V,V2,Out) 


### Specialization of cost equations fun1/2 
* CE 29 is refined into CE [72,73] 
* CE 30 is refined into CE [74] 


### Cost equations --> "Loop" of fun1/2 
* CEs [72] --> Loop 42 
* CEs [73] --> Loop 43 
* CEs [74] --> Loop 44 

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

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


### Specialization of cost equations fun2/2 
* CE 31 is refined into CE [75,76,77] 
* CE 32 is refined into CE [78] 


### Cost equations --> "Loop" of fun2/2 
* CEs [76] --> Loop 45 
* CEs [75,77,78] --> Loop 46 

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

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


### Specialization of cost equations fun3/2 
* CE 33 is refined into CE [79,80,81] 
* CE 34 is refined into CE [82] 


### Cost equations --> "Loop" of fun3/2 
* CEs [80] --> Loop 47 
* CEs [81] --> Loop 48 
* CEs [79,82] --> Loop 49 

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

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


### Specialization of cost equations fun4/1 
* CE 35 is refined into CE [83,84] 
* CE 36 is refined into CE [85] 


### Cost equations --> "Loop" of fun4/1 
* CEs [84] --> Loop 50 
* CEs [83,85] --> Loop 51 

### Ranking functions of CR fun4(Out) 

#### Partial ranking functions of CR fun4(Out) 


### Specialization of cost equations start/3 
* CE 1 is refined into CE [86] 
* CE 2 is refined into CE [87,88] 
* CE 3 is refined into CE [89,90] 
* CE 4 is refined into CE [91,92] 
* CE 5 is refined into CE [93,94] 
* CE 6 is refined into CE [95] 
* CE 7 is refined into CE [96,97,98] 
* CE 8 is refined into CE [99,100] 
* CE 9 is refined into CE [101,102,103] 
* CE 10 is refined into CE [104,105] 
* CE 11 is refined into CE [106] 


### Cost equations --> "Loop" of start/3 
* CEs [86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106] --> Loop 52 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


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

#### Cost of chains of g(V1,Out):
* Chain [20]: 1
  with precondition: [V1=1,Out=0] 

* Chain [19]: 1
  with precondition: [V1+1=Out,V1>=0] 


#### Cost of chains of activate(V1,Out):
* Chain [22]: 2
  with precondition: [V1=2,Out=0] 

* Chain [21]: 2
  with precondition: [V1=Out,V1>=0] 


#### Cost of chains of b(Out):
* Chain [24]: 1
  with precondition: [Out=0] 

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


#### Cost of chains of f(V1,V,V2,Out):
* Chain [33]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [32,33]: 7
  with precondition: [V2=2,Out=0,V1+1=V,V1>=0] 

* Chain [31,33]: 7
  with precondition: [V2=2,Out=0,V1+1=V,V1>=0] 

* Chain [30,33]: 7
  with precondition: [V2=2,Out=0,V1+1=V,V1>=0] 

* Chain [29,33]: 7
  with precondition: [V2=2,Out=0,V1+1=V,V1>=0] 

* Chain [28,33]: 7
  with precondition: [V2=2,Out=0,V1+1=V,V1>=0] 

* Chain [27,33]: 7
  with precondition: [V2=2,Out=0,V1+1=V,V1>=0] 

* Chain [26,33]: 7
  with precondition: [V2=2,Out=0,V1+1=V,V1>=0] 

* Chain [25,33]: 7
  with precondition: [Out=0,V1+1=V,V1>=0,V2>=0] 


#### Cost of chains of encArg(V1,Out):
* Chain [40]: 1
  with precondition: [Out=0,V1>=0] 

* Chain [39]: 0
  with precondition: [V1=2,Out=1] 

* Chain [multiple([34,35,36,37,38],[[40],[39]])]: 13*it(34)+1*it([40])+0
  Such that:it([40]) =< 2*V1+1
aux(1) =< V1
it(34) =< aux(1)

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


#### Cost of chains of fun(V1,V,V2,Out):
* Chain [41]: 4*s(5)+52*s(6)+4*s(8)+52*s(9)+4*s(11)+52*s(12)+10
  Such that:aux(2) =< V1
aux(3) =< 2*V1+1
aux(4) =< V
aux(5) =< 2*V+1
aux(6) =< V2
aux(7) =< 2*V2+1
s(5) =< aux(3)
s(8) =< aux(5)
s(11) =< aux(7)
s(12) =< aux(6)
s(9) =< aux(4)
s(6) =< aux(2)

  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 


#### Cost of chains of fun1(V1,Out):
* Chain [44]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [43]: 1
  with precondition: [Out=1,V1>=0] 

* Chain [42]: 1*s(41)+13*s(42)+0
  Such that:s(40) =< V1
s(41) =< 2*V1+1
s(42) =< s(40)

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


#### Cost of chains of fun2(V1,Out):
* Chain [46]: 1*s(44)+13*s(45)+3
  Such that:s(43) =< V1
s(44) =< 2*V1+1
s(45) =< s(43)

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

* Chain [45]: 1*s(47)+13*s(48)+2
  Such that:s(46) =< V1
s(47) =< 2*V1+1
s(48) =< s(46)

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


#### Cost of chains of fun3(V1,Out):
* Chain [49]: 1*s(50)+13*s(51)+1
  Such that:s(49) =< V1
s(50) =< 2*V1+1
s(51) =< s(49)

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

* Chain [48]: 2
  with precondition: [Out=1,V1>=0] 

* Chain [47]: 1*s(53)+13*s(54)+1
  Such that:s(52) =< V1
s(53) =< 2*V1+1
s(54) =< s(52)

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


#### Cost of chains of fun4(Out):
* Chain [51]: 1
  with precondition: [Out=0] 

* Chain [50]: 0
  with precondition: [Out=1] 


#### Cost of chains of start(V1,V,V2):
* Chain [52]: 10*s(56)+130*s(57)+4*s(65)+4*s(66)+52*s(67)+52*s(68)+10
  Such that:s(60) =< V
s(61) =< 2*V+1
s(62) =< V2
s(63) =< 2*V2+1
aux(8) =< V1
aux(9) =< 2*V1+1
s(56) =< aux(9)
s(57) =< aux(8)
s(65) =< s(61)
s(66) =< s(63)
s(67) =< s(62)
s(68) =< s(60)

  with precondition: [] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [52] with precondition: [] 
    - Upper bound: nat(V1)*130+10+nat(V)*52+nat(V2)*52+nat(2*V1+1)*10+nat(2*V+1)*4+nat(2*V2+1)*4 
    - Complexity: n 

### Maximum cost of start(V1,V,V2): nat(V1)*130+10+nat(V)*52+nat(V2)*52+nat(2*V1+1)*10+nat(2*V+1)*4+nat(2*V2+1)*4 
Asymptotic class: n 
* Total analysis performed in 406 ms.


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

(16)
BOUNDS(1, n^1)