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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(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) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 7 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 702 ms]
    (18) BOUNDS(1, n^2)
(19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 279 ms]
    (26) proven lower bound
    (27) LowerBoundPropagationProof [FINISHED, 0 ms]
    (28) BOUNDS(n^1, INF)


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   U11(tt) -> U12(tt)
   U12(tt) -> tt
   isNePal(__(I, __(P, I))) -> U11(tt)
   activate(X) -> X

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

(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(nil) -> nil
   encArg(tt) -> tt
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(cons_U11(x_1)) -> U11(encArg(x_1))
   encArg(cons_U12(x_1)) -> U12(encArg(x_1))
   encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_tt -> tt
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))


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

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


The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   U11(tt) -> U12(tt)
   U12(tt) -> tt
   isNePal(__(I, __(P, I))) -> U11(tt)
   activate(X) -> X

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

   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(cons_U11(x_1)) -> U11(encArg(x_1))
   encArg(cons_U12(x_1)) -> U12(encArg(x_1))
   encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_tt -> tt
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   U11(tt) -> U12(tt)
   U12(tt) -> tt
   isNePal(__(I, __(P, I))) -> U11(tt)
   activate(X) -> X

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

   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(cons_U11(x_1)) -> U11(encArg(x_1))
   encArg(cons_U12(x_1)) -> U12(encArg(x_1))
   encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_tt -> tt
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

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

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   __(X, nil) -> X
   __(nil, X) -> X
   U11(tt) -> U12(tt)
   U12(tt) -> tt
   activate(X) -> X
   isNePal(c___(I, c___(P, I))) -> U11(tt)
   __(c___(X, Y), Z) -> __(X, __(Y, Z))

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

   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(cons_U11(x_1)) -> U11(encArg(x_1))
   encArg(cons_U12(x_1)) -> U12(encArg(x_1))
   encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_tt -> tt
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   __(x0, x1) -> c___(x0, x1)

Rewrite Strategy: FULL
----------------------------------------

(7) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   __(X, nil) -> X
   __(nil, X) -> X
   U11(tt) -> U12(tt)
   U12(tt) -> tt
   activate(X) -> X
   isNePal(c___(I, c___(P, I))) -> U11(tt)
   __(c___(X, Y), Z) -> __(X, __(Y, Z))

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

   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(cons_U11(x_1)) -> U11(encArg(x_1))
   encArg(cons_U12(x_1)) -> U12(encArg(x_1))
   encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_tt -> tt
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   __(x0, x1) -> c___(x0, x1)

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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   __(X, nil) -> X [1]
   __(nil, X) -> X [1]
   U11(tt) -> U12(tt) [1]
   U12(tt) -> tt [1]
   activate(X) -> X [1]
   isNePal(c___(I, c___(P, I))) -> U11(tt) [1]
   __(c___(X, Y), Z) -> __(X, __(Y, Z)) [1]
   encArg(nil) -> nil [0]
   encArg(tt) -> tt [0]
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_U11(x_1)) -> U11(encArg(x_1)) [0]
   encArg(cons_U12(x_1)) -> U12(encArg(x_1)) [0]
   encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1)) [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_U11(x_1) -> U11(encArg(x_1)) [0]
   encode_tt -> tt [0]
   encode_U12(x_1) -> U12(encArg(x_1)) [0]
   encode_isNePal(x_1) -> isNePal(encArg(x_1)) [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   __(x0, x1) -> c___(x0, x1) [0]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   __(X, nil) -> X [1]
   __(nil, X) -> X [1]
   U11(tt) -> U12(tt) [1]
   U12(tt) -> tt [1]
   activate(X) -> X [1]
   isNePal(c___(I, c___(P, I))) -> U11(tt) [1]
   __(c___(X, Y), Z) -> __(X, __(Y, Z)) [1]
   encArg(nil) -> nil [0]
   encArg(tt) -> tt [0]
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_U11(x_1)) -> U11(encArg(x_1)) [0]
   encArg(cons_U12(x_1)) -> U12(encArg(x_1)) [0]
   encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1)) [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_U11(x_1) -> U11(encArg(x_1)) [0]
   encode_tt -> tt [0]
   encode_U12(x_1) -> U12(encArg(x_1)) [0]
   encode_isNePal(x_1) -> isNePal(encArg(x_1)) [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   __(x0, x1) -> c___(x0, x1) [0]

The TRS has the following type information:
__ :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
nil :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
U11 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
tt :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
U12 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
activate :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
isNePal :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
c___ :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encArg :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons___ :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_U11 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_U12 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_isNePal :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_activate :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode___ :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_nil :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_U11 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_tt :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_U12 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_isNePal :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_activate :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate

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

(13) 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___(v0, v1) -> null_encode___ [0]
   encode_nil -> null_encode_nil [0]
   encode_U11(v0) -> null_encode_U11 [0]
   encode_tt -> null_encode_tt [0]
   encode_U12(v0) -> null_encode_U12 [0]
   encode_isNePal(v0) -> null_encode_isNePal [0]
   encode_activate(v0) -> null_encode_activate [0]
   __(v0, v1) -> null___ [0]
   U11(v0) -> null_U11 [0]
   U12(v0) -> null_U12 [0]
   isNePal(v0) -> null_isNePal [0]

And the following fresh constants:    null_encArg, null_encode___, null_encode_nil, null_encode_U11, null_encode_tt, null_encode_U12, null_encode_isNePal, null_encode_activate, null___, null_U11, null_U12, null_isNePal

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

(14)
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:

   __(X, nil) -> X [1]
   __(nil, X) -> X [1]
   U11(tt) -> U12(tt) [1]
   U12(tt) -> tt [1]
   activate(X) -> X [1]
   isNePal(c___(I, c___(P, I))) -> U11(tt) [1]
   __(c___(X, Y), Z) -> __(X, __(Y, Z)) [1]
   encArg(nil) -> nil [0]
   encArg(tt) -> tt [0]
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_U11(x_1)) -> U11(encArg(x_1)) [0]
   encArg(cons_U12(x_1)) -> U12(encArg(x_1)) [0]
   encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1)) [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_U11(x_1) -> U11(encArg(x_1)) [0]
   encode_tt -> tt [0]
   encode_U12(x_1) -> U12(encArg(x_1)) [0]
   encode_isNePal(x_1) -> isNePal(encArg(x_1)) [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   __(x0, x1) -> c___(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode___(v0, v1) -> null_encode___ [0]
   encode_nil -> null_encode_nil [0]
   encode_U11(v0) -> null_encode_U11 [0]
   encode_tt -> null_encode_tt [0]
   encode_U12(v0) -> null_encode_U12 [0]
   encode_isNePal(v0) -> null_encode_isNePal [0]
   encode_activate(v0) -> null_encode_activate [0]
   __(v0, v1) -> null___ [0]
   U11(v0) -> null_U11 [0]
   U12(v0) -> null_U12 [0]
   isNePal(v0) -> null_isNePal [0]

The TRS has the following type information:
__ :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
nil :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
U11 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
tt :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
U12 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
activate :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
isNePal :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
c___ :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
encArg :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
cons___ :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
cons_U11 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
cons_U12 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
cons_isNePal :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
cons_activate :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
encode___ :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
encode_nil :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
encode_U11 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
encode_tt :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
encode_U12 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
encode_isNePal :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
encode_activate :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal -> nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_encArg :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_encode___ :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_encode_nil :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_encode_U11 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_encode_tt :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_encode_U12 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_encode_isNePal :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_encode_activate :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null___ :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_U11 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_U12 :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal
null_isNePal :: nil:tt:c___:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_isNePal:null_encode_activate:null___:null_U11:null_U12:null_isNePal

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

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

   nil => 0
   tt => 1
   null_encArg => 0
   null_encode___ => 0
   null_encode_nil => 0
   null_encode_U11 => 0
   null_encode_tt => 0
   null_encode_U12 => 0
   null_encode_isNePal => 0
   null_encode_activate => 0
   null___ => 0
   null_U11 => 0
   null_U12 => 0
   null_isNePal => 0

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

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

   U11(z) -{ 1 }-> U12(1) :|: z = 1
   U11(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   U12(z) -{ 1 }-> 1 :|: z = 1
   U12(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   __(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0
   __(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0
   __(z, z') -{ 1 }-> __(X, __(Y, Z)) :|: Z >= 0, Y >= 0, z = 1 + X + Y, X >= 0, z' = Z
   __(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   __(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1
   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   encArg(z) -{ 0 }-> isNePal(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> activate(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> __(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_U11(z) -{ 0 }-> U11(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_U11(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_U12(z) -{ 0 }-> U12(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_U12(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode___(z, z') -{ 0 }-> __(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode___(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_activate(z) -{ 0 }-> activate(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_isNePal(z) -{ 0 }-> isNePal(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_isNePal(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_nil -{ 0 }-> 0 :|: 
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   isNePal(z) -{ 1 }-> U11(1) :|: z = 1 + I + (1 + P + I), I >= 0, P >= 0
   isNePal(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun2(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[activate(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[isNePal(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun3(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun4(Out)],[]).
eq(start(V1, V),0,[fun5(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun6(Out)],[]).
eq(start(V1, V),0,[fun7(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun8(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun9(V1, Out)],[V1 >= 0]).
eq(fun(V1, V, Out),1,[],[Out = X1,X1 >= 0,V1 = X1,V = 0]).
eq(fun(V1, V, Out),1,[],[Out = X2,V = X2,X2 >= 0,V1 = 0]).
eq(fun1(V1, Out),1,[fun2(1, Ret)],[Out = Ret,V1 = 1]).
eq(fun2(V1, Out),1,[],[Out = 1,V1 = 1]).
eq(activate(V1, Out),1,[],[Out = X3,X3 >= 0,V1 = X3]).
eq(isNePal(V1, Out),1,[fun1(1, Ret1)],[Out = Ret1,V1 = 2 + 2*I1 + P1,I1 >= 0,P1 >= 0]).
eq(fun(V1, V, Out),1,[fun(Y1, Z1, Ret11),fun(X4, Ret11, Ret2)],[Out = Ret2,Z1 >= 0,Y1 >= 0,V1 = 1 + X4 + Y1,X4 >= 0,V = Z1]).
eq(encArg(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(encArg(V1, Out),0,[],[Out = 1,V1 = 1]).
eq(encArg(V1, Out),0,[encArg(V3, Ret0),encArg(V2, Ret12),fun(Ret0, Ret12, Ret3)],[Out = Ret3,V3 >= 0,V1 = 1 + V2 + V3,V2 >= 0]).
eq(encArg(V1, Out),0,[encArg(V4, Ret01),fun1(Ret01, Ret4)],[Out = Ret4,V1 = 1 + V4,V4 >= 0]).
eq(encArg(V1, Out),0,[encArg(V5, Ret02),fun2(Ret02, Ret5)],[Out = Ret5,V1 = 1 + V5,V5 >= 0]).
eq(encArg(V1, Out),0,[encArg(V6, Ret03),isNePal(Ret03, Ret6)],[Out = Ret6,V1 = 1 + V6,V6 >= 0]).
eq(encArg(V1, Out),0,[encArg(V7, Ret04),activate(Ret04, Ret7)],[Out = Ret7,V1 = 1 + V7,V7 >= 0]).
eq(fun3(V1, V, Out),0,[encArg(V9, Ret05),encArg(V8, Ret13),fun(Ret05, Ret13, Ret8)],[Out = Ret8,V9 >= 0,V8 >= 0,V1 = V9,V = V8]).
eq(fun4(Out),0,[],[Out = 0]).
eq(fun5(V1, Out),0,[encArg(V10, Ret06),fun1(Ret06, Ret9)],[Out = Ret9,V10 >= 0,V1 = V10]).
eq(fun6(Out),0,[],[Out = 1]).
eq(fun7(V1, Out),0,[encArg(V11, Ret07),fun2(Ret07, Ret10)],[Out = Ret10,V11 >= 0,V1 = V11]).
eq(fun8(V1, Out),0,[encArg(V12, Ret08),isNePal(Ret08, Ret14)],[Out = Ret14,V12 >= 0,V1 = V12]).
eq(fun9(V1, Out),0,[encArg(V13, Ret09),activate(Ret09, Ret15)],[Out = Ret15,V13 >= 0,V1 = V13]).
eq(fun(V1, V, Out),0,[],[Out = 1 + V14 + V15,V1 = V15,V15 >= 0,V14 >= 0,V = V14]).
eq(encArg(V1, Out),0,[],[Out = 0,V16 >= 0,V1 = V16]).
eq(fun3(V1, V, Out),0,[],[Out = 0,V18 >= 0,V17 >= 0,V1 = V18,V = V17]).
eq(fun5(V1, Out),0,[],[Out = 0,V19 >= 0,V1 = V19]).
eq(fun6(Out),0,[],[Out = 0]).
eq(fun7(V1, Out),0,[],[Out = 0,V20 >= 0,V1 = V20]).
eq(fun8(V1, Out),0,[],[Out = 0,V21 >= 0,V1 = V21]).
eq(fun9(V1, Out),0,[],[Out = 0,V22 >= 0,V1 = V22]).
eq(fun(V1, V, Out),0,[],[Out = 0,V23 >= 0,V24 >= 0,V1 = V23,V = V24]).
eq(fun1(V1, Out),0,[],[Out = 0,V25 >= 0,V1 = V25]).
eq(fun2(V1, Out),0,[],[Out = 0,V26 >= 0,V1 = V26]).
eq(isNePal(V1, Out),0,[],[Out = 0,V27 >= 0,V1 = V27]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,Out),[V1],[Out]).
input_output_vars(fun2(V1,Out),[V1],[Out]).
input_output_vars(activate(V1,Out),[V1],[Out]).
input_output_vars(isNePal(V1,Out),[V1],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun3(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun4(Out),[],[Out]).
input_output_vars(fun5(V1,Out),[V1],[Out]).
input_output_vars(fun6(Out),[],[Out]).
input_output_vars(fun7(V1,Out),[V1],[Out]).
input_output_vars(fun8(V1,Out),[V1],[Out]).
input_output_vars(fun9(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [activate/2]
1. recursive [multiple] : [fun/3]
2. non_recursive  : [fun2/2]
3. non_recursive  : [fun1/2]
4. non_recursive  : [isNePal/2]
5. recursive [non_tail,multiple] : [encArg/2]
6. non_recursive  : [fun3/3]
7. non_recursive  : [fun4/1]
8. non_recursive  : [fun5/2]
9. non_recursive  : [fun6/1]
10. non_recursive  : [fun7/2]
11. non_recursive  : [fun8/2]
12. non_recursive  : [fun9/2]
13. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is completely evaluated into other SCCs
1. SCC is partially evaluated into fun/3
2. SCC is partially evaluated into fun2/2
3. SCC is partially evaluated into fun1/2
4. SCC is partially evaluated into isNePal/2
5. SCC is partially evaluated into encArg/2
6. SCC is partially evaluated into fun3/3
7. SCC is completely evaluated into other SCCs
8. SCC is partially evaluated into fun5/2
9. SCC is partially evaluated into fun6/1
10. SCC is partially evaluated into fun7/2
11. SCC is partially evaluated into fun8/2
12. SCC is partially evaluated into fun9/2
13. SCC is partially evaluated into start/2

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

### Specialization of cost equations fun/3 
* CE 17 is refined into CE [44] 
* CE 18 is refined into CE [45] 
* CE 14 is refined into CE [46] 
* CE 15 is refined into CE [47] 
* CE 16 is refined into CE [48] 


### Cost equations --> "Loop" of fun/3 
* CEs [48] --> Loop 24 
* CEs [44] --> Loop 25 
* CEs [45] --> Loop 26 
* CEs [46] --> Loop 27 
* CEs [47] --> Loop 28 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [24]: [V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [24]:
  - RF of loop [24:1,24:2]:
    V1


### Specialization of cost equations fun2/2 
* CE 22 is refined into CE [49] 
* CE 21 is refined into CE [50] 


### Cost equations --> "Loop" of fun2/2 
* CEs [49] --> Loop 29 
* CEs [50] --> Loop 30 

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

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


### Specialization of cost equations fun1/2 
* CE 20 is refined into CE [51] 
* CE 19 is refined into CE [52,53] 


### Cost equations --> "Loop" of fun1/2 
* CEs [52] --> Loop 31 
* CEs [51,53] --> Loop 32 

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

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


### Specialization of cost equations isNePal/2 
* CE 23 is refined into CE [54,55] 
* CE 24 is refined into CE [56] 


### Cost equations --> "Loop" of isNePal/2 
* CEs [54] --> Loop 33 
* CEs [55,56] --> Loop 34 

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

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


### Specialization of cost equations encArg/2 
* CE 25 is refined into CE [57] 
* CE 26 is refined into CE [58] 
* CE 28 is refined into CE [59,60] 
* CE 29 is refined into CE [61,62] 
* CE 30 is refined into CE [63,64] 
* CE 31 is refined into CE [65] 
* CE 27 is refined into CE [66,67,68,69,70] 


### Cost equations --> "Loop" of encArg/2 
* CEs [70] --> Loop 35 
* CEs [69] --> Loop 36 
* CEs [67] --> Loop 37 
* CEs [66] --> Loop 38 
* CEs [68] --> Loop 39 
* CEs [64] --> Loop 40 
* CEs [59,61,65] --> Loop 41 
* CEs [60,62,63] --> Loop 42 
* CEs [57] --> Loop 43 
* CEs [58] --> Loop 44 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [35,36,37,38,39,40,41,42]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [35,36,37,38,39,40,41,42]:
  - RF of loop [35:1,35:2,36:1,36:2,37:1,37:2,38:1,38:2,39:1,39:2,40:1,41:1,42:1]:
    V1


### Specialization of cost equations fun3/3 
* CE 32 is refined into CE [71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88] 
* CE 33 is refined into CE [89] 


### Cost equations --> "Loop" of fun3/3 
* CEs [74,75] --> Loop 45 
* CEs [72,77,79,80] --> Loop 46 
* CEs [84] --> Loop 47 
* CEs [71,81] --> Loop 48 
* CEs [88] --> Loop 49 
* CEs [73,76,78,82,83,85,86,87,89] --> Loop 50 

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

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


### Specialization of cost equations fun5/2 
* CE 34 is refined into CE [90,91,92] 
* CE 35 is refined into CE [93] 


### Cost equations --> "Loop" of fun5/2 
* CEs [90] --> Loop 51 
* CEs [91,92,93] --> Loop 52 

### Ranking functions of CR fun5(V1,Out) 

#### Partial ranking functions of CR fun5(V1,Out) 


### Specialization of cost equations fun6/1 
* CE 36 is refined into CE [94] 
* CE 37 is refined into CE [95] 


### Cost equations --> "Loop" of fun6/1 
* CEs [94] --> Loop 53 
* CEs [95] --> Loop 54 

### Ranking functions of CR fun6(Out) 

#### Partial ranking functions of CR fun6(Out) 


### Specialization of cost equations fun7/2 
* CE 38 is refined into CE [96,97,98] 
* CE 39 is refined into CE [99] 


### Cost equations --> "Loop" of fun7/2 
* CEs [96] --> Loop 55 
* CEs [97,98,99] --> Loop 56 

### Ranking functions of CR fun7(V1,Out) 

#### Partial ranking functions of CR fun7(V1,Out) 


### Specialization of cost equations fun8/2 
* CE 40 is refined into CE [100,101,102] 
* CE 41 is refined into CE [103] 


### Cost equations --> "Loop" of fun8/2 
* CEs [101] --> Loop 57 
* CEs [100,102,103] --> Loop 58 

### Ranking functions of CR fun8(V1,Out) 

#### Partial ranking functions of CR fun8(V1,Out) 


### Specialization of cost equations fun9/2 
* CE 42 is refined into CE [104,105] 
* CE 43 is refined into CE [106] 


### Cost equations --> "Loop" of fun9/2 
* CEs [104] --> Loop 59 
* CEs [105,106] --> Loop 60 

### Ranking functions of CR fun9(V1,Out) 

#### Partial ranking functions of CR fun9(V1,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [107,108,109,110,111] 
* CE 2 is refined into CE [112,113] 
* CE 3 is refined into CE [114,115] 
* CE 4 is refined into CE [116] 
* CE 5 is refined into CE [117,118] 
* CE 6 is refined into CE [119,120] 
* CE 7 is refined into CE [121,122,123,124,125,126] 
* CE 8 is refined into CE [127] 
* CE 9 is refined into CE [128,129] 
* CE 10 is refined into CE [130,131] 
* CE 11 is refined into CE [132,133] 
* CE 12 is refined into CE [134,135] 
* CE 13 is refined into CE [136,137] 


### Cost equations --> "Loop" of start/2 
* CEs [107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137] --> Loop 61 

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

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


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

#### Cost of chains of fun(V1,V,Out):
* Chain [28]: 1
  with precondition: [V1=0,V=Out,V>=0] 

* Chain [27]: 1
  with precondition: [V=0,V1=Out,V1>=0] 

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

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

* Chain [multiple([24],[[28],[27],[26],[25]])]: 1*it(24)+2*it([27])+0
  Such that:it(24) =< V1
aux(1) =< V1+1
it([27]) =< aux(1)

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


#### Cost of chains of fun2(V1,Out):
* Chain [30]: 1
  with precondition: [V1=1,Out=1] 

* Chain [29]: 0
  with precondition: [Out=0,V1>=0] 


#### Cost of chains of fun1(V1,Out):
* Chain [32]: 1
  with precondition: [Out=0,V1>=0] 

* Chain [31]: 2
  with precondition: [V1=1,Out=1] 


#### Cost of chains of isNePal(V1,Out):
* Chain [34]: 2
  with precondition: [Out=0,V1>=0] 

* Chain [33]: 3
  with precondition: [Out=1,V1>=2] 


#### Cost of chains of encArg(V1,Out):
* Chain [44]: 0
  with precondition: [V1=1,Out=1] 

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

* Chain [multiple([35,36,37,38,39,40,41,42],[[44],[43]])]: 9*it(37)+3*s(6)+0
  Such that:aux(4) =< V1
aux(6) =< 2*V1+1
it(37) =< aux(4)
it(35) =< aux(6)
it(37) =< aux(6)
s(7) =< it(35)*aux(4)
s(6) =< s(7)

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


#### Cost of chains of fun3(V1,V,Out):
* Chain [50]: 27*s(16)+9*s(18)+27*s(21)+9*s(23)+1
  Such that:aux(7) =< V1
aux(8) =< 2*V1+1
aux(9) =< V
aux(10) =< 2*V+1
s(21) =< aux(9)
s(21) =< aux(10)
s(22) =< aux(10)*aux(9)
s(23) =< s(22)
s(16) =< aux(7)
s(16) =< aux(8)
s(17) =< aux(8)*aux(7)
s(18) =< s(17)

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

* Chain [49]: 0
  with precondition: [Out=1,V1>=0,V>=0] 

* Chain [48]: 9*s(46)+3*s(48)+18*s(51)+6*s(53)+1
  Such that:s(44) =< V1
s(45) =< 2*V1+1
aux(11) =< V
aux(12) =< 2*V+1
s(51) =< aux(11)
s(51) =< aux(12)
s(52) =< aux(12)*aux(11)
s(53) =< s(52)
s(46) =< s(44)
s(46) =< s(45)
s(47) =< s(45)*s(44)
s(48) =< s(47)

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

* Chain [47]: 9*s(61)+3*s(63)+0
  Such that:s(59) =< V
s(60) =< 2*V+1
s(61) =< s(59)
s(61) =< s(60)
s(62) =< s(60)*s(59)
s(63) =< s(62)

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

* Chain [46]: 36*s(66)+12*s(68)+9*s(71)+3*s(73)+1*s(89)+2*s(91)+1
  Such that:s(90) =< V1+1
s(69) =< V
s(70) =< 2*V+1
aux(14) =< V1
aux(15) =< 2*V1+1
s(71) =< s(69)
s(71) =< s(70)
s(72) =< s(70)*s(69)
s(73) =< s(72)
s(66) =< aux(14)
s(66) =< aux(15)
s(67) =< aux(15)*aux(14)
s(68) =< s(67)
s(89) =< aux(14)
s(91) =< s(90)

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

* Chain [45]: 18*s(94)+6*s(96)+18*s(99)+6*s(101)+1*s(112)+2*s(114)+0
  Such that:s(113) =< V1+1
aux(17) =< V1
aux(18) =< 2*V1+1
aux(19) =< V
aux(20) =< 2*V+1
s(99) =< aux(19)
s(99) =< aux(20)
s(100) =< aux(20)*aux(19)
s(101) =< s(100)
s(94) =< aux(17)
s(94) =< aux(18)
s(95) =< aux(18)*aux(17)
s(96) =< s(95)
s(112) =< aux(17)
s(114) =< s(113)

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


#### Cost of chains of fun5(V1,Out):
* Chain [52]: 9*s(117)+3*s(119)+1
  Such that:s(115) =< V1
s(116) =< 2*V1+1
s(117) =< s(115)
s(117) =< s(116)
s(118) =< s(116)*s(115)
s(119) =< s(118)

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

* Chain [51]: 9*s(122)+3*s(124)+2
  Such that:s(120) =< V1
s(121) =< 2*V1+1
s(122) =< s(120)
s(122) =< s(121)
s(123) =< s(121)*s(120)
s(124) =< s(123)

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


#### Cost of chains of fun6(Out):
* Chain [54]: 0
  with precondition: [Out=0] 

* Chain [53]: 0
  with precondition: [Out=1] 


#### Cost of chains of fun7(V1,Out):
* Chain [56]: 9*s(127)+3*s(129)+0
  Such that:s(125) =< V1
s(126) =< 2*V1+1
s(127) =< s(125)
s(127) =< s(126)
s(128) =< s(126)*s(125)
s(129) =< s(128)

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

* Chain [55]: 9*s(132)+3*s(134)+1
  Such that:s(130) =< V1
s(131) =< 2*V1+1
s(132) =< s(130)
s(132) =< s(131)
s(133) =< s(131)*s(130)
s(134) =< s(133)

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


#### Cost of chains of fun8(V1,Out):
* Chain [58]: 9*s(137)+3*s(139)+2
  Such that:s(135) =< V1
s(136) =< 2*V1+1
s(137) =< s(135)
s(137) =< s(136)
s(138) =< s(136)*s(135)
s(139) =< s(138)

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

* Chain [57]: 9*s(142)+3*s(144)+3
  Such that:s(140) =< V1
s(141) =< 2*V1+1
s(142) =< s(140)
s(142) =< s(141)
s(143) =< s(141)*s(140)
s(144) =< s(143)

  with precondition: [Out=1,V1>=2] 


#### Cost of chains of fun9(V1,Out):
* Chain [60]: 1
  with precondition: [Out=0,V1>=0] 

* Chain [59]: 9*s(147)+3*s(149)+1
  Such that:s(145) =< V1
s(146) =< 2*V1+1
s(147) =< s(145)
s(147) =< s(146)
s(148) =< s(146)*s(145)
s(149) =< s(148)

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


#### Cost of chains of start(V1,V):
* Chain [61]: 3*s(150)+6*s(152)+162*s(155)+54*s(157)+81*s(162)+27*s(164)+3
  Such that:aux(21) =< V1
aux(22) =< V1+1
aux(23) =< 2*V1+1
aux(24) =< V
aux(25) =< 2*V+1
s(150) =< aux(21)
s(152) =< aux(22)
s(155) =< aux(21)
s(155) =< aux(23)
s(156) =< aux(23)*aux(21)
s(157) =< s(156)
s(162) =< aux(24)
s(162) =< aux(25)
s(163) =< aux(25)*aux(24)
s(164) =< s(163)

  with precondition: [] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [61] with precondition: [] 
    - Upper bound: nat(V1)*165+3+nat(V1)*54*nat(2*V1+1)+nat(V)*81+nat(V)*27*nat(2*V+1)+nat(V1+1)*6 
    - Complexity: n^2 

### Maximum cost of start(V1,V): nat(V1)*165+3+nat(V1)*54*nat(2*V1+1)+nat(V)*81+nat(V)*27*nat(2*V+1)+nat(V1+1)*6 
Asymptotic class: n^2 
* Total analysis performed in 594 ms.


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

(18)
BOUNDS(1, n^2)

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

(19) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(20)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   U11(tt) -> U12(tt)
   U12(tt) -> tt
   isNePal(__(I, __(P, I))) -> U11(tt)
   activate(X) -> X

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

   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(cons_U11(x_1)) -> U11(encArg(x_1))
   encArg(cons_U12(x_1)) -> U12(encArg(x_1))
   encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_tt -> tt
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

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

(22)
Obligation:
TRS:
Rules:
__(__(X, Y), Z) -> __(X, __(Y, Z))
__(X, nil) -> X
__(nil, X) -> X
U11(tt) -> U12(tt)
U12(tt) -> tt
isNePal(__(I, __(P, I))) -> U11(tt)
activate(X) -> X
encArg(nil) -> nil
encArg(tt) -> tt
encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
encArg(cons_U11(x_1)) -> U11(encArg(x_1))
encArg(cons_U12(x_1)) -> U12(encArg(x_1))
encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_U11(x_1) -> U11(encArg(x_1))
encode_tt -> tt
encode_U12(x_1) -> U12(encArg(x_1))
encode_isNePal(x_1) -> isNePal(encArg(x_1))
encode_activate(x_1) -> activate(encArg(x_1))

Types:
__ :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
nil :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
U11 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
tt :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
U12 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
isNePal :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
activate :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encArg :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons___ :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_U11 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_U12 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_isNePal :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_activate :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode___ :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_nil :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_U11 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_tt :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_U12 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_isNePal :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_activate :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
hole_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate1_0 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0 :: Nat -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
__, encArg

They will be analysed ascendingly in the following order:
__ < encArg

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

(24)
Obligation:
TRS:
Rules:
__(__(X, Y), Z) -> __(X, __(Y, Z))
__(X, nil) -> X
__(nil, X) -> X
U11(tt) -> U12(tt)
U12(tt) -> tt
isNePal(__(I, __(P, I))) -> U11(tt)
activate(X) -> X
encArg(nil) -> nil
encArg(tt) -> tt
encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
encArg(cons_U11(x_1)) -> U11(encArg(x_1))
encArg(cons_U12(x_1)) -> U12(encArg(x_1))
encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_U11(x_1) -> U11(encArg(x_1))
encode_tt -> tt
encode_U12(x_1) -> U12(encArg(x_1))
encode_isNePal(x_1) -> isNePal(encArg(x_1))
encode_activate(x_1) -> activate(encArg(x_1))

Types:
__ :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
nil :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
U11 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
tt :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
U12 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
isNePal :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
activate :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encArg :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons___ :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_U11 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_U12 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_isNePal :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_activate :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode___ :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_nil :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_U11 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_tt :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_U12 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_isNePal :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_activate :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
hole_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate1_0 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0 :: Nat -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate


Generator Equations:
gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(0) <=> nil
gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(+(x, 1)) <=> cons___(nil, gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(x))


The following defined symbols remain to be analysed:
__, encArg

They will be analysed ascendingly in the following order:
__ < encArg

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(n15_0)) -> gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(0), rt in Omega(n15_0)

Induction Base:
encArg(gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(0)) ->_R^Omega(0)
nil

Induction Step:
encArg(gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(+(n15_0, 1))) ->_R^Omega(0)
__(encArg(nil), encArg(gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(n15_0))) ->_R^Omega(0)
__(nil, encArg(gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(n15_0))) ->_IH
__(nil, gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(0)) ->_R^Omega(1)
nil

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(26)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
__(__(X, Y), Z) -> __(X, __(Y, Z))
__(X, nil) -> X
__(nil, X) -> X
U11(tt) -> U12(tt)
U12(tt) -> tt
isNePal(__(I, __(P, I))) -> U11(tt)
activate(X) -> X
encArg(nil) -> nil
encArg(tt) -> tt
encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
encArg(cons_U11(x_1)) -> U11(encArg(x_1))
encArg(cons_U12(x_1)) -> U12(encArg(x_1))
encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
encode_nil -> nil
encode_U11(x_1) -> U11(encArg(x_1))
encode_tt -> tt
encode_U12(x_1) -> U12(encArg(x_1))
encode_isNePal(x_1) -> isNePal(encArg(x_1))
encode_activate(x_1) -> activate(encArg(x_1))

Types:
__ :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
nil :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
U11 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
tt :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
U12 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
isNePal :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
activate :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encArg :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons___ :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_U11 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_U12 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_isNePal :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
cons_activate :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode___ :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_nil :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_U11 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_tt :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_U12 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_isNePal :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
encode_activate :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
hole_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate1_0 :: nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate
gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0 :: Nat -> nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate


Generator Equations:
gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(0) <=> nil
gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(+(x, 1)) <=> cons___(nil, gen_nil:tt:cons___:cons_U11:cons_U12:cons_isNePal:cons_activate2_0(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(27) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(28)
BOUNDS(n^1, INF)