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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 185 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), 0 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 732 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), 246 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
   and(tt, X) -> activate(X)
   isNePal(__(I, __(P, I))) -> 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_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_tt -> tt
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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
   and(tt, X) -> activate(X)
   isNePal(__(I, __(P, I))) -> 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_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_tt -> tt
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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
   and(tt, X) -> activate(X)
   isNePal(__(I, __(P, I))) -> 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_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_tt -> tt
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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
   and(tt, X) -> activate(X)
   activate(X) -> X
   __(c___(X, Y), Z) -> __(X, __(Y, Z))
   isNePal(c___(I, c___(P, I))) -> tt

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_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_tt -> tt
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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
   and(tt, X) -> activate(X)
   activate(X) -> X
   __(c___(X, Y), Z) -> __(X, __(Y, Z))
   isNePal(c___(I, c___(P, I))) -> tt

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_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_tt -> tt
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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]
   and(tt, X) -> activate(X) [1]
   activate(X) -> X [1]
   __(c___(X, Y), Z) -> __(X, __(Y, Z)) [1]
   isNePal(c___(I, c___(P, I))) -> tt [1]
   encArg(nil) -> nil [0]
   encArg(tt) -> tt [0]
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0]
   encode_tt -> tt [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   encode_isNePal(x_1) -> isNePal(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]
   and(tt, X) -> activate(X) [1]
   activate(X) -> X [1]
   __(c___(X, Y), Z) -> __(X, __(Y, Z)) [1]
   isNePal(c___(I, c___(P, I))) -> tt [1]
   encArg(nil) -> nil [0]
   encArg(tt) -> tt [0]
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0]
   encode_tt -> tt [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   encode_isNePal(x_1) -> isNePal(encArg(x_1)) [0]
   __(x0, x1) -> c___(x0, x1) [0]

The TRS has the following type information:
__ :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
nil :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
and :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
tt :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
activate :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
c___ :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
isNePal :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
encArg :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
cons___ :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
cons_and :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
cons_isNePal :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
cons_activate :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
encode___ :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
encode_nil :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
encode_and :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
encode_tt :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
encode_activate :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate
encode_isNePal :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:c___:cons___:cons_and: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_and(v0, v1) -> null_encode_and [0]
   encode_tt -> null_encode_tt [0]
   encode_activate(v0) -> null_encode_activate [0]
   encode_isNePal(v0) -> null_encode_isNePal [0]
   __(v0, v1) -> null___ [0]
   and(v0, v1) -> null_and [0]
   isNePal(v0) -> null_isNePal [0]

And the following fresh constants:    null_encArg, null_encode___, null_encode_nil, null_encode_and, null_encode_tt, null_encode_activate, null_encode_isNePal, null___, null_and, 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]
   and(tt, X) -> activate(X) [1]
   activate(X) -> X [1]
   __(c___(X, Y), Z) -> __(X, __(Y, Z)) [1]
   isNePal(c___(I, c___(P, I))) -> tt [1]
   encArg(nil) -> nil [0]
   encArg(tt) -> tt [0]
   encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0]
   encode_tt -> tt [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   encode_isNePal(x_1) -> isNePal(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_and(v0, v1) -> null_encode_and [0]
   encode_tt -> null_encode_tt [0]
   encode_activate(v0) -> null_encode_activate [0]
   encode_isNePal(v0) -> null_encode_isNePal [0]
   __(v0, v1) -> null___ [0]
   and(v0, v1) -> null_and [0]
   isNePal(v0) -> null_isNePal [0]

The TRS has the following type information:
__ :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
nil :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
and :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
tt :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
activate :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
c___ :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
isNePal :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
encArg :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
cons___ :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
cons_and :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
cons_isNePal :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
cons_activate :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
encode___ :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
encode_nil :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
encode_and :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
encode_tt :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
encode_activate :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
encode_isNePal :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal -> nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
null_encArg :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
null_encode___ :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
null_encode_nil :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
null_encode_and :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
null_encode_tt :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
null_encode_activate :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
null_encode_isNePal :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
null___ :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
null_and :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and:null_isNePal
null_isNePal :: nil:tt:c___:cons___:cons_and:cons_isNePal:cons_activate:null_encArg:null_encode___:null_encode_nil:null_encode_and:null_encode_tt:null_encode_activate:null_encode_isNePal:null___:null_and: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_and => 0
   null_encode_tt => 0
   null_encode_activate => 0
   null_encode_isNePal => 0
   null___ => 0
   null_and => 0
   null_isNePal => 0

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

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

   __(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
   and(z, z') -{ 1 }-> activate(X) :|: z' = X, z = 1, X >= 0
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encArg(z) -{ 0 }-> isNePal(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> __(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(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_and(z, z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   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 }-> 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,[and(V1, V, Out)],[V1 >= 0,V >= 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,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun2(Out)],[]).
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(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(and(V1, V, Out),1,[activate(X3, Ret)],[Out = Ret,V = X3,V1 = 1,X3 >= 0]).
eq(activate(V1, Out),1,[],[Out = X4,X4 >= 0,V1 = X4]).
eq(fun(V1, V, Out),1,[fun(Y1, Z1, Ret1),fun(X5, Ret1, Ret2)],[Out = Ret2,Z1 >= 0,Y1 >= 0,V1 = 1 + X5 + Y1,X5 >= 0,V = Z1]).
eq(isNePal(V1, Out),1,[],[Out = 1,V1 = 2 + 2*I1 + P1,I1 >= 0,P1 >= 0]).
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, Ret11),fun(Ret0, Ret11, Ret3)],[Out = Ret3,V3 >= 0,V1 = 1 + V2 + V3,V2 >= 0]).
eq(encArg(V1, Out),0,[encArg(V4, Ret01),encArg(V5, Ret12),and(Ret01, Ret12, Ret4)],[Out = Ret4,V4 >= 0,V1 = 1 + V4 + V5,V5 >= 0]).
eq(encArg(V1, Out),0,[encArg(V6, Ret02),isNePal(Ret02, Ret5)],[Out = Ret5,V1 = 1 + V6,V6 >= 0]).
eq(encArg(V1, Out),0,[encArg(V7, Ret03),activate(Ret03, Ret6)],[Out = Ret6,V1 = 1 + V7,V7 >= 0]).
eq(fun1(V1, V, Out),0,[encArg(V9, Ret04),encArg(V8, Ret13),fun(Ret04, Ret13, Ret7)],[Out = Ret7,V9 >= 0,V8 >= 0,V1 = V9,V = V8]).
eq(fun2(Out),0,[],[Out = 0]).
eq(fun3(V1, V, Out),0,[encArg(V11, Ret05),encArg(V10, Ret14),and(Ret05, Ret14, Ret8)],[Out = Ret8,V11 >= 0,V10 >= 0,V1 = V11,V = V10]).
eq(fun4(Out),0,[],[Out = 1]).
eq(fun5(V1, Out),0,[encArg(V12, Ret06),activate(Ret06, Ret9)],[Out = Ret9,V12 >= 0,V1 = V12]).
eq(fun6(V1, Out),0,[encArg(V13, Ret07),isNePal(Ret07, Ret10)],[Out = Ret10,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(fun1(V1, V, Out),0,[],[Out = 0,V18 >= 0,V17 >= 0,V1 = V18,V = V17]).
eq(fun3(V1, V, Out),0,[],[Out = 0,V20 >= 0,V19 >= 0,V1 = V20,V = V19]).
eq(fun4(Out),0,[],[Out = 0]).
eq(fun5(V1, Out),0,[],[Out = 0,V21 >= 0,V1 = V21]).
eq(fun6(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(and(V1, V, Out),0,[],[Out = 0,V25 >= 0,V26 >= 0,V1 = V25,V = 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(and(V1,V,Out),[V1,V],[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(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(Out),[],[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(V1,Out),[V1],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is completely evaluated into other SCCs
1. SCC is partially evaluated into and/3
2. SCC is partially evaluated into fun/3
3. SCC is partially evaluated into isNePal/2
4. SCC is partially evaluated into encArg/2
5. SCC is partially evaluated into fun1/3
6. SCC is completely evaluated into other SCCs
7. SCC is partially evaluated into fun3/3
8. SCC is partially evaluated into fun4/1
9. SCC is partially evaluated into fun5/2
10. SCC is partially evaluated into fun6/2
11. SCC is partially evaluated into start/2

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

### Specialization of cost equations and/3 
* CE 18 is refined into CE [37] 
* CE 17 is refined into CE [38] 


### Cost equations --> "Loop" of and/3 
* CEs [37] --> Loop 21 
* CEs [38] --> Loop 22 

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

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


### Specialization of cost equations fun/3 
* CE 15 is refined into CE [39] 
* CE 16 is refined into CE [40] 
* CE 12 is refined into CE [41] 
* CE 13 is refined into CE [42] 
* CE 14 is refined into CE [43] 


### Cost equations --> "Loop" of fun/3 
* CEs [43] --> Loop 23 
* CEs [39] --> Loop 24 
* CEs [40] --> Loop 25 
* CEs [41] --> Loop 26 
* CEs [42] --> Loop 27 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [23]: [V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [23]:
  - RF of loop [23:1,23:2]:
    V1


### Specialization of cost equations isNePal/2 
* CE 19 is refined into CE [44] 
* CE 20 is refined into CE [45] 


### Cost equations --> "Loop" of isNePal/2 
* CEs [44] --> Loop 28 
* CEs [45] --> Loop 29 

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

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


### Specialization of cost equations encArg/2 
* CE 21 is refined into CE [46] 
* CE 22 is refined into CE [47] 
* CE 25 is refined into CE [48,49] 
* CE 26 is refined into CE [50] 
* CE 23 is refined into CE [51,52,53,54,55] 
* CE 24 is refined into CE [56,57] 


### Cost equations --> "Loop" of encArg/2 
* CEs [54] --> Loop 30 
* CEs [52] --> Loop 31 
* CEs [55,56] --> Loop 32 
* CEs [51] --> Loop 33 
* CEs [53,57] --> Loop 34 
* CEs [50] --> Loop 35 
* CEs [49] --> Loop 36 
* CEs [48] --> Loop 37 
* CEs [46] --> Loop 38 
* CEs [47] --> Loop 39 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [30,31,32,33,34,35,36,37]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [30,31,32,33,34,35,36,37]:
  - RF of loop [30:1,30:2,31:1,31:2,32:1,32:2,33:1,33:2,34:1,34:2,35:1,36:1,37:1]:
    V1


### Specialization of cost equations fun1/3 
* CE 27 is refined into CE [58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75] 
* CE 28 is refined into CE [76] 


### Cost equations --> "Loop" of fun1/3 
* CEs [61,62] --> Loop 40 
* CEs [59,64,66,67] --> Loop 41 
* CEs [71] --> Loop 42 
* CEs [58,68] --> Loop 43 
* CEs [75] --> Loop 44 
* CEs [60,63,65,69,70,72,73,74,76] --> Loop 45 

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

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


### Specialization of cost equations fun3/3 
* CE 29 is refined into CE [77,78,79,80,81,82] 
* CE 30 is refined into CE [83] 


### Cost equations --> "Loop" of fun3/3 
* CEs [77] --> Loop 46 
* CEs [78,79,80,81,82,83] --> Loop 47 

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

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


### Specialization of cost equations fun4/1 
* CE 31 is refined into CE [84] 
* CE 32 is refined into CE [85] 


### Cost equations --> "Loop" of fun4/1 
* CEs [84] --> Loop 48 
* CEs [85] --> Loop 49 

### Ranking functions of CR fun4(Out) 

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


### Specialization of cost equations fun5/2 
* CE 33 is refined into CE [86,87] 
* CE 34 is refined into CE [88] 


### Cost equations --> "Loop" of fun5/2 
* CEs [86] --> Loop 50 
* CEs [87,88] --> Loop 51 

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

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


### Specialization of cost equations fun6/2 
* CE 35 is refined into CE [89,90,91] 
* CE 36 is refined into CE [92] 


### Cost equations --> "Loop" of fun6/2 
* CEs [90] --> Loop 52 
* CEs [89,91,92] --> Loop 53 

### Ranking functions of CR fun6(V1,Out) 

#### Partial ranking functions of CR fun6(V1,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [93,94,95,96,97] 
* CE 2 is refined into CE [98,99] 
* CE 3 is refined into CE [100] 
* CE 4 is refined into CE [101,102] 
* CE 5 is refined into CE [103,104] 
* CE 6 is refined into CE [105,106,107,108,109,110] 
* CE 7 is refined into CE [111] 
* CE 8 is refined into CE [112,113] 
* CE 9 is refined into CE [114,115] 
* CE 10 is refined into CE [116,117] 
* CE 11 is refined into CE [118,119] 


### Cost equations --> "Loop" of start/2 
* CEs [93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119] --> Loop 54 

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

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


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

#### Cost of chains of and(V1,V,Out):
* Chain [22]: 2
  with precondition: [V1=1,V=Out,V>=0] 

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


#### Cost of chains of fun(V1,V,Out):
* Chain [27]: 1
  with precondition: [V1=0,V=Out,V>=0] 

* Chain [26]: 1
  with precondition: [V=0,V1=Out,V1>=0] 

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

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

* Chain [multiple([23],[[27],[26],[25],[24]])]: 1*it(23)+2*it([26])+0
  Such that:it(23) =< V1
aux(1) =< V1+1
it([26]) =< aux(1)

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


#### Cost of chains of isNePal(V1,Out):
* Chain [29]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [28]: 1
  with precondition: [Out=1,V1>=2] 


#### Cost of chains of encArg(V1,Out):
* Chain [39]: 0
  with precondition: [V1=1,Out=1] 

* Chain [38]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([30,31,32,33,34,35,36,37],[[39],[38]])]: 1*it(31)+5*it(32)+3*s(6)+0
  Such that:aux(4) =< V1
aux(6) =< 2*V1+1
it(31) =< aux(4)
it(32) =< aux(4)
it(32) =< aux(6)
s(7) =< it(32)*aux(4)
s(6) =< s(7)

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


#### Cost of chains of fun1(V1,V,Out):
* Chain [45]: 3*s(16)+15*s(17)+9*s(19)+3*s(22)+15*s(23)+9*s(25)+1
  Such that:aux(7) =< V1
aux(8) =< 2*V1+1
aux(9) =< V
aux(10) =< 2*V+1
s(22) =< aux(9)
s(23) =< aux(9)
s(23) =< aux(10)
s(24) =< s(23)*aux(9)
s(25) =< s(24)
s(16) =< aux(7)
s(17) =< aux(7)
s(17) =< aux(8)
s(18) =< s(17)*aux(7)
s(19) =< s(18)

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

* Chain [44]: 0
  with precondition: [Out=1,V1>=0,V>=0] 

* Chain [43]: 1*s(52)+5*s(53)+3*s(55)+2*s(58)+10*s(59)+6*s(61)+1
  Such that:s(50) =< V1
s(51) =< 2*V1+1
aux(11) =< V
aux(12) =< 2*V+1
s(58) =< aux(11)
s(59) =< aux(11)
s(59) =< aux(12)
s(60) =< s(59)*aux(11)
s(61) =< s(60)
s(52) =< s(50)
s(53) =< s(50)
s(53) =< s(51)
s(54) =< s(53)*s(50)
s(55) =< s(54)

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

* Chain [42]: 1*s(70)+5*s(71)+3*s(73)+0
  Such that:s(68) =< V
s(69) =< 2*V+1
s(70) =< s(68)
s(71) =< s(68)
s(71) =< s(69)
s(72) =< s(71)*s(68)
s(73) =< s(72)

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

* Chain [41]: 5*s(76)+20*s(77)+12*s(79)+1*s(82)+5*s(83)+3*s(85)+2*s(106)+1
  Such that:s(105) =< V1+1
s(80) =< V
s(81) =< 2*V+1
aux(14) =< V1
aux(15) =< 2*V1+1
s(82) =< s(80)
s(83) =< s(80)
s(83) =< s(81)
s(84) =< s(83)*s(80)
s(85) =< s(84)
s(76) =< aux(14)
s(77) =< aux(14)
s(77) =< aux(15)
s(78) =< s(77)*aux(14)
s(79) =< s(78)
s(106) =< s(105)

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

* Chain [40]: 3*s(109)+10*s(110)+6*s(112)+2*s(115)+10*s(116)+6*s(118)+2*s(133)+0
  Such that:s(132) =< V1+1
aux(17) =< V1
aux(18) =< 2*V1+1
aux(19) =< V
aux(20) =< 2*V+1
s(115) =< aux(19)
s(116) =< aux(19)
s(116) =< aux(20)
s(117) =< s(116)*aux(19)
s(118) =< s(117)
s(109) =< aux(17)
s(110) =< aux(17)
s(110) =< aux(18)
s(111) =< s(110)*aux(17)
s(112) =< s(111)
s(133) =< s(132)

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


#### Cost of chains of fun3(V1,V,Out):
* Chain [47]: 3*s(136)+15*s(137)+9*s(139)+2*s(142)+10*s(143)+6*s(145)+2
  Such that:aux(21) =< V1
aux(22) =< 2*V1+1
aux(23) =< V
aux(24) =< 2*V+1
s(142) =< aux(23)
s(143) =< aux(23)
s(143) =< aux(24)
s(144) =< s(143)*aux(23)
s(145) =< s(144)
s(136) =< aux(21)
s(137) =< aux(21)
s(137) =< aux(22)
s(138) =< s(137)*aux(21)
s(139) =< s(138)

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

* Chain [46]: 1*s(166)+5*s(167)+3*s(169)+1*s(172)+5*s(173)+3*s(175)+2
  Such that:s(164) =< V1
s(165) =< 2*V1+1
s(170) =< V
s(171) =< 2*V+1
s(172) =< s(170)
s(173) =< s(170)
s(173) =< s(171)
s(174) =< s(173)*s(170)
s(175) =< s(174)
s(166) =< s(164)
s(167) =< s(164)
s(167) =< s(165)
s(168) =< s(167)*s(164)
s(169) =< s(168)

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


#### Cost of chains of fun4(Out):
* Chain [49]: 0
  with precondition: [Out=0] 

* Chain [48]: 0
  with precondition: [Out=1] 


#### Cost of chains of fun5(V1,Out):
* Chain [51]: 1
  with precondition: [Out=0,V1>=0] 

* Chain [50]: 1*s(178)+5*s(179)+3*s(181)+1
  Such that:s(176) =< V1
s(177) =< 2*V1+1
s(178) =< s(176)
s(179) =< s(176)
s(179) =< s(177)
s(180) =< s(179)*s(176)
s(181) =< s(180)

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


#### Cost of chains of fun6(V1,Out):
* Chain [53]: 1*s(184)+5*s(185)+3*s(187)+0
  Such that:s(182) =< V1
s(183) =< 2*V1+1
s(184) =< s(182)
s(185) =< s(182)
s(185) =< s(183)
s(186) =< s(185)*s(182)
s(187) =< s(186)

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

* Chain [52]: 1*s(190)+5*s(191)+3*s(193)+1
  Such that:s(188) =< V1
s(189) =< 2*V1+1
s(190) =< s(188)
s(191) =< s(188)
s(191) =< s(189)
s(192) =< s(191)*s(188)
s(193) =< s(192)

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


#### Cost of chains of start(V1,V):
* Chain [54]: 21*s(194)+6*s(196)+90*s(200)+54*s(202)+12*s(207)+60*s(208)+36*s(210)+2
  Such that:aux(25) =< V1
aux(26) =< V1+1
aux(27) =< 2*V1+1
aux(28) =< V
aux(29) =< 2*V+1
s(194) =< aux(25)
s(196) =< aux(26)
s(200) =< aux(25)
s(200) =< aux(27)
s(201) =< s(200)*aux(25)
s(202) =< s(201)
s(207) =< aux(28)
s(208) =< aux(28)
s(208) =< aux(29)
s(209) =< s(208)*aux(28)
s(210) =< s(209)

  with precondition: [] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [54] with precondition: [] 
    - Upper bound: nat(V1)*111+2+nat(V1)*54*nat(V1)+nat(V)*72+nat(V)*36*nat(V)+nat(V1+1)*6 
    - Complexity: n^2 

### Maximum cost of start(V1,V): nat(V1)*111+2+nat(V1)*54*nat(V1)+nat(V)*72+nat(V)*36*nat(V)+nat(V1+1)*6 
Asymptotic class: n^2 
* Total analysis performed in 614 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
   and(tt, X) -> activate(X)
   isNePal(__(I, __(P, I))) -> 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_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_tt -> tt
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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
and(tt, X) -> activate(X)
isNePal(__(I, __(P, I))) -> tt
activate(X) -> X
encArg(nil) -> nil
encArg(tt) -> tt
encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
encode_tt -> tt
encode_activate(x_1) -> activate(encArg(x_1))
encode_isNePal(x_1) -> isNePal(encArg(x_1))

Types:
__ :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
nil :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
and :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
tt :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
activate :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
isNePal :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encArg :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons___ :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons_and :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons_isNePal :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons_activate :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode___ :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_nil :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_and :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_tt :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_activate :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_isNePal :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
hole_nil:tt:cons___:cons_and:cons_isNePal:cons_activate1_0 :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
gen_nil:tt:cons___:cons_and:cons_isNePal:cons_activate2_0 :: Nat -> nil:tt:cons___:cons_and: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
and(tt, X) -> activate(X)
isNePal(__(I, __(P, I))) -> tt
activate(X) -> X
encArg(nil) -> nil
encArg(tt) -> tt
encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
encode_tt -> tt
encode_activate(x_1) -> activate(encArg(x_1))
encode_isNePal(x_1) -> isNePal(encArg(x_1))

Types:
__ :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
nil :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
and :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
tt :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
activate :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
isNePal :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encArg :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons___ :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons_and :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons_isNePal :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons_activate :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode___ :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_nil :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_and :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_tt :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_activate :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_isNePal :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
hole_nil:tt:cons___:cons_and:cons_isNePal:cons_activate1_0 :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
gen_nil:tt:cons___:cons_and:cons_isNePal:cons_activate2_0 :: Nat -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate


Generator Equations:
gen_nil:tt:cons___:cons_and:cons_isNePal:cons_activate2_0(0) <=> nil
gen_nil:tt:cons___:cons_and:cons_isNePal:cons_activate2_0(+(x, 1)) <=> cons___(nil, gen_nil:tt:cons___:cons_and: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_and:cons_isNePal:cons_activate2_0(n15_0)) -> gen_nil:tt:cons___:cons_and:cons_isNePal:cons_activate2_0(0), rt in Omega(n15_0)

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

Induction Step:
encArg(gen_nil:tt:cons___:cons_and:cons_isNePal:cons_activate2_0(+(n15_0, 1))) ->_R^Omega(0)
__(encArg(nil), encArg(gen_nil:tt:cons___:cons_and:cons_isNePal:cons_activate2_0(n15_0))) ->_R^Omega(0)
__(nil, encArg(gen_nil:tt:cons___:cons_and:cons_isNePal:cons_activate2_0(n15_0))) ->_IH
__(nil, gen_nil:tt:cons___:cons_and: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
and(tt, X) -> activate(X)
isNePal(__(I, __(P, I))) -> tt
activate(X) -> X
encArg(nil) -> nil
encArg(tt) -> tt
encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
encode_tt -> tt
encode_activate(x_1) -> activate(encArg(x_1))
encode_isNePal(x_1) -> isNePal(encArg(x_1))

Types:
__ :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
nil :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
and :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
tt :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
activate :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
isNePal :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encArg :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons___ :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons_and :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons_isNePal :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
cons_activate :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode___ :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_nil :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_and :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_tt :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_activate :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
encode_isNePal :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate
hole_nil:tt:cons___:cons_and:cons_isNePal:cons_activate1_0 :: nil:tt:cons___:cons_and:cons_isNePal:cons_activate
gen_nil:tt:cons___:cons_and:cons_isNePal:cons_activate2_0 :: Nat -> nil:tt:cons___:cons_and:cons_isNePal:cons_activate


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