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


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WORST_CASE(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 195 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) CompleteCoflocoProof [FINISHED, 817 ms]
    (16) BOUNDS(1, n^2)
(17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxRelTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 0 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 440 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 30 ms]
            (30) BOUNDS(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:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   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(tt) -> tt
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(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:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   activate(X) -> X

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

   encArg(tt) -> tt
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(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:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   activate(X) -> X

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

   encArg(tt) -> tt
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

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

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

(6)
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:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   activate(X) -> X

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

   encArg(tt) -> tt
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   activate(X) -> X [1]
   encArg(tt) -> tt [0]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

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

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

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

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   activate(X) -> X [1]
   encArg(tt) -> tt [0]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

The TRS has the following type information:
and :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
tt :: tt:0:s:cons_and:cons_plus:cons_activate
activate :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
plus :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
0 :: tt:0:s:cons_and:cons_plus:cons_activate
s :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
encArg :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
cons_and :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
cons_plus :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
cons_activate :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
encode_and :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
encode_tt :: tt:0:s:cons_and:cons_plus:cons_activate
encode_activate :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
encode_plus :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate
encode_0 :: tt:0:s:cons_and:cons_plus:cons_activate
encode_s :: tt:0:s:cons_and:cons_plus:cons_activate -> tt:0:s:cons_and:cons_plus:cons_activate

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

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

   encArg(v0) -> null_encArg [0]
   encode_and(v0, v1) -> null_encode_and [0]
   encode_tt -> null_encode_tt [0]
   encode_activate(v0) -> null_encode_activate [0]
   encode_plus(v0, v1) -> null_encode_plus [0]
   encode_0 -> null_encode_0 [0]
   encode_s(v0) -> null_encode_s [0]
   and(v0, v1) -> null_and [0]
   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_encArg, null_encode_and, null_encode_tt, null_encode_activate, null_encode_plus, null_encode_0, null_encode_s, null_and, null_plus

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

(12)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   activate(X) -> X [1]
   encArg(tt) -> tt [0]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_and(v0, v1) -> null_encode_and [0]
   encode_tt -> null_encode_tt [0]
   encode_activate(v0) -> null_encode_activate [0]
   encode_plus(v0, v1) -> null_encode_plus [0]
   encode_0 -> null_encode_0 [0]
   encode_s(v0) -> null_encode_s [0]
   and(v0, v1) -> null_and [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
and :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
tt :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
activate :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
plus :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
0 :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
s :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
encArg :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
cons_and :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
cons_plus :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
cons_activate :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
encode_and :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
encode_tt :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
encode_activate :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
encode_plus :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
encode_0 :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
encode_s :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus -> tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
null_encArg :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
null_encode_and :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
null_encode_tt :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
null_encode_activate :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
null_encode_plus :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
null_encode_0 :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
null_encode_s :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
null_and :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus
null_plus :: tt:0:s:cons_and:cons_plus:cons_activate:null_encArg:null_encode_and:null_encode_tt:null_encode_activate:null_encode_plus:null_encode_0:null_encode_s:null_and:null_plus

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

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

   tt => 1
   0 => 0
   null_encArg => 0
   null_encode_and => 0
   null_encode_tt => 0
   null_encode_activate => 0
   null_encode_plus => 0
   null_encode_0 => 0
   null_encode_s => 0
   null_and => 0
   null_plus => 0

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

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

   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 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 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 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_activate(z) -{ 0 }-> activate(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_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_plus(z, z') -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[and(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[activate(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(Out)],[]).
eq(start(V1, V),0,[fun2(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(and(V1, V, Out),1,[activate(X1, Ret)],[Out = Ret,V = X1,V1 = 1,X1 >= 0]).
eq(plus(V1, V, Out),1,[],[Out = N1,V1 = N1,V = 0,N1 >= 0]).
eq(plus(V1, V, Out),1,[plus(N2, M1, Ret1)],[Out = 1 + Ret1,V = 1 + M1,V1 = N2,M1 >= 0,N2 >= 0]).
eq(activate(V1, Out),1,[],[Out = X2,X2 >= 0,V1 = X2]).
eq(encArg(V1, Out),0,[],[Out = 1,V1 = 1]).
eq(encArg(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(encArg(V1, Out),0,[encArg(V2, Ret11)],[Out = 1 + Ret11,V1 = 1 + V2,V2 >= 0]).
eq(encArg(V1, Out),0,[encArg(V3, Ret0),encArg(V4, Ret12),and(Ret0, Ret12, Ret2)],[Out = Ret2,V3 >= 0,V1 = 1 + V3 + V4,V4 >= 0]).
eq(encArg(V1, Out),0,[encArg(V5, Ret01),encArg(V6, Ret13),plus(Ret01, Ret13, Ret3)],[Out = Ret3,V5 >= 0,V1 = 1 + V5 + V6,V6 >= 0]).
eq(encArg(V1, Out),0,[encArg(V7, Ret02),activate(Ret02, Ret4)],[Out = Ret4,V1 = 1 + V7,V7 >= 0]).
eq(fun(V1, V, Out),0,[encArg(V9, Ret03),encArg(V8, Ret14),and(Ret03, Ret14, Ret5)],[Out = Ret5,V9 >= 0,V8 >= 0,V1 = V9,V = V8]).
eq(fun1(Out),0,[],[Out = 1]).
eq(fun2(V1, Out),0,[encArg(V10, Ret04),activate(Ret04, Ret6)],[Out = Ret6,V10 >= 0,V1 = V10]).
eq(fun3(V1, V, Out),0,[encArg(V12, Ret05),encArg(V11, Ret15),plus(Ret05, Ret15, Ret7)],[Out = Ret7,V12 >= 0,V11 >= 0,V1 = V12,V = V11]).
eq(fun4(Out),0,[],[Out = 0]).
eq(fun5(V1, Out),0,[encArg(V13, Ret16)],[Out = 1 + Ret16,V13 >= 0,V1 = V13]).
eq(encArg(V1, Out),0,[],[Out = 0,V14 >= 0,V1 = V14]).
eq(fun(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
eq(fun1(Out),0,[],[Out = 0]).
eq(fun2(V1, Out),0,[],[Out = 0,V17 >= 0,V1 = V17]).
eq(fun3(V1, V, Out),0,[],[Out = 0,V18 >= 0,V19 >= 0,V1 = V18,V = V19]).
eq(fun5(V1, Out),0,[],[Out = 0,V20 >= 0,V1 = V20]).
eq(and(V1, V, Out),0,[],[Out = 0,V21 >= 0,V22 >= 0,V1 = V21,V = V22]).
eq(plus(V1, V, Out),0,[],[Out = 0,V23 >= 0,V24 >= 0,V1 = V23,V = V24]).
input_output_vars(and(V1,V,Out),[V1,V],[Out]).
input_output_vars(plus(V1,V,Out),[V1,V],[Out]).
input_output_vars(activate(V1,Out),[V1],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(Out),[],[Out]).
input_output_vars(fun2(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]).


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

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

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

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

### Specialization of cost equations and/3 
* CE 12 is refined into CE [32] 
* CE 11 is refined into CE [33] 


### Cost equations --> "Loop" of and/3 
* CEs [32] --> Loop 18 
* CEs [33] --> Loop 19 

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

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


### Specialization of cost equations plus/3 
* CE 15 is refined into CE [34] 
* CE 13 is refined into CE [35] 
* CE 14 is refined into CE [36] 


### Cost equations --> "Loop" of plus/3 
* CEs [36] --> Loop 20 
* CEs [34] --> Loop 21 
* CEs [35] --> Loop 22 

### Ranking functions of CR plus(V1,V,Out) 
* RF of phase [20]: [V]

#### Partial ranking functions of CR plus(V1,V,Out) 
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V


### Specialization of cost equations encArg/2 
* CE 17 is refined into CE [37] 
* CE 16 is refined into CE [38] 
* CE 18 is refined into CE [39] 
* CE 21 is refined into CE [40] 
* CE 19 is refined into CE [41,42] 
* CE 20 is refined into CE [43,44,45,46] 


### Cost equations --> "Loop" of encArg/2 
* CEs [46] --> Loop 23 
* CEs [45] --> Loop 24 
* CEs [43] --> Loop 25 
* CEs [41] --> Loop 26 
* CEs [42,44] --> Loop 27 
* CEs [39] --> Loop 28 
* CEs [40] --> Loop 29 
* CEs [37] --> Loop 30 
* CEs [38] --> Loop 31 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [23,24,25,26,27,28,29]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [23,24,25,26,27,28,29]:
  - RF of loop [23:1,23:2,24:1,24:2,25:1,25:2,26:1,26:2,27:1,27:2,28:1,29:1]:
    V1


### Specialization of cost equations fun/3 
* CE 22 is refined into CE [47,48,49,50,51,52] 
* CE 23 is refined into CE [53] 


### Cost equations --> "Loop" of fun/3 
* CEs [47] --> Loop 32 
* CEs [48,49,50,51,52,53] --> Loop 33 

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

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


### Specialization of cost equations fun1/1 
* CE 24 is refined into CE [54] 
* CE 25 is refined into CE [55] 


### Cost equations --> "Loop" of fun1/1 
* CEs [54] --> Loop 34 
* CEs [55] --> Loop 35 

### Ranking functions of CR fun1(Out) 

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


### Specialization of cost equations fun2/2 
* CE 26 is refined into CE [56,57] 
* CE 27 is refined into CE [58] 


### Cost equations --> "Loop" of fun2/2 
* CEs [56] --> Loop 36 
* CEs [57,58] --> Loop 37 

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

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


### Specialization of cost equations fun3/3 
* CE 28 is refined into CE [59,60,61,62,63,64,65,66,67,68,69,70] 
* CE 29 is refined into CE [71] 


### Cost equations --> "Loop" of fun3/3 
* CEs [61] --> Loop 38 
* CEs [59,63] --> Loop 39 
* CEs [62,67,68] --> Loop 40 
* CEs [60,64,65,66,69,70,71] --> Loop 41 

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

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


### Specialization of cost equations fun5/2 
* CE 30 is refined into CE [72,73] 
* CE 31 is refined into CE [74] 


### Cost equations --> "Loop" of fun5/2 
* CEs [72] --> Loop 42 
* CEs [73] --> Loop 43 
* CEs [74] --> Loop 44 

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

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


### Specialization of cost equations start/2 
* CE 1 is refined into CE [75,76] 
* CE 2 is refined into CE [77,78,79,80] 
* CE 3 is refined into CE [81] 
* CE 4 is refined into CE [82,83] 
* CE 5 is refined into CE [84,85] 
* CE 6 is refined into CE [86,87] 
* CE 7 is refined into CE [88,89] 
* CE 8 is refined into CE [90,91,92,93] 
* CE 9 is refined into CE [94] 
* CE 10 is refined into CE [95,96,97] 


### Cost equations --> "Loop" of start/2 
* CEs [75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97] --> Loop 45 

### 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 [19]: 2
  with precondition: [V1=1,V=Out,V>=0] 

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


#### Cost of chains of plus(V1,V,Out):
* Chain [[20],22]: 1*it(20)+1
  Such that:it(20) =< V

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

* Chain [[20],21]: 1*it(20)+0
  Such that:it(20) =< Out

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

* Chain [22]: 1
  with precondition: [V=0,V1=Out,V1>=0] 

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


#### Cost of chains of encArg(V1,Out):
* Chain [31]: 0
  with precondition: [V1=1,Out=1] 

* Chain [30]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([23,24,25,26,27,28,29],[[31],[30]])]: 1*it(24)+1*it(25)+2*it(26)+1*it(29)+1*s(5)+1*s(6)+0
  Such that:aux(6) =< V1
aux(7) =< V1+1
aux(8) =< 2/3*V1
it(25) =< aux(6)
it(26) =< aux(6)
it(29) =< aux(6)
it([30]) =< aux(7)
it(23) =< aux(8)
it(24) =< aux(8)
it(26) =< aux(8)
aux(4) =< aux(6)
it(23) =< it([30])*(1/3)+aux(8)
it(24) =< it([30])*(1/3)+aux(8)
it(25) =< it([30])*(1/3)+aux(8)
it(26) =< it([30])*(1/3)+aux(8)
s(6) =< it(24)*aux(4)
s(5) =< it(23)*aux(6)

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


#### Cost of chains of fun(V1,V,Out):
* Chain [33]: 3*s(22)+6*s(23)+3*s(24)+3*s(26)+3*s(28)+3*s(29)+2*s(33)+4*s(34)+2*s(35)+2*s(37)+2*s(39)+2*s(40)+2
  Such that:aux(9) =< V1
aux(10) =< V1+1
aux(11) =< 2/3*V1
aux(12) =< V
aux(13) =< V+1
aux(14) =< 2/3*V
s(33) =< aux(12)
s(34) =< aux(12)
s(35) =< aux(12)
s(36) =< aux(14)
s(37) =< aux(14)
s(34) =< aux(14)
s(38) =< aux(12)
s(36) =< aux(13)*(1/3)+aux(14)
s(37) =< aux(13)*(1/3)+aux(14)
s(33) =< aux(13)*(1/3)+aux(14)
s(34) =< aux(13)*(1/3)+aux(14)
s(39) =< s(37)*s(38)
s(40) =< s(36)*aux(12)
s(22) =< aux(9)
s(23) =< aux(9)
s(24) =< aux(9)
s(25) =< aux(11)
s(26) =< aux(11)
s(23) =< aux(11)
s(27) =< aux(9)
s(25) =< aux(10)*(1/3)+aux(11)
s(26) =< aux(10)*(1/3)+aux(11)
s(22) =< aux(10)*(1/3)+aux(11)
s(23) =< aux(10)*(1/3)+aux(11)
s(28) =< s(26)*s(27)
s(29) =< s(25)*aux(9)

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

* Chain [32]: 1*s(77)+2*s(78)+1*s(79)+1*s(81)+1*s(83)+1*s(84)+1*s(88)+2*s(89)+1*s(90)+1*s(92)+1*s(94)+1*s(95)+2
  Such that:s(74) =< V1
s(75) =< V1+1
s(76) =< 2/3*V1
s(85) =< V
s(86) =< V+1
s(87) =< 2/3*V
s(88) =< s(85)
s(89) =< s(85)
s(90) =< s(85)
s(91) =< s(87)
s(92) =< s(87)
s(89) =< s(87)
s(93) =< s(85)
s(91) =< s(86)*(1/3)+s(87)
s(92) =< s(86)*(1/3)+s(87)
s(88) =< s(86)*(1/3)+s(87)
s(89) =< s(86)*(1/3)+s(87)
s(94) =< s(92)*s(93)
s(95) =< s(91)*s(85)
s(77) =< s(74)
s(78) =< s(74)
s(79) =< s(74)
s(80) =< s(76)
s(81) =< s(76)
s(78) =< s(76)
s(82) =< s(74)
s(80) =< s(75)*(1/3)+s(76)
s(81) =< s(75)*(1/3)+s(76)
s(77) =< s(75)*(1/3)+s(76)
s(78) =< s(75)*(1/3)+s(76)
s(83) =< s(81)*s(82)
s(84) =< s(80)*s(74)

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


#### Cost of chains of fun1(Out):
* Chain [35]: 0
  with precondition: [Out=0] 

* Chain [34]: 0
  with precondition: [Out=1] 


#### Cost of chains of fun2(V1,Out):
* Chain [37]: 1
  with precondition: [Out=0,V1>=0] 

* Chain [36]: 1*s(99)+2*s(100)+1*s(101)+1*s(103)+1*s(105)+1*s(106)+1
  Such that:s(96) =< V1
s(97) =< V1+1
s(98) =< 2/3*V1
s(99) =< s(96)
s(100) =< s(96)
s(101) =< s(96)
s(102) =< s(98)
s(103) =< s(98)
s(100) =< s(98)
s(104) =< s(96)
s(102) =< s(97)*(1/3)+s(98)
s(103) =< s(97)*(1/3)+s(98)
s(99) =< s(97)*(1/3)+s(98)
s(100) =< s(97)*(1/3)+s(98)
s(105) =< s(103)*s(104)
s(106) =< s(102)*s(96)

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


#### Cost of chains of fun3(V1,V,Out):
* Chain [41]: 2*s(110)+4*s(111)+2*s(112)+2*s(114)+2*s(116)+2*s(117)+3*s(121)+6*s(122)+3*s(123)+3*s(125)+3*s(127)+3*s(128)+1
  Such that:aux(15) =< V1
aux(16) =< V1+1
aux(17) =< 2/3*V1
aux(18) =< V
aux(19) =< V+1
aux(20) =< 2/3*V
s(121) =< aux(18)
s(122) =< aux(18)
s(123) =< aux(18)
s(124) =< aux(20)
s(125) =< aux(20)
s(122) =< aux(20)
s(126) =< aux(18)
s(124) =< aux(19)*(1/3)+aux(20)
s(125) =< aux(19)*(1/3)+aux(20)
s(121) =< aux(19)*(1/3)+aux(20)
s(122) =< aux(19)*(1/3)+aux(20)
s(127) =< s(125)*s(126)
s(128) =< s(124)*aux(18)
s(110) =< aux(15)
s(111) =< aux(15)
s(112) =< aux(15)
s(113) =< aux(17)
s(114) =< aux(17)
s(111) =< aux(17)
s(115) =< aux(15)
s(113) =< aux(16)*(1/3)+aux(17)
s(114) =< aux(16)*(1/3)+aux(17)
s(110) =< aux(16)*(1/3)+aux(17)
s(111) =< aux(16)*(1/3)+aux(17)
s(116) =< s(114)*s(115)
s(117) =< s(113)*aux(15)

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

* Chain [40]: 1*s(165)+2*s(166)+1*s(167)+1*s(169)+1*s(171)+1*s(172)+3*s(176)+6*s(177)+6*s(178)+3*s(180)+3*s(182)+3*s(183)+1
  Such that:s(162) =< V1
s(163) =< V1+1
s(164) =< 2/3*V1
aux(24) =< V
aux(25) =< V+1
aux(26) =< 2/3*V
s(178) =< aux(24)
s(176) =< aux(24)
s(177) =< aux(24)
s(179) =< aux(26)
s(180) =< aux(26)
s(177) =< aux(26)
s(181) =< aux(24)
s(179) =< aux(25)*(1/3)+aux(26)
s(180) =< aux(25)*(1/3)+aux(26)
s(176) =< aux(25)*(1/3)+aux(26)
s(177) =< aux(25)*(1/3)+aux(26)
s(182) =< s(180)*s(181)
s(183) =< s(179)*aux(24)
s(165) =< s(162)
s(166) =< s(162)
s(167) =< s(162)
s(168) =< s(164)
s(169) =< s(164)
s(166) =< s(164)
s(170) =< s(162)
s(168) =< s(163)*(1/3)+s(164)
s(169) =< s(163)*(1/3)+s(164)
s(165) =< s(163)*(1/3)+s(164)
s(166) =< s(163)*(1/3)+s(164)
s(171) =< s(169)*s(170)
s(172) =< s(168)*s(162)

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

* Chain [39]: 2*s(212)+4*s(213)+2*s(214)+2*s(216)+2*s(218)+2*s(219)+1*s(223)+2*s(224)+1*s(225)+1*s(227)+1*s(229)+1*s(230)+1
  Such that:s(220) =< V
s(221) =< V+1
s(222) =< 2/3*V
aux(27) =< V1
aux(28) =< V1+1
aux(29) =< 2/3*V1
s(223) =< s(220)
s(224) =< s(220)
s(225) =< s(220)
s(226) =< s(222)
s(227) =< s(222)
s(224) =< s(222)
s(228) =< s(220)
s(226) =< s(221)*(1/3)+s(222)
s(227) =< s(221)*(1/3)+s(222)
s(223) =< s(221)*(1/3)+s(222)
s(224) =< s(221)*(1/3)+s(222)
s(229) =< s(227)*s(228)
s(230) =< s(226)*s(220)
s(212) =< aux(27)
s(213) =< aux(27)
s(214) =< aux(27)
s(215) =< aux(29)
s(216) =< aux(29)
s(213) =< aux(29)
s(217) =< aux(27)
s(215) =< aux(28)*(1/3)+aux(29)
s(216) =< aux(28)*(1/3)+aux(29)
s(212) =< aux(28)*(1/3)+aux(29)
s(213) =< aux(28)*(1/3)+aux(29)
s(218) =< s(216)*s(217)
s(219) =< s(215)*aux(27)

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

* Chain [38]: 1*s(245)+2*s(246)+1*s(247)+1*s(249)+1*s(251)+1*s(252)+1*s(256)+2*s(257)+2*s(258)+1*s(260)+1*s(262)+1*s(263)+1
  Such that:s(242) =< V1
s(243) =< V1+1
s(244) =< 2/3*V1
s(254) =< V+1
s(255) =< 2/3*V
aux(30) =< V
s(258) =< aux(30)
s(256) =< aux(30)
s(257) =< aux(30)
s(259) =< s(255)
s(260) =< s(255)
s(257) =< s(255)
s(261) =< aux(30)
s(259) =< s(254)*(1/3)+s(255)
s(260) =< s(254)*(1/3)+s(255)
s(256) =< s(254)*(1/3)+s(255)
s(257) =< s(254)*(1/3)+s(255)
s(262) =< s(260)*s(261)
s(263) =< s(259)*aux(30)
s(245) =< s(242)
s(246) =< s(242)
s(247) =< s(242)
s(248) =< s(244)
s(249) =< s(244)
s(246) =< s(244)
s(250) =< s(242)
s(248) =< s(243)*(1/3)+s(244)
s(249) =< s(243)*(1/3)+s(244)
s(245) =< s(243)*(1/3)+s(244)
s(246) =< s(243)*(1/3)+s(244)
s(251) =< s(249)*s(250)
s(252) =< s(248)*s(242)

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


#### Cost of chains of fun5(V1,Out):
* Chain [44]: 0
  with precondition: [Out=0,V1>=0] 

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

* Chain [42]: 1*s(268)+2*s(269)+1*s(270)+1*s(272)+1*s(274)+1*s(275)+0
  Such that:s(265) =< V1
s(266) =< V1+1
s(267) =< 2/3*V1
s(268) =< s(265)
s(269) =< s(265)
s(270) =< s(265)
s(271) =< s(267)
s(272) =< s(267)
s(269) =< s(267)
s(273) =< s(265)
s(271) =< s(266)*(1/3)+s(267)
s(272) =< s(266)*(1/3)+s(267)
s(268) =< s(266)*(1/3)+s(267)
s(269) =< s(266)*(1/3)+s(267)
s(274) =< s(272)*s(273)
s(275) =< s(271)*s(265)

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


#### Cost of chains of start(V1,V):
* Chain [45]: 17*s(276)+13*s(281)+26*s(282)+13*s(283)+13*s(285)+13*s(287)+13*s(288)+11*s(295)+22*s(296)+11*s(299)+11*s(301)+11*s(302)+2
  Such that:aux(31) =< V1
aux(32) =< V1+1
aux(33) =< 2/3*V1
aux(34) =< V
aux(35) =< V+1
aux(36) =< 2/3*V
s(276) =< aux(34)
s(281) =< aux(31)
s(282) =< aux(31)
s(283) =< aux(31)
s(284) =< aux(33)
s(285) =< aux(33)
s(282) =< aux(33)
s(286) =< aux(31)
s(284) =< aux(32)*(1/3)+aux(33)
s(285) =< aux(32)*(1/3)+aux(33)
s(281) =< aux(32)*(1/3)+aux(33)
s(282) =< aux(32)*(1/3)+aux(33)
s(287) =< s(285)*s(286)
s(288) =< s(284)*aux(31)
s(295) =< aux(34)
s(296) =< aux(34)
s(298) =< aux(36)
s(299) =< aux(36)
s(296) =< aux(36)
s(300) =< aux(34)
s(298) =< aux(35)*(1/3)+aux(36)
s(299) =< aux(35)*(1/3)+aux(36)
s(295) =< aux(35)*(1/3)+aux(36)
s(296) =< aux(35)*(1/3)+aux(36)
s(301) =< s(299)*s(300)
s(302) =< s(298)*aux(34)

  with precondition: [] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [45] with precondition: [] 
    - Upper bound: nat(V1)*52+2+nat(V1)*26*nat(2/3*V1)+nat(V)*50+nat(V)*22*nat(2/3*V)+nat(2/3*V1)*13+nat(2/3*V)*11 
    - Complexity: n^2 

### Maximum cost of start(V1,V): nat(V1)*52+2+nat(V1)*26*nat(2/3*V1)+nat(V)*50+nat(V)*22*nat(2/3*V)+nat(2/3*V1)*13+nat(2/3*V)*11 
Asymptotic class: n^2 
* Total analysis performed in 687 ms.


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

(16)
BOUNDS(1, n^2)

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

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

(18)
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:

   and(tt, X) -> activate(X)
   plus(N, 0') -> N
   plus(N, s(M)) -> s(plus(N, M))
   activate(X) -> X

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

   encArg(tt) -> tt
   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))

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

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

(20)
Obligation:
TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
activate(X) -> X
encArg(tt) -> tt
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
tt :: tt:0':s:cons_and:cons_plus:cons_activate
activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
0' :: tt:0':s:cons_and:cons_plus:cons_activate
s :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encArg :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_tt :: tt:0':s:cons_and:cons_plus:cons_activate
encode_activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_0 :: tt:0':s:cons_and:cons_plus:cons_activate
encode_s :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
hole_tt:0':s:cons_and:cons_plus:cons_activate1_3 :: tt:0':s:cons_and:cons_plus:cons_activate
gen_tt:0':s:cons_and:cons_plus:cons_activate2_3 :: Nat -> tt:0':s:cons_and:cons_plus:cons_activate

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

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

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

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

(22)
Obligation:
TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
activate(X) -> X
encArg(tt) -> tt
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
tt :: tt:0':s:cons_and:cons_plus:cons_activate
activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
0' :: tt:0':s:cons_and:cons_plus:cons_activate
s :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encArg :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_tt :: tt:0':s:cons_and:cons_plus:cons_activate
encode_activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_0 :: tt:0':s:cons_and:cons_plus:cons_activate
encode_s :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
hole_tt:0':s:cons_and:cons_plus:cons_activate1_3 :: tt:0':s:cons_and:cons_plus:cons_activate
gen_tt:0':s:cons_and:cons_plus:cons_activate2_3 :: Nat -> tt:0':s:cons_and:cons_plus:cons_activate


Generator Equations:
gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(0) <=> tt
gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(+(x, 1)) <=> s(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(x))


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

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(a), gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3)

Induction Base:
plus(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(a), gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(+(1, 0)))

Induction Step:
plus(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(a), gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(+(1, +(n4_3, 1)))) ->_R^Omega(1)
s(plus(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(a), gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(+(1, n4_3)))) ->_IH
s(*3_3)

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

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
activate(X) -> X
encArg(tt) -> tt
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
tt :: tt:0':s:cons_and:cons_plus:cons_activate
activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
0' :: tt:0':s:cons_and:cons_plus:cons_activate
s :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encArg :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_tt :: tt:0':s:cons_and:cons_plus:cons_activate
encode_activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_0 :: tt:0':s:cons_and:cons_plus:cons_activate
encode_s :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
hole_tt:0':s:cons_and:cons_plus:cons_activate1_3 :: tt:0':s:cons_and:cons_plus:cons_activate
gen_tt:0':s:cons_and:cons_plus:cons_activate2_3 :: Nat -> tt:0':s:cons_and:cons_plus:cons_activate


Generator Equations:
gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(0) <=> tt
gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(+(x, 1)) <=> s(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(x))


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

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

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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
activate(X) -> X
encArg(tt) -> tt
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
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_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
tt :: tt:0':s:cons_and:cons_plus:cons_activate
activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
0' :: tt:0':s:cons_and:cons_plus:cons_activate
s :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encArg :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
cons_activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_and :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_tt :: tt:0':s:cons_and:cons_plus:cons_activate
encode_activate :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_plus :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
encode_0 :: tt:0':s:cons_and:cons_plus:cons_activate
encode_s :: tt:0':s:cons_and:cons_plus:cons_activate -> tt:0':s:cons_and:cons_plus:cons_activate
hole_tt:0':s:cons_and:cons_plus:cons_activate1_3 :: tt:0':s:cons_and:cons_plus:cons_activate
gen_tt:0':s:cons_and:cons_plus:cons_activate2_3 :: Nat -> tt:0':s:cons_and:cons_plus:cons_activate


Lemmas:
plus(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(a), gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3)


Generator Equations:
gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(0) <=> tt
gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(+(x, 1)) <=> s(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(x))


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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(n614_3)) -> gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(n614_3), rt in Omega(0)

Induction Base:
encArg(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(0)) ->_R^Omega(0)
tt

Induction Step:
encArg(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(+(n614_3, 1))) ->_R^Omega(0)
s(encArg(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(n614_3))) ->_IH
s(gen_tt:0':s:cons_and:cons_plus:cons_activate2_3(c615_3))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(30)
BOUNDS(1, INF)