/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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), 197 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) CompleteCoflocoProof [FINISHED, 7338 ms]
    (14) BOUNDS(1, n^2)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 1018 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 17 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 408 ms]
            (30) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_pred(x_1)) -> pred(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_pred(x_1) -> pred(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_log(x_1) -> log(encArg(x_1))


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

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


The TRS R consists of the following rules:

   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_pred(x_1)) -> pred(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_pred(x_1) -> pred(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_log(x_1) -> log(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_pred(x_1)) -> pred(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_pred(x_1) -> pred(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_log(x_1) -> log(encArg(x_1))

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

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

(6)
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:

   pred(s(x)) -> x [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> pred(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [1]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(0) -> 0 [0]
   encArg(cons_pred(x_1)) -> pred(encArg(x_1)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [0]
   encode_pred(x_1) -> pred(encArg(x_1)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [0]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   pred(s(x)) -> x [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> pred(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [1]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(0) -> 0 [0]
   encArg(cons_pred(x_1)) -> pred(encArg(x_1)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [0]
   encode_pred(x_1) -> pred(encArg(x_1)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [0]

The TRS has the following type information:
pred :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
s :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
minus :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
0 :: s:0:cons_pred:cons_minus:cons_quot:cons_log
quot :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
log :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
encArg :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
cons_pred :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
cons_minus :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
cons_quot :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
cons_log :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
encode_pred :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
encode_s :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
encode_minus :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
encode_0 :: s:0:cons_pred:cons_minus:cons_quot:cons_log
encode_quot :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log
encode_log :: s:0:cons_pred:cons_minus:cons_quot:cons_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log

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

(9) 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_pred(v0) -> null_encode_pred [0]
   encode_s(v0) -> null_encode_s [0]
   encode_minus(v0, v1) -> null_encode_minus [0]
   encode_0 -> null_encode_0 [0]
   encode_quot(v0, v1) -> null_encode_quot [0]
   encode_log(v0) -> null_encode_log [0]
   pred(v0) -> null_pred [0]
   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   log(v0) -> null_log [0]

And the following fresh constants:    null_encArg, null_encode_pred, null_encode_s, null_encode_minus, null_encode_0, null_encode_quot, null_encode_log, null_pred, null_minus, null_quot, null_log

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

(10)
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:

   pred(s(x)) -> x [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> pred(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [1]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(0) -> 0 [0]
   encArg(cons_pred(x_1)) -> pred(encArg(x_1)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [0]
   encode_pred(x_1) -> pred(encArg(x_1)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_pred(v0) -> null_encode_pred [0]
   encode_s(v0) -> null_encode_s [0]
   encode_minus(v0, v1) -> null_encode_minus [0]
   encode_0 -> null_encode_0 [0]
   encode_quot(v0, v1) -> null_encode_quot [0]
   encode_log(v0) -> null_encode_log [0]
   pred(v0) -> null_pred [0]
   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   log(v0) -> null_log [0]

The TRS has the following type information:
pred :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
s :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
minus :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
0 :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
quot :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
log :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
encArg :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
cons_pred :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
cons_minus :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
cons_quot :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
cons_log :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
encode_pred :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
encode_s :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
encode_minus :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
encode_0 :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
encode_quot :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
encode_log :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log -> s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_encArg :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_encode_pred :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_encode_s :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_encode_minus :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_encode_0 :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_encode_quot :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_encode_log :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_pred :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_minus :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_quot :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log
null_log :: s:0:cons_pred:cons_minus:cons_quot:cons_log:null_encArg:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_log:null_pred:null_minus:null_quot:null_log

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

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

   0 => 0
   null_encArg => 0
   null_encode_pred => 0
   null_encode_s => 0
   null_encode_minus => 0
   null_encode_0 => 0
   null_encode_quot => 0
   null_encode_log => 0
   null_pred => 0
   null_minus => 0
   null_quot => 0
   null_log => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> quot(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> pred(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> log(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_log(z) -{ 0 }-> log(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_minus(z, z') -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_pred(z) -{ 0 }-> pred(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_quot(z, z') -{ 0 }-> quot(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_quot(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
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   log(z) -{ 1 }-> 1 + log(1 + quot(x, 1 + (1 + 0))) :|: x >= 0, z = 1 + (1 + x)
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> pred(minus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   pred(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V2),0,[pred(V, Out)],[V >= 0]).
eq(start(V, V2),0,[minus(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[quot(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[log(V, Out)],[V >= 0]).
eq(start(V, V2),0,[encArg(V, Out)],[V >= 0]).
eq(start(V, V2),0,[fun(V, Out)],[V >= 0]).
eq(start(V, V2),0,[fun1(V, Out)],[V >= 0]).
eq(start(V, V2),0,[fun2(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[fun3(Out)],[]).
eq(start(V, V2),0,[fun4(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[fun5(V, Out)],[V >= 0]).
eq(pred(V, Out),1,[],[Out = V1,V1 >= 0,V = 1 + V1]).
eq(minus(V, V2, Out),1,[],[Out = V3,V3 >= 0,V = V3,V2 = 0]).
eq(minus(V, V2, Out),1,[minus(V4, V5, Ret0),pred(Ret0, Ret)],[Out = Ret,V2 = 1 + V5,V4 >= 0,V5 >= 0,V = V4]).
eq(quot(V, V2, Out),1,[],[Out = 0,V2 = 1 + V6,V6 >= 0,V = 0]).
eq(quot(V, V2, Out),1,[minus(V7, V8, Ret10),quot(Ret10, 1 + V8, Ret1)],[Out = 1 + Ret1,V2 = 1 + V8,V7 >= 0,V8 >= 0,V = 1 + V7]).
eq(log(V, Out),1,[],[Out = 0,V = 1]).
eq(log(V, Out),1,[quot(V9, 1 + (1 + 0), Ret101),log(1 + Ret101, Ret11)],[Out = 1 + Ret11,V9 >= 0,V = 2 + V9]).
eq(encArg(V, Out),0,[encArg(V10, Ret12)],[Out = 1 + Ret12,V = 1 + V10,V10 >= 0]).
eq(encArg(V, Out),0,[],[Out = 0,V = 0]).
eq(encArg(V, Out),0,[encArg(V11, Ret01),pred(Ret01, Ret2)],[Out = Ret2,V = 1 + V11,V11 >= 0]).
eq(encArg(V, Out),0,[encArg(V12, Ret02),encArg(V13, Ret13),minus(Ret02, Ret13, Ret3)],[Out = Ret3,V12 >= 0,V = 1 + V12 + V13,V13 >= 0]).
eq(encArg(V, Out),0,[encArg(V15, Ret03),encArg(V14, Ret14),quot(Ret03, Ret14, Ret4)],[Out = Ret4,V15 >= 0,V = 1 + V14 + V15,V14 >= 0]).
eq(encArg(V, Out),0,[encArg(V16, Ret04),log(Ret04, Ret5)],[Out = Ret5,V = 1 + V16,V16 >= 0]).
eq(fun(V, Out),0,[encArg(V17, Ret05),pred(Ret05, Ret6)],[Out = Ret6,V17 >= 0,V = V17]).
eq(fun1(V, Out),0,[encArg(V18, Ret15)],[Out = 1 + Ret15,V18 >= 0,V = V18]).
eq(fun2(V, V2, Out),0,[encArg(V20, Ret06),encArg(V19, Ret16),minus(Ret06, Ret16, Ret7)],[Out = Ret7,V20 >= 0,V19 >= 0,V = V20,V2 = V19]).
eq(fun3(Out),0,[],[Out = 0]).
eq(fun4(V, V2, Out),0,[encArg(V22, Ret07),encArg(V21, Ret17),quot(Ret07, Ret17, Ret8)],[Out = Ret8,V22 >= 0,V21 >= 0,V = V22,V2 = V21]).
eq(fun5(V, Out),0,[encArg(V23, Ret08),log(Ret08, Ret9)],[Out = Ret9,V23 >= 0,V = V23]).
eq(encArg(V, Out),0,[],[Out = 0,V24 >= 0,V = V24]).
eq(fun(V, Out),0,[],[Out = 0,V25 >= 0,V = V25]).
eq(fun1(V, Out),0,[],[Out = 0,V26 >= 0,V = V26]).
eq(fun2(V, V2, Out),0,[],[Out = 0,V27 >= 0,V28 >= 0,V = V27,V2 = V28]).
eq(fun4(V, V2, Out),0,[],[Out = 0,V29 >= 0,V30 >= 0,V = V29,V2 = V30]).
eq(fun5(V, Out),0,[],[Out = 0,V31 >= 0,V = V31]).
eq(pred(V, Out),0,[],[Out = 0,V32 >= 0,V = V32]).
eq(minus(V, V2, Out),0,[],[Out = 0,V33 >= 0,V34 >= 0,V = V33,V2 = V34]).
eq(quot(V, V2, Out),0,[],[Out = 0,V35 >= 0,V36 >= 0,V = V35,V2 = V36]).
eq(log(V, Out),0,[],[Out = 0,V37 >= 0,V = V37]).
input_output_vars(pred(V,Out),[V],[Out]).
input_output_vars(minus(V,V2,Out),[V,V2],[Out]).
input_output_vars(quot(V,V2,Out),[V,V2],[Out]).
input_output_vars(log(V,Out),[V],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(fun1(V,Out),[V],[Out]).
input_output_vars(fun2(V,V2,Out),[V,V2],[Out]).
input_output_vars(fun3(Out),[],[Out]).
input_output_vars(fun4(V,V2,Out),[V,V2],[Out]).
input_output_vars(fun5(V,Out),[V],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into pred/2
1. SCC is partially evaluated into minus/3
2. SCC is partially evaluated into quot/3
3. SCC is partially evaluated into log/2
4. SCC is partially evaluated into encArg/2
5. SCC is partially evaluated into fun/2
6. SCC is partially evaluated into fun1/2
7. SCC is partially evaluated into fun2/3
8. SCC is completely evaluated into other SCCs
9. SCC is partially evaluated into fun4/3
10. SCC is partially evaluated into fun5/2
11. SCC is partially evaluated into start/2

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

### Specialization of cost equations pred/2 
* CE 12 is refined into CE [39] 
* CE 13 is refined into CE [40] 


### Cost equations --> "Loop" of pred/2 
* CEs [39] --> Loop 19 
* CEs [40] --> Loop 20 

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

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


### Specialization of cost equations minus/3 
* CE 16 is refined into CE [41] 
* CE 14 is refined into CE [42] 
* CE 15 is refined into CE [43,44] 


### Cost equations --> "Loop" of minus/3 
* CEs [44] --> Loop 21 
* CEs [43] --> Loop 22 
* CEs [41] --> Loop 23 
* CEs [42] --> Loop 24 

### Ranking functions of CR minus(V,V2,Out) 
* RF of phase [21]: [V2]
* RF of phase [22]: [V2]

#### Partial ranking functions of CR minus(V,V2,Out) 
* Partial RF of phase [21]:
  - RF of loop [21:1]:
    V2
* Partial RF of phase [22]:
  - RF of loop [22:1]:
    V2


### Specialization of cost equations quot/3 
* CE 17 is refined into CE [45] 
* CE 19 is refined into CE [46] 
* CE 18 is refined into CE [47,48,49] 


### Cost equations --> "Loop" of quot/3 
* CEs [49] --> Loop 25 
* CEs [48] --> Loop 26 
* CEs [47] --> Loop 27 
* CEs [45,46] --> Loop 28 

### Ranking functions of CR quot(V,V2,Out) 
* RF of phase [25]: [V-1,V-V2+1]
* RF of phase [27]: [V]

#### Partial ranking functions of CR quot(V,V2,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V-1
    V-V2+1
* Partial RF of phase [27]:
  - RF of loop [27:1]:
    V


### Specialization of cost equations log/2 
* CE 20 is refined into CE [50] 
* CE 22 is refined into CE [51] 
* CE 21 is refined into CE [52,53,54,55] 


### Cost equations --> "Loop" of log/2 
* CEs [55] --> Loop 29 
* CEs [54] --> Loop 30 
* CEs [53] --> Loop 31 
* CEs [52] --> Loop 32 
* CEs [50,51] --> Loop 33 

### Ranking functions of CR log(V,Out) 
* RF of phase [29,30]: [V-3,V/2-3/2]

#### Partial ranking functions of CR log(V,Out) 
* Partial RF of phase [29,30]:
  - RF of loop [29:1]:
    V/2-2
  - RF of loop [30:1]:
    V-3


### Specialization of cost equations encArg/2 
* CE 24 is refined into CE [56] 
* CE 23 is refined into CE [57] 
* CE 25 is refined into CE [58,59] 
* CE 28 is refined into CE [60,61,62,63,64,65] 
* CE 26 is refined into CE [66,67,68] 
* CE 27 is refined into CE [69,70,71,72,73] 


### Cost equations --> "Loop" of encArg/2 
* CEs [73] --> Loop 34 
* CEs [72] --> Loop 35 
* CEs [68] --> Loop 36 
* CEs [69] --> Loop 37 
* CEs [66] --> Loop 38 
* CEs [71] --> Loop 39 
* CEs [67,70] --> Loop 40 
* CEs [65] --> Loop 41 
* CEs [64] --> Loop 42 
* CEs [63] --> Loop 43 
* CEs [57] --> Loop 44 
* CEs [59] --> Loop 45 
* CEs [62] --> Loop 46 
* CEs [61] --> Loop 47 
* CEs [58,60] --> Loop 48 
* CEs [56] --> Loop 49 

### Ranking functions of CR encArg(V,Out) 
* RF of phase [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]: [V]

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]:
  - RF of loop [34:1,34:2,35:1,35:2,36:1,36:2,37:1,37:2,38:1,38:2,39:1,39:2,40:1,40:2,41:1,42:1,43:1,44:1,45:1,46:1,47:1,48:1]:
    V


### Specialization of cost equations fun/2 
* CE 29 is refined into CE [74,75,76] 
* CE 30 is refined into CE [77] 


### Cost equations --> "Loop" of fun/2 
* CEs [76] --> Loop 50 
* CEs [74,75,77] --> Loop 51 

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

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


### Specialization of cost equations fun1/2 
* CE 31 is refined into CE [78,79] 
* CE 32 is refined into CE [80] 


### Cost equations --> "Loop" of fun1/2 
* CEs [79] --> Loop 52 
* CEs [78] --> Loop 53 
* CEs [80] --> Loop 54 

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

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


### Specialization of cost equations fun2/3 
* CE 33 is refined into CE [81,82,83,84,85,86,87,88,89] 
* CE 34 is refined into CE [90] 


### Cost equations --> "Loop" of fun2/3 
* CEs [85,87,89] --> Loop 55 
* CEs [81,82,83,84,86,88,90] --> Loop 56 

### Ranking functions of CR fun2(V,V2,Out) 

#### Partial ranking functions of CR fun2(V,V2,Out) 


### Specialization of cost equations fun4/3 
* CE 35 is refined into CE [91,92,93,94,95,96,97,98] 
* CE 36 is refined into CE [99] 


### Cost equations --> "Loop" of fun4/3 
* CEs [94,96,97,98] --> Loop 57 
* CEs [91,92,93,95,99] --> Loop 58 

### Ranking functions of CR fun4(V,V2,Out) 

#### Partial ranking functions of CR fun4(V,V2,Out) 


### Specialization of cost equations fun5/2 
* CE 37 is refined into CE [100,101,102,103,104,105,106] 
* CE 38 is refined into CE [107] 


### Cost equations --> "Loop" of fun5/2 
* CEs [106] --> Loop 59 
* CEs [105] --> Loop 60 
* CEs [104] --> Loop 61 
* CEs [103] --> Loop 62 
* CEs [102] --> Loop 63 
* CEs [100,101,107] --> Loop 64 

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

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


### Specialization of cost equations start/2 
* CE 1 is refined into CE [108,109] 
* CE 2 is refined into CE [110,111,112] 
* CE 3 is refined into CE [113,114,115,116,117] 
* CE 4 is refined into CE [118,119,120,121,122,123] 
* CE 5 is refined into CE [124,125] 
* CE 6 is refined into CE [126,127] 
* CE 7 is refined into CE [128,129,130] 
* CE 8 is refined into CE [131,132] 
* CE 9 is refined into CE [133] 
* CE 10 is refined into CE [134,135] 
* CE 11 is refined into CE [136,137,138,139,140,141] 


### Cost equations --> "Loop" of start/2 
* CEs [108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141] --> Loop 65 

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

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


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

#### Cost of chains of pred(V,Out):
* Chain [20]: 0
  with precondition: [Out=0,V>=0] 

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


#### Cost of chains of minus(V,V2,Out):
* Chain [[22],[21],24]: 3*it(21)+1
  Such that:aux(1) =< V2
it(21) =< aux(1)

  with precondition: [Out=0,V>=1,V2>=2] 

* Chain [[22],24]: 1*it(22)+1
  Such that:it(22) =< V2

  with precondition: [Out=0,V>=0,V2>=1] 

* Chain [[22],23]: 1*it(22)+0
  Such that:it(22) =< V2

  with precondition: [Out=0,V>=0,V2>=1] 

* Chain [[21],24]: 2*it(21)+1
  Such that:it(21) =< V2

  with precondition: [V=Out+V2,V2>=1,V>=V2] 

* Chain [24]: 1
  with precondition: [V2=0,V=Out,V>=0] 

* Chain [23]: 0
  with precondition: [Out=0,V>=0,V2>=0] 


#### Cost of chains of quot(V,V2,Out):
* Chain [[27],28]: 2*it(27)+1
  Such that:it(27) =< Out

  with precondition: [V2=1,Out>=1,V>=Out] 

* Chain [[27],26,28]: 2*it(27)+5*s(6)+3
  Such that:s(5) =< 1
it(27) =< Out
s(6) =< s(5)

  with precondition: [V2=1,Out>=2,V>=Out] 

* Chain [[25],28]: 2*it(25)+2*s(9)+1
  Such that:it(25) =< V-V2+1
aux(5) =< V
it(25) =< aux(5)
s(9) =< aux(5)

  with precondition: [V2>=2,Out>=1,V+2>=2*Out+V2] 

* Chain [[25],26,28]: 2*it(25)+5*s(6)+2*s(9)+3
  Such that:it(25) =< V-V2+1
s(5) =< V2
aux(6) =< V
s(6) =< s(5)
it(25) =< aux(6)
s(9) =< aux(6)

  with precondition: [V2>=2,Out>=2,V+3>=2*Out+V2] 

* Chain [28]: 1
  with precondition: [Out=0,V>=0,V2>=0] 

* Chain [26,28]: 5*s(6)+3
  Such that:s(5) =< V2
s(6) =< s(5)

  with precondition: [Out=1,V>=1,V2>=1] 


#### Cost of chains of log(V,Out):
* Chain [[29,30],33]: 4*it(29)+2*it(30)+4*s(28)+5*s(29)+4*s(32)+1
  Such that:s(33) =< 2*V
aux(16) =< 5/2*V
aux(15) =< 5/2*V+27/2
aux(17) =< V
aux(18) =< V/2
aux(10) =< aux(17)
it(29) =< aux(17)
it(30) =< aux(17)
aux(10) =< aux(18)
it(29) =< aux(18)
it(30) =< aux(18)
it(30) =< aux(15)
s(31) =< aux(15)
it(30) =< aux(16)
s(31) =< aux(16)
s(30) =< aux(10)*2
s(32) =< s(33)
s(28) =< s(31)
s(29) =< s(30)

  with precondition: [Out>=1,V>=3*Out+1] 

* Chain [[29,30],32,33]: 4*it(29)+2*it(30)+4*s(28)+5*s(29)+4*s(32)+3
  Such that:s(33) =< 2*V
aux(16) =< 5/2*V
aux(15) =< 5/2*V+27/2
aux(19) =< V
aux(20) =< V/2
aux(10) =< aux(19)
it(29) =< aux(19)
it(30) =< aux(19)
aux(10) =< aux(20)
it(29) =< aux(20)
it(30) =< aux(20)
it(30) =< aux(15)
s(31) =< aux(15)
it(30) =< aux(16)
s(31) =< aux(16)
s(30) =< aux(10)*2
s(32) =< s(33)
s(28) =< s(31)
s(29) =< s(30)

  with precondition: [Out>=2,V+2>=3*Out] 

* Chain [[29,30],31,33]: 4*it(29)+2*it(30)+4*s(28)+5*s(29)+4*s(32)+5*s(35)+5
  Such that:s(34) =< 2
s(33) =< 2*V
aux(16) =< 5/2*V
aux(15) =< 5/2*V+27/2
aux(21) =< V
aux(22) =< V/2
s(35) =< s(34)
aux(10) =< aux(21)
it(29) =< aux(21)
it(30) =< aux(21)
aux(10) =< aux(22)
it(29) =< aux(22)
it(30) =< aux(22)
it(30) =< aux(15)
s(31) =< aux(15)
it(30) =< aux(16)
s(31) =< aux(16)
s(30) =< aux(10)*2
s(32) =< s(33)
s(28) =< s(31)
s(29) =< s(30)

  with precondition: [Out>=2,V+3>=4*Out] 

* Chain [[29,30],31,32,33]: 4*it(29)+2*it(30)+4*s(28)+5*s(29)+4*s(32)+5*s(35)+7
  Such that:s(34) =< 2
s(33) =< 2*V
aux(16) =< 5/2*V
aux(15) =< 5/2*V+27/2
aux(23) =< V
aux(24) =< V/2
s(35) =< s(34)
aux(10) =< aux(23)
it(29) =< aux(23)
it(30) =< aux(23)
aux(10) =< aux(24)
it(29) =< aux(24)
it(30) =< aux(24)
it(30) =< aux(15)
s(31) =< aux(15)
it(30) =< aux(16)
s(31) =< aux(16)
s(30) =< aux(10)*2
s(32) =< s(33)
s(28) =< s(31)
s(29) =< s(30)

  with precondition: [Out>=3,V+7>=4*Out] 

* Chain [33]: 1
  with precondition: [Out=0,V>=0] 

* Chain [32,33]: 3
  with precondition: [Out=1,V>=2] 

* Chain [31,33]: 5*s(35)+5
  Such that:s(34) =< 2
s(35) =< s(34)

  with precondition: [Out=1,V>=3] 

* Chain [31,32,33]: 5*s(35)+7
  Such that:s(34) =< 2
s(35) =< s(34)

  with precondition: [Out=2,V>=3] 


#### Cost of chains of encArg(V,Out):
* Chain [49]: 0
  with precondition: [Out=0,V>=0] 

* Chain [multiple([34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[[49]])]: 3*it(34)+1*it(35)+4*it(36)+2*it(38)+3*it(39)+37*it(41)+9*s(186)+4*s(188)+2*s(190)+4*s(191)+5*s(192)+5*s(195)+5*s(197)+15*s(199)+16*s(200)+8*s(201)+16*s(202)+16*s(203)+20*s(204)+0
  Such that:it([49]) =< V+1
aux(62) =< V
aux(63) =< V/2
aux(64) =< V/3
aux(65) =< 2/7*V
it(38) =< aux(62)
it(39) =< aux(62)
it(41) =< aux(62)
aux(39) =< aux(63)
it(36) =< aux(63)
it(39) =< aux(63)
it(35) =< aux(64)
it(34) =< aux(65)
aux(49) =< aux(62)*(5/2)+16
aux(40) =< aux(62)+1
aux(35) =< aux(62)
s(213) =< aux(62)*2
it(36) =< it([49])*(1/2)+aux(63)
it(38) =< it([49])*(1/2)+aux(63)
it(39) =< it([49])*(1/2)+aux(63)
aux(39) =< it([49])*(1/2)+aux(63)
it(35) =< it([49])*(2/3)+aux(64)
it(36) =< it([49])*(2/3)+aux(64)
it(38) =< it([49])*(2/3)+aux(64)
it(39) =< it([49])*(2/3)+aux(64)
it(34) =< it([49])*(5/7)+aux(65)
it(35) =< it([49])*(5/7)+aux(65)
it(36) =< it([49])*(5/7)+aux(65)
it(38) =< it([49])*(5/7)+aux(65)
it(39) =< it([49])*(5/7)+aux(65)
s(210) =< it(41)*aux(49)
aux(46) =< it(41)*aux(40)
s(198) =< it(38)*aux(40)
s(196) =< it(39)*aux(35)
s(194) =< it(36)*aux(40)
s(190) =< it(36)*aux(35)
s(189) =< it(35)*aux(35)
s(187) =< it(34)*aux(62)
s(209) =< aux(46)*(5/2)
s(211) =< aux(46)*(1/2)
s(207) =< aux(46)*2
s(199) =< s(213)
s(208) =< aux(46)
s(200) =< aux(46)
s(201) =< aux(46)
s(208) =< s(211)
s(200) =< s(211)
s(201) =< s(211)
s(201) =< s(210)
s(206) =< s(210)
s(201) =< s(209)
s(206) =< s(209)
s(205) =< s(208)*2
s(202) =< s(207)
s(203) =< s(206)
s(204) =< s(205)
s(197) =< s(198)
s(195) =< s(196)
s(191) =< s(194)
s(192) =< aux(39)
s(188) =< s(189)
s(186) =< s(187)

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


#### Cost of chains of fun(V,Out):
* Chain [51]: 2*s(249)+3*s(250)+37*s(251)+4*s(253)+1*s(254)+3*s(255)+2*s(265)+15*s(271)+16*s(273)+8*s(274)+16*s(277)+16*s(278)+20*s(279)+5*s(280)+5*s(281)+4*s(282)+5*s(283)+4*s(284)+9*s(285)+0
  Such that:s(245) =< V
s(244) =< V+1
s(246) =< V/2
s(247) =< V/3
s(248) =< 2/7*V
s(249) =< s(245)
s(250) =< s(245)
s(251) =< s(245)
s(252) =< s(246)
s(253) =< s(246)
s(250) =< s(246)
s(254) =< s(247)
s(255) =< s(248)
s(256) =< s(245)*(5/2)+16
s(257) =< s(245)+1
s(258) =< s(245)
s(259) =< s(245)*2
s(253) =< s(244)*(1/2)+s(246)
s(249) =< s(244)*(1/2)+s(246)
s(250) =< s(244)*(1/2)+s(246)
s(252) =< s(244)*(1/2)+s(246)
s(254) =< s(244)*(2/3)+s(247)
s(253) =< s(244)*(2/3)+s(247)
s(249) =< s(244)*(2/3)+s(247)
s(250) =< s(244)*(2/3)+s(247)
s(255) =< s(244)*(5/7)+s(248)
s(254) =< s(244)*(5/7)+s(248)
s(253) =< s(244)*(5/7)+s(248)
s(249) =< s(244)*(5/7)+s(248)
s(250) =< s(244)*(5/7)+s(248)
s(260) =< s(251)*s(256)
s(261) =< s(251)*s(257)
s(262) =< s(249)*s(257)
s(263) =< s(250)*s(258)
s(264) =< s(253)*s(257)
s(265) =< s(253)*s(258)
s(266) =< s(254)*s(258)
s(267) =< s(255)*s(245)
s(268) =< s(261)*(5/2)
s(269) =< s(261)*(1/2)
s(270) =< s(261)*2
s(271) =< s(259)
s(272) =< s(261)
s(273) =< s(261)
s(274) =< s(261)
s(272) =< s(269)
s(273) =< s(269)
s(274) =< s(269)
s(274) =< s(260)
s(275) =< s(260)
s(274) =< s(268)
s(275) =< s(268)
s(276) =< s(272)*2
s(277) =< s(270)
s(278) =< s(275)
s(279) =< s(276)
s(280) =< s(262)
s(281) =< s(263)
s(282) =< s(264)
s(283) =< s(252)
s(284) =< s(266)
s(285) =< s(267)

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

* Chain [50]: 2*s(291)+3*s(292)+37*s(293)+4*s(295)+1*s(296)+3*s(297)+2*s(307)+15*s(313)+16*s(315)+8*s(316)+16*s(319)+16*s(320)+20*s(321)+5*s(322)+5*s(323)+4*s(324)+5*s(325)+4*s(326)+9*s(327)+1
  Such that:s(287) =< V
s(286) =< V+1
s(288) =< V/2
s(289) =< V/3
s(290) =< 2/7*V
s(291) =< s(287)
s(292) =< s(287)
s(293) =< s(287)
s(294) =< s(288)
s(295) =< s(288)
s(292) =< s(288)
s(296) =< s(289)
s(297) =< s(290)
s(298) =< s(287)*(5/2)+16
s(299) =< s(287)+1
s(300) =< s(287)
s(301) =< s(287)*2
s(295) =< s(286)*(1/2)+s(288)
s(291) =< s(286)*(1/2)+s(288)
s(292) =< s(286)*(1/2)+s(288)
s(294) =< s(286)*(1/2)+s(288)
s(296) =< s(286)*(2/3)+s(289)
s(295) =< s(286)*(2/3)+s(289)
s(291) =< s(286)*(2/3)+s(289)
s(292) =< s(286)*(2/3)+s(289)
s(297) =< s(286)*(5/7)+s(290)
s(296) =< s(286)*(5/7)+s(290)
s(295) =< s(286)*(5/7)+s(290)
s(291) =< s(286)*(5/7)+s(290)
s(292) =< s(286)*(5/7)+s(290)
s(302) =< s(293)*s(298)
s(303) =< s(293)*s(299)
s(304) =< s(291)*s(299)
s(305) =< s(292)*s(300)
s(306) =< s(295)*s(299)
s(307) =< s(295)*s(300)
s(308) =< s(296)*s(300)
s(309) =< s(297)*s(287)
s(310) =< s(303)*(5/2)
s(311) =< s(303)*(1/2)
s(312) =< s(303)*2
s(313) =< s(301)
s(314) =< s(303)
s(315) =< s(303)
s(316) =< s(303)
s(314) =< s(311)
s(315) =< s(311)
s(316) =< s(311)
s(316) =< s(302)
s(317) =< s(302)
s(316) =< s(310)
s(317) =< s(310)
s(318) =< s(314)*2
s(319) =< s(312)
s(320) =< s(317)
s(321) =< s(318)
s(322) =< s(304)
s(323) =< s(305)
s(324) =< s(306)
s(325) =< s(294)
s(326) =< s(308)
s(327) =< s(309)

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


#### Cost of chains of fun1(V,Out):
* Chain [54]: 0
  with precondition: [Out=0,V>=0] 

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

* Chain [52]: 2*s(333)+3*s(334)+37*s(335)+4*s(337)+1*s(338)+3*s(339)+2*s(349)+15*s(355)+16*s(357)+8*s(358)+16*s(361)+16*s(362)+20*s(363)+5*s(364)+5*s(365)+4*s(366)+5*s(367)+4*s(368)+9*s(369)+0
  Such that:s(329) =< V
s(328) =< V+1
s(330) =< V/2
s(331) =< V/3
s(332) =< 2/7*V
s(333) =< s(329)
s(334) =< s(329)
s(335) =< s(329)
s(336) =< s(330)
s(337) =< s(330)
s(334) =< s(330)
s(338) =< s(331)
s(339) =< s(332)
s(340) =< s(329)*(5/2)+16
s(341) =< s(329)+1
s(342) =< s(329)
s(343) =< s(329)*2
s(337) =< s(328)*(1/2)+s(330)
s(333) =< s(328)*(1/2)+s(330)
s(334) =< s(328)*(1/2)+s(330)
s(336) =< s(328)*(1/2)+s(330)
s(338) =< s(328)*(2/3)+s(331)
s(337) =< s(328)*(2/3)+s(331)
s(333) =< s(328)*(2/3)+s(331)
s(334) =< s(328)*(2/3)+s(331)
s(339) =< s(328)*(5/7)+s(332)
s(338) =< s(328)*(5/7)+s(332)
s(337) =< s(328)*(5/7)+s(332)
s(333) =< s(328)*(5/7)+s(332)
s(334) =< s(328)*(5/7)+s(332)
s(344) =< s(335)*s(340)
s(345) =< s(335)*s(341)
s(346) =< s(333)*s(341)
s(347) =< s(334)*s(342)
s(348) =< s(337)*s(341)
s(349) =< s(337)*s(342)
s(350) =< s(338)*s(342)
s(351) =< s(339)*s(329)
s(352) =< s(345)*(5/2)
s(353) =< s(345)*(1/2)
s(354) =< s(345)*2
s(355) =< s(343)
s(356) =< s(345)
s(357) =< s(345)
s(358) =< s(345)
s(356) =< s(353)
s(357) =< s(353)
s(358) =< s(353)
s(358) =< s(344)
s(359) =< s(344)
s(358) =< s(352)
s(359) =< s(352)
s(360) =< s(356)*2
s(361) =< s(354)
s(362) =< s(359)
s(363) =< s(360)
s(364) =< s(346)
s(365) =< s(347)
s(366) =< s(348)
s(367) =< s(336)
s(368) =< s(350)
s(369) =< s(351)

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


#### Cost of chains of fun2(V,V2,Out):
* Chain [56]: 6*s(377)+9*s(378)+121*s(379)+12*s(381)+3*s(382)+9*s(383)+6*s(393)+45*s(399)+48*s(401)+24*s(402)+48*s(405)+48*s(406)+60*s(407)+15*s(408)+15*s(409)+12*s(410)+15*s(411)+12*s(412)+27*s(413)+4*s(463)+6*s(464)+74*s(465)+8*s(467)+2*s(468)+6*s(469)+4*s(479)+30*s(485)+32*s(487)+16*s(488)+32*s(491)+32*s(492)+40*s(493)+10*s(494)+10*s(495)+8*s(496)+10*s(497)+8*s(498)+18*s(499)+1
  Such that:aux(68) =< V
aux(69) =< V+1
aux(70) =< V/2
aux(71) =< V/3
aux(72) =< 2/7*V
aux(73) =< V2
aux(74) =< V2+1
aux(75) =< V2/2
aux(76) =< V2/3
aux(77) =< 2/7*V2
s(463) =< aux(68)
s(464) =< aux(68)
s(465) =< aux(68)
s(466) =< aux(70)
s(467) =< aux(70)
s(464) =< aux(70)
s(468) =< aux(71)
s(469) =< aux(72)
s(470) =< aux(68)*(5/2)+16
s(471) =< aux(68)+1
s(472) =< aux(68)
s(473) =< aux(68)*2
s(467) =< aux(69)*(1/2)+aux(70)
s(463) =< aux(69)*(1/2)+aux(70)
s(464) =< aux(69)*(1/2)+aux(70)
s(466) =< aux(69)*(1/2)+aux(70)
s(468) =< aux(69)*(2/3)+aux(71)
s(467) =< aux(69)*(2/3)+aux(71)
s(463) =< aux(69)*(2/3)+aux(71)
s(464) =< aux(69)*(2/3)+aux(71)
s(469) =< aux(69)*(5/7)+aux(72)
s(468) =< aux(69)*(5/7)+aux(72)
s(467) =< aux(69)*(5/7)+aux(72)
s(463) =< aux(69)*(5/7)+aux(72)
s(464) =< aux(69)*(5/7)+aux(72)
s(474) =< s(465)*s(470)
s(475) =< s(465)*s(471)
s(476) =< s(463)*s(471)
s(477) =< s(464)*s(472)
s(478) =< s(467)*s(471)
s(479) =< s(467)*s(472)
s(480) =< s(468)*s(472)
s(481) =< s(469)*aux(68)
s(482) =< s(475)*(5/2)
s(483) =< s(475)*(1/2)
s(484) =< s(475)*2
s(485) =< s(473)
s(486) =< s(475)
s(487) =< s(475)
s(488) =< s(475)
s(486) =< s(483)
s(487) =< s(483)
s(488) =< s(483)
s(488) =< s(474)
s(489) =< s(474)
s(488) =< s(482)
s(489) =< s(482)
s(490) =< s(486)*2
s(491) =< s(484)
s(492) =< s(489)
s(493) =< s(490)
s(494) =< s(476)
s(495) =< s(477)
s(496) =< s(478)
s(497) =< s(466)
s(498) =< s(480)
s(499) =< s(481)
s(379) =< aux(73)
s(377) =< aux(73)
s(378) =< aux(73)
s(380) =< aux(75)
s(381) =< aux(75)
s(378) =< aux(75)
s(382) =< aux(76)
s(383) =< aux(77)
s(384) =< aux(73)*(5/2)+16
s(385) =< aux(73)+1
s(386) =< aux(73)
s(387) =< aux(73)*2
s(381) =< aux(74)*(1/2)+aux(75)
s(377) =< aux(74)*(1/2)+aux(75)
s(378) =< aux(74)*(1/2)+aux(75)
s(380) =< aux(74)*(1/2)+aux(75)
s(382) =< aux(74)*(2/3)+aux(76)
s(381) =< aux(74)*(2/3)+aux(76)
s(377) =< aux(74)*(2/3)+aux(76)
s(378) =< aux(74)*(2/3)+aux(76)
s(383) =< aux(74)*(5/7)+aux(77)
s(382) =< aux(74)*(5/7)+aux(77)
s(381) =< aux(74)*(5/7)+aux(77)
s(377) =< aux(74)*(5/7)+aux(77)
s(378) =< aux(74)*(5/7)+aux(77)
s(388) =< s(379)*s(384)
s(389) =< s(379)*s(385)
s(390) =< s(377)*s(385)
s(391) =< s(378)*s(386)
s(392) =< s(381)*s(385)
s(393) =< s(381)*s(386)
s(394) =< s(382)*s(386)
s(395) =< s(383)*aux(73)
s(396) =< s(389)*(5/2)
s(397) =< s(389)*(1/2)
s(398) =< s(389)*2
s(399) =< s(387)
s(400) =< s(389)
s(401) =< s(389)
s(402) =< s(389)
s(400) =< s(397)
s(401) =< s(397)
s(402) =< s(397)
s(402) =< s(388)
s(403) =< s(388)
s(402) =< s(396)
s(403) =< s(396)
s(404) =< s(400)*2
s(405) =< s(398)
s(406) =< s(403)
s(407) =< s(404)
s(408) =< s(390)
s(409) =< s(391)
s(410) =< s(392)
s(411) =< s(380)
s(412) =< s(394)
s(413) =< s(395)

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

* Chain [55]: 6*s(593)+9*s(594)+111*s(595)+12*s(597)+3*s(598)+9*s(599)+6*s(609)+45*s(615)+48*s(617)+24*s(618)+48*s(621)+48*s(622)+60*s(623)+15*s(624)+15*s(625)+12*s(626)+15*s(627)+12*s(628)+27*s(629)+4*s(677)+6*s(678)+76*s(679)+8*s(681)+2*s(682)+6*s(683)+4*s(693)+30*s(699)+32*s(701)+16*s(702)+32*s(705)+32*s(706)+40*s(707)+10*s(708)+10*s(709)+8*s(710)+10*s(711)+8*s(712)+18*s(713)+1
  Such that:aux(79) =< V
aux(80) =< V+1
aux(81) =< V/2
aux(82) =< V/3
aux(83) =< 2/7*V
aux(84) =< V2
aux(85) =< V2+1
aux(86) =< V2/2
aux(87) =< V2/3
aux(88) =< 2/7*V2
s(593) =< aux(79)
s(594) =< aux(79)
s(595) =< aux(79)
s(596) =< aux(81)
s(597) =< aux(81)
s(594) =< aux(81)
s(598) =< aux(82)
s(599) =< aux(83)
s(600) =< aux(79)*(5/2)+16
s(601) =< aux(79)+1
s(602) =< aux(79)
s(603) =< aux(79)*2
s(597) =< aux(80)*(1/2)+aux(81)
s(593) =< aux(80)*(1/2)+aux(81)
s(594) =< aux(80)*(1/2)+aux(81)
s(596) =< aux(80)*(1/2)+aux(81)
s(598) =< aux(80)*(2/3)+aux(82)
s(597) =< aux(80)*(2/3)+aux(82)
s(593) =< aux(80)*(2/3)+aux(82)
s(594) =< aux(80)*(2/3)+aux(82)
s(599) =< aux(80)*(5/7)+aux(83)
s(598) =< aux(80)*(5/7)+aux(83)
s(597) =< aux(80)*(5/7)+aux(83)
s(593) =< aux(80)*(5/7)+aux(83)
s(594) =< aux(80)*(5/7)+aux(83)
s(604) =< s(595)*s(600)
s(605) =< s(595)*s(601)
s(606) =< s(593)*s(601)
s(607) =< s(594)*s(602)
s(608) =< s(597)*s(601)
s(609) =< s(597)*s(602)
s(610) =< s(598)*s(602)
s(611) =< s(599)*aux(79)
s(612) =< s(605)*(5/2)
s(613) =< s(605)*(1/2)
s(614) =< s(605)*2
s(615) =< s(603)
s(616) =< s(605)
s(617) =< s(605)
s(618) =< s(605)
s(616) =< s(613)
s(617) =< s(613)
s(618) =< s(613)
s(618) =< s(604)
s(619) =< s(604)
s(618) =< s(612)
s(619) =< s(612)
s(620) =< s(616)*2
s(621) =< s(614)
s(622) =< s(619)
s(623) =< s(620)
s(624) =< s(606)
s(625) =< s(607)
s(626) =< s(608)
s(627) =< s(596)
s(628) =< s(610)
s(629) =< s(611)
s(677) =< aux(84)
s(678) =< aux(84)
s(679) =< aux(84)
s(680) =< aux(86)
s(681) =< aux(86)
s(678) =< aux(86)
s(682) =< aux(87)
s(683) =< aux(88)
s(684) =< aux(84)*(5/2)+16
s(685) =< aux(84)+1
s(686) =< aux(84)
s(687) =< aux(84)*2
s(681) =< aux(85)*(1/2)+aux(86)
s(677) =< aux(85)*(1/2)+aux(86)
s(678) =< aux(85)*(1/2)+aux(86)
s(680) =< aux(85)*(1/2)+aux(86)
s(682) =< aux(85)*(2/3)+aux(87)
s(681) =< aux(85)*(2/3)+aux(87)
s(677) =< aux(85)*(2/3)+aux(87)
s(678) =< aux(85)*(2/3)+aux(87)
s(683) =< aux(85)*(5/7)+aux(88)
s(682) =< aux(85)*(5/7)+aux(88)
s(681) =< aux(85)*(5/7)+aux(88)
s(677) =< aux(85)*(5/7)+aux(88)
s(678) =< aux(85)*(5/7)+aux(88)
s(688) =< s(679)*s(684)
s(689) =< s(679)*s(685)
s(690) =< s(677)*s(685)
s(691) =< s(678)*s(686)
s(692) =< s(681)*s(685)
s(693) =< s(681)*s(686)
s(694) =< s(682)*s(686)
s(695) =< s(683)*aux(84)
s(696) =< s(689)*(5/2)
s(697) =< s(689)*(1/2)
s(698) =< s(689)*2
s(699) =< s(687)
s(700) =< s(689)
s(701) =< s(689)
s(702) =< s(689)
s(700) =< s(697)
s(701) =< s(697)
s(702) =< s(697)
s(702) =< s(688)
s(703) =< s(688)
s(702) =< s(696)
s(703) =< s(696)
s(704) =< s(700)*2
s(705) =< s(698)
s(706) =< s(703)
s(707) =< s(704)
s(708) =< s(690)
s(709) =< s(691)
s(710) =< s(692)
s(711) =< s(680)
s(712) =< s(694)
s(713) =< s(695)

  with precondition: [V>=1,V2>=0,Out>=0,V>=Out] 


#### Cost of chains of fun4(V,V2,Out):
* Chain [58]: 4*s(804)+6*s(805)+74*s(806)+8*s(808)+2*s(809)+6*s(810)+4*s(820)+30*s(826)+32*s(828)+16*s(829)+32*s(832)+32*s(833)+40*s(834)+10*s(835)+10*s(836)+8*s(837)+10*s(838)+8*s(839)+18*s(840)+4*s(846)+6*s(847)+74*s(848)+8*s(850)+2*s(851)+6*s(852)+4*s(862)+30*s(868)+32*s(870)+16*s(871)+32*s(874)+32*s(875)+40*s(876)+10*s(877)+10*s(878)+8*s(879)+10*s(880)+8*s(881)+18*s(882)+1
  Such that:aux(89) =< V
aux(90) =< V+1
aux(91) =< V/2
aux(92) =< V/3
aux(93) =< 2/7*V
aux(94) =< V2
aux(95) =< V2+1
aux(96) =< V2/2
aux(97) =< V2/3
aux(98) =< 2/7*V2
s(846) =< aux(89)
s(847) =< aux(89)
s(848) =< aux(89)
s(849) =< aux(91)
s(850) =< aux(91)
s(847) =< aux(91)
s(851) =< aux(92)
s(852) =< aux(93)
s(853) =< aux(89)*(5/2)+16
s(854) =< aux(89)+1
s(855) =< aux(89)
s(856) =< aux(89)*2
s(850) =< aux(90)*(1/2)+aux(91)
s(846) =< aux(90)*(1/2)+aux(91)
s(847) =< aux(90)*(1/2)+aux(91)
s(849) =< aux(90)*(1/2)+aux(91)
s(851) =< aux(90)*(2/3)+aux(92)
s(850) =< aux(90)*(2/3)+aux(92)
s(846) =< aux(90)*(2/3)+aux(92)
s(847) =< aux(90)*(2/3)+aux(92)
s(852) =< aux(90)*(5/7)+aux(93)
s(851) =< aux(90)*(5/7)+aux(93)
s(850) =< aux(90)*(5/7)+aux(93)
s(846) =< aux(90)*(5/7)+aux(93)
s(847) =< aux(90)*(5/7)+aux(93)
s(857) =< s(848)*s(853)
s(858) =< s(848)*s(854)
s(859) =< s(846)*s(854)
s(860) =< s(847)*s(855)
s(861) =< s(850)*s(854)
s(862) =< s(850)*s(855)
s(863) =< s(851)*s(855)
s(864) =< s(852)*aux(89)
s(865) =< s(858)*(5/2)
s(866) =< s(858)*(1/2)
s(867) =< s(858)*2
s(868) =< s(856)
s(869) =< s(858)
s(870) =< s(858)
s(871) =< s(858)
s(869) =< s(866)
s(870) =< s(866)
s(871) =< s(866)
s(871) =< s(857)
s(872) =< s(857)
s(871) =< s(865)
s(872) =< s(865)
s(873) =< s(869)*2
s(874) =< s(867)
s(875) =< s(872)
s(876) =< s(873)
s(877) =< s(859)
s(878) =< s(860)
s(879) =< s(861)
s(880) =< s(849)
s(881) =< s(863)
s(882) =< s(864)
s(804) =< aux(94)
s(805) =< aux(94)
s(806) =< aux(94)
s(807) =< aux(96)
s(808) =< aux(96)
s(805) =< aux(96)
s(809) =< aux(97)
s(810) =< aux(98)
s(811) =< aux(94)*(5/2)+16
s(812) =< aux(94)+1
s(813) =< aux(94)
s(814) =< aux(94)*2
s(808) =< aux(95)*(1/2)+aux(96)
s(804) =< aux(95)*(1/2)+aux(96)
s(805) =< aux(95)*(1/2)+aux(96)
s(807) =< aux(95)*(1/2)+aux(96)
s(809) =< aux(95)*(2/3)+aux(97)
s(808) =< aux(95)*(2/3)+aux(97)
s(804) =< aux(95)*(2/3)+aux(97)
s(805) =< aux(95)*(2/3)+aux(97)
s(810) =< aux(95)*(5/7)+aux(98)
s(809) =< aux(95)*(5/7)+aux(98)
s(808) =< aux(95)*(5/7)+aux(98)
s(804) =< aux(95)*(5/7)+aux(98)
s(805) =< aux(95)*(5/7)+aux(98)
s(815) =< s(806)*s(811)
s(816) =< s(806)*s(812)
s(817) =< s(804)*s(812)
s(818) =< s(805)*s(813)
s(819) =< s(808)*s(812)
s(820) =< s(808)*s(813)
s(821) =< s(809)*s(813)
s(822) =< s(810)*aux(94)
s(823) =< s(816)*(5/2)
s(824) =< s(816)*(1/2)
s(825) =< s(816)*2
s(826) =< s(814)
s(827) =< s(816)
s(828) =< s(816)
s(829) =< s(816)
s(827) =< s(824)
s(828) =< s(824)
s(829) =< s(824)
s(829) =< s(815)
s(830) =< s(815)
s(829) =< s(823)
s(830) =< s(823)
s(831) =< s(827)*2
s(832) =< s(825)
s(833) =< s(830)
s(834) =< s(831)
s(835) =< s(817)
s(836) =< s(818)
s(837) =< s(819)
s(838) =< s(807)
s(839) =< s(821)
s(840) =< s(822)

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

* Chain [57]: 8*s(972)+12*s(973)+158*s(974)+16*s(976)+4*s(977)+12*s(978)+8*s(988)+60*s(994)+64*s(996)+32*s(997)+64*s(1000)+64*s(1001)+80*s(1002)+20*s(1003)+20*s(1004)+16*s(1005)+20*s(1006)+16*s(1007)+36*s(1008)+8*s(1014)+12*s(1015)+153*s(1016)+16*s(1018)+4*s(1019)+12*s(1020)+8*s(1030)+60*s(1036)+64*s(1038)+32*s(1039)+64*s(1042)+64*s(1043)+80*s(1044)+20*s(1045)+20*s(1046)+16*s(1047)+20*s(1048)+16*s(1049)+36*s(1050)+5*s(1054)+2*s(1312)+5*s(1315)+3
  Such that:s(1051) =< 1
aux(104) =< V
aux(105) =< V+1
aux(106) =< V/2
aux(107) =< V/3
aux(108) =< 2/7*V
aux(109) =< V2
aux(110) =< V2+1
aux(111) =< V2/2
aux(112) =< V2/3
aux(113) =< 2/7*V2
s(974) =< aux(104)
s(1054) =< s(1051)
s(1014) =< aux(109)
s(1015) =< aux(109)
s(1016) =< aux(109)
s(1017) =< aux(111)
s(1018) =< aux(111)
s(1015) =< aux(111)
s(1019) =< aux(112)
s(1020) =< aux(113)
s(1021) =< aux(109)*(5/2)+16
s(1022) =< aux(109)+1
s(1023) =< aux(109)
s(1024) =< aux(109)*2
s(1018) =< aux(110)*(1/2)+aux(111)
s(1014) =< aux(110)*(1/2)+aux(111)
s(1015) =< aux(110)*(1/2)+aux(111)
s(1017) =< aux(110)*(1/2)+aux(111)
s(1019) =< aux(110)*(2/3)+aux(112)
s(1018) =< aux(110)*(2/3)+aux(112)
s(1014) =< aux(110)*(2/3)+aux(112)
s(1015) =< aux(110)*(2/3)+aux(112)
s(1020) =< aux(110)*(5/7)+aux(113)
s(1019) =< aux(110)*(5/7)+aux(113)
s(1018) =< aux(110)*(5/7)+aux(113)
s(1014) =< aux(110)*(5/7)+aux(113)
s(1015) =< aux(110)*(5/7)+aux(113)
s(1025) =< s(1016)*s(1021)
s(1026) =< s(1016)*s(1022)
s(1027) =< s(1014)*s(1022)
s(1028) =< s(1015)*s(1023)
s(1029) =< s(1018)*s(1022)
s(1030) =< s(1018)*s(1023)
s(1031) =< s(1019)*s(1023)
s(1032) =< s(1020)*aux(109)
s(1033) =< s(1026)*(5/2)
s(1034) =< s(1026)*(1/2)
s(1035) =< s(1026)*2
s(1036) =< s(1024)
s(1037) =< s(1026)
s(1038) =< s(1026)
s(1039) =< s(1026)
s(1037) =< s(1034)
s(1038) =< s(1034)
s(1039) =< s(1034)
s(1039) =< s(1025)
s(1040) =< s(1025)
s(1039) =< s(1033)
s(1040) =< s(1033)
s(1041) =< s(1037)*2
s(1042) =< s(1035)
s(1043) =< s(1040)
s(1044) =< s(1041)
s(1045) =< s(1027)
s(1046) =< s(1028)
s(1047) =< s(1029)
s(1048) =< s(1017)
s(1049) =< s(1031)
s(1050) =< s(1032)
s(972) =< aux(104)
s(973) =< aux(104)
s(975) =< aux(106)
s(976) =< aux(106)
s(973) =< aux(106)
s(977) =< aux(107)
s(978) =< aux(108)
s(979) =< aux(104)*(5/2)+16
s(980) =< aux(104)+1
s(981) =< aux(104)
s(982) =< aux(104)*2
s(976) =< aux(105)*(1/2)+aux(106)
s(972) =< aux(105)*(1/2)+aux(106)
s(973) =< aux(105)*(1/2)+aux(106)
s(975) =< aux(105)*(1/2)+aux(106)
s(977) =< aux(105)*(2/3)+aux(107)
s(976) =< aux(105)*(2/3)+aux(107)
s(972) =< aux(105)*(2/3)+aux(107)
s(973) =< aux(105)*(2/3)+aux(107)
s(978) =< aux(105)*(5/7)+aux(108)
s(977) =< aux(105)*(5/7)+aux(108)
s(976) =< aux(105)*(5/7)+aux(108)
s(972) =< aux(105)*(5/7)+aux(108)
s(973) =< aux(105)*(5/7)+aux(108)
s(983) =< s(974)*s(979)
s(984) =< s(974)*s(980)
s(985) =< s(972)*s(980)
s(986) =< s(973)*s(981)
s(987) =< s(976)*s(980)
s(988) =< s(976)*s(981)
s(989) =< s(977)*s(981)
s(990) =< s(978)*aux(104)
s(991) =< s(984)*(5/2)
s(992) =< s(984)*(1/2)
s(993) =< s(984)*2
s(994) =< s(982)
s(995) =< s(984)
s(996) =< s(984)
s(997) =< s(984)
s(995) =< s(992)
s(996) =< s(992)
s(997) =< s(992)
s(997) =< s(983)
s(998) =< s(983)
s(997) =< s(991)
s(998) =< s(991)
s(999) =< s(995)*2
s(1000) =< s(993)
s(1001) =< s(998)
s(1002) =< s(999)
s(1003) =< s(985)
s(1004) =< s(986)
s(1005) =< s(987)
s(1006) =< s(975)
s(1007) =< s(989)
s(1008) =< s(990)
s(1312) =< aux(105)
s(1315) =< aux(105)
s(1312) =< aux(104)

  with precondition: [V2>=1,Out>=1,V>=Out] 


#### Cost of chains of fun5(V,Out):
* Chain [64]: 2*s(1322)+3*s(1323)+37*s(1324)+4*s(1326)+1*s(1327)+3*s(1328)+2*s(1338)+15*s(1344)+16*s(1346)+8*s(1347)+16*s(1350)+16*s(1351)+20*s(1352)+5*s(1353)+5*s(1354)+4*s(1355)+5*s(1356)+4*s(1357)+9*s(1358)+1
  Such that:s(1318) =< V
s(1317) =< V+1
s(1319) =< V/2
s(1320) =< V/3
s(1321) =< 2/7*V
s(1322) =< s(1318)
s(1323) =< s(1318)
s(1324) =< s(1318)
s(1325) =< s(1319)
s(1326) =< s(1319)
s(1323) =< s(1319)
s(1327) =< s(1320)
s(1328) =< s(1321)
s(1329) =< s(1318)*(5/2)+16
s(1330) =< s(1318)+1
s(1331) =< s(1318)
s(1332) =< s(1318)*2
s(1326) =< s(1317)*(1/2)+s(1319)
s(1322) =< s(1317)*(1/2)+s(1319)
s(1323) =< s(1317)*(1/2)+s(1319)
s(1325) =< s(1317)*(1/2)+s(1319)
s(1327) =< s(1317)*(2/3)+s(1320)
s(1326) =< s(1317)*(2/3)+s(1320)
s(1322) =< s(1317)*(2/3)+s(1320)
s(1323) =< s(1317)*(2/3)+s(1320)
s(1328) =< s(1317)*(5/7)+s(1321)
s(1327) =< s(1317)*(5/7)+s(1321)
s(1326) =< s(1317)*(5/7)+s(1321)
s(1322) =< s(1317)*(5/7)+s(1321)
s(1323) =< s(1317)*(5/7)+s(1321)
s(1333) =< s(1324)*s(1329)
s(1334) =< s(1324)*s(1330)
s(1335) =< s(1322)*s(1330)
s(1336) =< s(1323)*s(1331)
s(1337) =< s(1326)*s(1330)
s(1338) =< s(1326)*s(1331)
s(1339) =< s(1327)*s(1331)
s(1340) =< s(1328)*s(1318)
s(1341) =< s(1334)*(5/2)
s(1342) =< s(1334)*(1/2)
s(1343) =< s(1334)*2
s(1344) =< s(1332)
s(1345) =< s(1334)
s(1346) =< s(1334)
s(1347) =< s(1334)
s(1345) =< s(1342)
s(1346) =< s(1342)
s(1347) =< s(1342)
s(1347) =< s(1333)
s(1348) =< s(1333)
s(1347) =< s(1341)
s(1348) =< s(1341)
s(1349) =< s(1345)*2
s(1350) =< s(1343)
s(1351) =< s(1348)
s(1352) =< s(1349)
s(1353) =< s(1335)
s(1354) =< s(1336)
s(1355) =< s(1337)
s(1356) =< s(1325)
s(1357) =< s(1339)
s(1358) =< s(1340)

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

* Chain [63]: 2*s(1364)+3*s(1365)+37*s(1366)+4*s(1368)+1*s(1369)+3*s(1370)+2*s(1380)+15*s(1386)+16*s(1388)+8*s(1389)+16*s(1392)+16*s(1393)+20*s(1394)+5*s(1395)+5*s(1396)+4*s(1397)+5*s(1398)+4*s(1399)+9*s(1400)+15
  Such that:s(1360) =< V
s(1359) =< V+1
s(1361) =< V/2
s(1362) =< V/3
s(1363) =< 2/7*V
s(1364) =< s(1360)
s(1365) =< s(1360)
s(1366) =< s(1360)
s(1367) =< s(1361)
s(1368) =< s(1361)
s(1365) =< s(1361)
s(1369) =< s(1362)
s(1370) =< s(1363)
s(1371) =< s(1360)*(5/2)+16
s(1372) =< s(1360)+1
s(1373) =< s(1360)
s(1374) =< s(1360)*2
s(1368) =< s(1359)*(1/2)+s(1361)
s(1364) =< s(1359)*(1/2)+s(1361)
s(1365) =< s(1359)*(1/2)+s(1361)
s(1367) =< s(1359)*(1/2)+s(1361)
s(1369) =< s(1359)*(2/3)+s(1362)
s(1368) =< s(1359)*(2/3)+s(1362)
s(1364) =< s(1359)*(2/3)+s(1362)
s(1365) =< s(1359)*(2/3)+s(1362)
s(1370) =< s(1359)*(5/7)+s(1363)
s(1369) =< s(1359)*(5/7)+s(1363)
s(1368) =< s(1359)*(5/7)+s(1363)
s(1364) =< s(1359)*(5/7)+s(1363)
s(1365) =< s(1359)*(5/7)+s(1363)
s(1375) =< s(1366)*s(1371)
s(1376) =< s(1366)*s(1372)
s(1377) =< s(1364)*s(1372)
s(1378) =< s(1365)*s(1373)
s(1379) =< s(1368)*s(1372)
s(1380) =< s(1368)*s(1373)
s(1381) =< s(1369)*s(1373)
s(1382) =< s(1370)*s(1360)
s(1383) =< s(1376)*(5/2)
s(1384) =< s(1376)*(1/2)
s(1385) =< s(1376)*2
s(1386) =< s(1374)
s(1387) =< s(1376)
s(1388) =< s(1376)
s(1389) =< s(1376)
s(1387) =< s(1384)
s(1388) =< s(1384)
s(1389) =< s(1384)
s(1389) =< s(1375)
s(1390) =< s(1375)
s(1389) =< s(1383)
s(1390) =< s(1383)
s(1391) =< s(1387)*2
s(1392) =< s(1385)
s(1393) =< s(1390)
s(1394) =< s(1391)
s(1395) =< s(1377)
s(1396) =< s(1378)
s(1397) =< s(1379)
s(1398) =< s(1367)
s(1399) =< s(1381)
s(1400) =< s(1382)

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

* Chain [62]: 2*s(1406)+3*s(1407)+37*s(1408)+4*s(1410)+1*s(1411)+3*s(1412)+2*s(1422)+15*s(1428)+16*s(1430)+8*s(1431)+16*s(1434)+16*s(1435)+20*s(1436)+5*s(1437)+5*s(1438)+4*s(1439)+5*s(1440)+4*s(1441)+9*s(1442)+5*s(1444)+7
  Such that:s(1443) =< 2
s(1402) =< V
s(1401) =< V+1
s(1403) =< V/2
s(1404) =< V/3
s(1405) =< 2/7*V
s(1444) =< s(1443)
s(1406) =< s(1402)
s(1407) =< s(1402)
s(1408) =< s(1402)
s(1409) =< s(1403)
s(1410) =< s(1403)
s(1407) =< s(1403)
s(1411) =< s(1404)
s(1412) =< s(1405)
s(1413) =< s(1402)*(5/2)+16
s(1414) =< s(1402)+1
s(1415) =< s(1402)
s(1416) =< s(1402)*2
s(1410) =< s(1401)*(1/2)+s(1403)
s(1406) =< s(1401)*(1/2)+s(1403)
s(1407) =< s(1401)*(1/2)+s(1403)
s(1409) =< s(1401)*(1/2)+s(1403)
s(1411) =< s(1401)*(2/3)+s(1404)
s(1410) =< s(1401)*(2/3)+s(1404)
s(1406) =< s(1401)*(2/3)+s(1404)
s(1407) =< s(1401)*(2/3)+s(1404)
s(1412) =< s(1401)*(5/7)+s(1405)
s(1411) =< s(1401)*(5/7)+s(1405)
s(1410) =< s(1401)*(5/7)+s(1405)
s(1406) =< s(1401)*(5/7)+s(1405)
s(1407) =< s(1401)*(5/7)+s(1405)
s(1417) =< s(1408)*s(1413)
s(1418) =< s(1408)*s(1414)
s(1419) =< s(1406)*s(1414)
s(1420) =< s(1407)*s(1415)
s(1421) =< s(1410)*s(1414)
s(1422) =< s(1410)*s(1415)
s(1423) =< s(1411)*s(1415)
s(1424) =< s(1412)*s(1402)
s(1425) =< s(1418)*(5/2)
s(1426) =< s(1418)*(1/2)
s(1427) =< s(1418)*2
s(1428) =< s(1416)
s(1429) =< s(1418)
s(1430) =< s(1418)
s(1431) =< s(1418)
s(1429) =< s(1426)
s(1430) =< s(1426)
s(1431) =< s(1426)
s(1431) =< s(1417)
s(1432) =< s(1417)
s(1431) =< s(1425)
s(1432) =< s(1425)
s(1433) =< s(1429)*2
s(1434) =< s(1427)
s(1435) =< s(1432)
s(1436) =< s(1433)
s(1437) =< s(1419)
s(1438) =< s(1420)
s(1439) =< s(1421)
s(1440) =< s(1409)
s(1441) =< s(1423)
s(1442) =< s(1424)

  with precondition: [Out=2,V>=3] 

* Chain [61]: 2*s(1450)+3*s(1451)+37*s(1452)+4*s(1454)+1*s(1455)+3*s(1456)+2*s(1466)+15*s(1472)+16*s(1474)+8*s(1475)+16*s(1478)+16*s(1479)+20*s(1480)+5*s(1481)+5*s(1482)+4*s(1483)+5*s(1484)+4*s(1485)+9*s(1486)+4*s(1493)+2*s(1494)+4*s(1497)+4*s(1498)+5*s(1499)+1
  Such that:s(1445) =< V+1
s(1487) =< 2*V
s(1448) =< V/3
s(1449) =< 2/7*V
s(1488) =< 5/2*V
s(1489) =< 5/2*V+27/2
aux(114) =< V
aux(115) =< V/2
s(1492) =< aux(114)
s(1493) =< aux(114)
s(1494) =< aux(114)
s(1492) =< aux(115)
s(1493) =< aux(115)
s(1494) =< aux(115)
s(1494) =< s(1489)
s(1495) =< s(1489)
s(1494) =< s(1488)
s(1495) =< s(1488)
s(1496) =< s(1492)*2
s(1497) =< s(1487)
s(1498) =< s(1495)
s(1499) =< s(1496)
s(1450) =< aux(114)
s(1451) =< aux(114)
s(1452) =< aux(114)
s(1453) =< aux(115)
s(1454) =< aux(115)
s(1451) =< aux(115)
s(1455) =< s(1448)
s(1456) =< s(1449)
s(1457) =< aux(114)*(5/2)+16
s(1458) =< aux(114)+1
s(1459) =< aux(114)
s(1460) =< aux(114)*2
s(1454) =< s(1445)*(1/2)+aux(115)
s(1450) =< s(1445)*(1/2)+aux(115)
s(1451) =< s(1445)*(1/2)+aux(115)
s(1453) =< s(1445)*(1/2)+aux(115)
s(1455) =< s(1445)*(2/3)+s(1448)
s(1454) =< s(1445)*(2/3)+s(1448)
s(1450) =< s(1445)*(2/3)+s(1448)
s(1451) =< s(1445)*(2/3)+s(1448)
s(1456) =< s(1445)*(5/7)+s(1449)
s(1455) =< s(1445)*(5/7)+s(1449)
s(1454) =< s(1445)*(5/7)+s(1449)
s(1450) =< s(1445)*(5/7)+s(1449)
s(1451) =< s(1445)*(5/7)+s(1449)
s(1461) =< s(1452)*s(1457)
s(1462) =< s(1452)*s(1458)
s(1463) =< s(1450)*s(1458)
s(1464) =< s(1451)*s(1459)
s(1465) =< s(1454)*s(1458)
s(1466) =< s(1454)*s(1459)
s(1467) =< s(1455)*s(1459)
s(1468) =< s(1456)*aux(114)
s(1469) =< s(1462)*(5/2)
s(1470) =< s(1462)*(1/2)
s(1471) =< s(1462)*2
s(1472) =< s(1460)
s(1473) =< s(1462)
s(1474) =< s(1462)
s(1475) =< s(1462)
s(1473) =< s(1470)
s(1474) =< s(1470)
s(1475) =< s(1470)
s(1475) =< s(1461)
s(1476) =< s(1461)
s(1475) =< s(1469)
s(1476) =< s(1469)
s(1477) =< s(1473)*2
s(1478) =< s(1471)
s(1479) =< s(1476)
s(1480) =< s(1477)
s(1481) =< s(1463)
s(1482) =< s(1464)
s(1483) =< s(1465)
s(1484) =< s(1453)
s(1485) =< s(1467)
s(1486) =< s(1468)

  with precondition: [Out>=1,V>=3*Out+1] 

* Chain [60]: 2*s(1505)+3*s(1506)+37*s(1507)+4*s(1509)+1*s(1510)+3*s(1511)+2*s(1521)+15*s(1527)+16*s(1529)+8*s(1530)+16*s(1533)+16*s(1534)+20*s(1535)+5*s(1536)+5*s(1537)+4*s(1538)+5*s(1539)+4*s(1540)+9*s(1541)+5*s(1548)+8*s(1550)+4*s(1551)+8*s(1554)+8*s(1555)+10*s(1556)+5
  Such that:s(1542) =< 2
s(1500) =< V+1
s(1544) =< 2*V
s(1503) =< V/3
s(1504) =< 2/7*V
s(1546) =< 5/2*V
s(1547) =< 5/2*V+27/2
aux(116) =< V
aux(117) =< V/2
s(1548) =< s(1542)
s(1549) =< aux(116)
s(1550) =< aux(116)
s(1551) =< aux(116)
s(1549) =< aux(117)
s(1550) =< aux(117)
s(1551) =< aux(117)
s(1551) =< s(1547)
s(1552) =< s(1547)
s(1551) =< s(1546)
s(1552) =< s(1546)
s(1553) =< s(1549)*2
s(1554) =< s(1544)
s(1555) =< s(1552)
s(1556) =< s(1553)
s(1505) =< aux(116)
s(1506) =< aux(116)
s(1507) =< aux(116)
s(1508) =< aux(117)
s(1509) =< aux(117)
s(1506) =< aux(117)
s(1510) =< s(1503)
s(1511) =< s(1504)
s(1512) =< aux(116)*(5/2)+16
s(1513) =< aux(116)+1
s(1514) =< aux(116)
s(1515) =< aux(116)*2
s(1509) =< s(1500)*(1/2)+aux(117)
s(1505) =< s(1500)*(1/2)+aux(117)
s(1506) =< s(1500)*(1/2)+aux(117)
s(1508) =< s(1500)*(1/2)+aux(117)
s(1510) =< s(1500)*(2/3)+s(1503)
s(1509) =< s(1500)*(2/3)+s(1503)
s(1505) =< s(1500)*(2/3)+s(1503)
s(1506) =< s(1500)*(2/3)+s(1503)
s(1511) =< s(1500)*(5/7)+s(1504)
s(1510) =< s(1500)*(5/7)+s(1504)
s(1509) =< s(1500)*(5/7)+s(1504)
s(1505) =< s(1500)*(5/7)+s(1504)
s(1506) =< s(1500)*(5/7)+s(1504)
s(1516) =< s(1507)*s(1512)
s(1517) =< s(1507)*s(1513)
s(1518) =< s(1505)*s(1513)
s(1519) =< s(1506)*s(1514)
s(1520) =< s(1509)*s(1513)
s(1521) =< s(1509)*s(1514)
s(1522) =< s(1510)*s(1514)
s(1523) =< s(1511)*aux(116)
s(1524) =< s(1517)*(5/2)
s(1525) =< s(1517)*(1/2)
s(1526) =< s(1517)*2
s(1527) =< s(1515)
s(1528) =< s(1517)
s(1529) =< s(1517)
s(1530) =< s(1517)
s(1528) =< s(1525)
s(1529) =< s(1525)
s(1530) =< s(1525)
s(1530) =< s(1516)
s(1531) =< s(1516)
s(1530) =< s(1524)
s(1531) =< s(1524)
s(1532) =< s(1528)*2
s(1533) =< s(1526)
s(1534) =< s(1531)
s(1535) =< s(1532)
s(1536) =< s(1518)
s(1537) =< s(1519)
s(1538) =< s(1520)
s(1539) =< s(1508)
s(1540) =< s(1522)
s(1541) =< s(1523)

  with precondition: [Out>=2,V+2>=3*Out] 

* Chain [59]: 2*s(1562)+3*s(1563)+37*s(1564)+4*s(1566)+1*s(1567)+3*s(1568)+2*s(1578)+15*s(1584)+16*s(1586)+8*s(1587)+16*s(1590)+16*s(1591)+20*s(1592)+5*s(1593)+5*s(1594)+4*s(1595)+5*s(1596)+4*s(1597)+9*s(1598)+5*s(1605)+4*s(1607)+2*s(1608)+4*s(1611)+4*s(1612)+5*s(1613)+7
  Such that:s(1599) =< 2
s(1557) =< V+1
s(1600) =< 2*V
s(1560) =< V/3
s(1561) =< 2/7*V
s(1601) =< 5/2*V
s(1602) =< 5/2*V+27/2
aux(118) =< V
aux(119) =< V/2
s(1605) =< s(1599)
s(1606) =< aux(118)
s(1607) =< aux(118)
s(1608) =< aux(118)
s(1606) =< aux(119)
s(1607) =< aux(119)
s(1608) =< aux(119)
s(1608) =< s(1602)
s(1609) =< s(1602)
s(1608) =< s(1601)
s(1609) =< s(1601)
s(1610) =< s(1606)*2
s(1611) =< s(1600)
s(1612) =< s(1609)
s(1613) =< s(1610)
s(1562) =< aux(118)
s(1563) =< aux(118)
s(1564) =< aux(118)
s(1565) =< aux(119)
s(1566) =< aux(119)
s(1563) =< aux(119)
s(1567) =< s(1560)
s(1568) =< s(1561)
s(1569) =< aux(118)*(5/2)+16
s(1570) =< aux(118)+1
s(1571) =< aux(118)
s(1572) =< aux(118)*2
s(1566) =< s(1557)*(1/2)+aux(119)
s(1562) =< s(1557)*(1/2)+aux(119)
s(1563) =< s(1557)*(1/2)+aux(119)
s(1565) =< s(1557)*(1/2)+aux(119)
s(1567) =< s(1557)*(2/3)+s(1560)
s(1566) =< s(1557)*(2/3)+s(1560)
s(1562) =< s(1557)*(2/3)+s(1560)
s(1563) =< s(1557)*(2/3)+s(1560)
s(1568) =< s(1557)*(5/7)+s(1561)
s(1567) =< s(1557)*(5/7)+s(1561)
s(1566) =< s(1557)*(5/7)+s(1561)
s(1562) =< s(1557)*(5/7)+s(1561)
s(1563) =< s(1557)*(5/7)+s(1561)
s(1573) =< s(1564)*s(1569)
s(1574) =< s(1564)*s(1570)
s(1575) =< s(1562)*s(1570)
s(1576) =< s(1563)*s(1571)
s(1577) =< s(1566)*s(1570)
s(1578) =< s(1566)*s(1571)
s(1579) =< s(1567)*s(1571)
s(1580) =< s(1568)*aux(118)
s(1581) =< s(1574)*(5/2)
s(1582) =< s(1574)*(1/2)
s(1583) =< s(1574)*2
s(1584) =< s(1572)
s(1585) =< s(1574)
s(1586) =< s(1574)
s(1587) =< s(1574)
s(1585) =< s(1582)
s(1586) =< s(1582)
s(1587) =< s(1582)
s(1587) =< s(1573)
s(1588) =< s(1573)
s(1587) =< s(1581)
s(1588) =< s(1581)
s(1589) =< s(1585)*2
s(1590) =< s(1583)
s(1591) =< s(1588)
s(1592) =< s(1589)
s(1593) =< s(1575)
s(1594) =< s(1576)
s(1595) =< s(1577)
s(1596) =< s(1565)
s(1597) =< s(1579)
s(1598) =< s(1580)

  with precondition: [Out>=3,V+7>=4*Out] 


#### Cost of chains of start(V,V2):
* Chain [65]: 441*s(1615)+795*s(1619)+10*s(1620)+4*s(1623)+30*s(1632)+32*s(1639)+16*s(1640)+32*s(1643)+32*s(1644)+40*s(1645)+42*s(1681)+63*s(1682)+84*s(1685)+21*s(1686)+63*s(1687)+42*s(1697)+315*s(1703)+336*s(1705)+168*s(1706)+336*s(1709)+336*s(1710)+420*s(1711)+105*s(1712)+105*s(1713)+84*s(1714)+105*s(1715)+84*s(1716)+189*s(1717)+22*s(1892)+33*s(1893)+44*s(1895)+11*s(1896)+33*s(1897)+22*s(1907)+165*s(1913)+176*s(1915)+88*s(1916)+176*s(1919)+176*s(1920)+220*s(1921)+55*s(1922)+55*s(1923)+44*s(1924)+55*s(1925)+44*s(1926)+99*s(1927)+2*s(2182)+5*s(2183)+15
  Such that:aux(120) =< 1
aux(121) =< 2
aux(122) =< V
aux(123) =< V+1
aux(124) =< V-V2+1
aux(125) =< 2*V
aux(126) =< V/2
aux(127) =< V/3
aux(128) =< 2/7*V
aux(129) =< 5/2*V
aux(130) =< 5/2*V+27/2
aux(131) =< V2
aux(132) =< V2+1
aux(133) =< V2/2
aux(134) =< V2/3
aux(135) =< 2/7*V2
s(1623) =< aux(124)
s(1615) =< aux(131)
s(1619) =< aux(122)
s(1620) =< aux(120)
s(1892) =< aux(131)
s(1893) =< aux(131)
s(1894) =< aux(133)
s(1895) =< aux(133)
s(1893) =< aux(133)
s(1896) =< aux(134)
s(1897) =< aux(135)
s(1898) =< aux(131)*(5/2)+16
s(1899) =< aux(131)+1
s(1900) =< aux(131)
s(1901) =< aux(131)*2
s(1895) =< aux(132)*(1/2)+aux(133)
s(1892) =< aux(132)*(1/2)+aux(133)
s(1893) =< aux(132)*(1/2)+aux(133)
s(1894) =< aux(132)*(1/2)+aux(133)
s(1896) =< aux(132)*(2/3)+aux(134)
s(1895) =< aux(132)*(2/3)+aux(134)
s(1892) =< aux(132)*(2/3)+aux(134)
s(1893) =< aux(132)*(2/3)+aux(134)
s(1897) =< aux(132)*(5/7)+aux(135)
s(1896) =< aux(132)*(5/7)+aux(135)
s(1895) =< aux(132)*(5/7)+aux(135)
s(1892) =< aux(132)*(5/7)+aux(135)
s(1893) =< aux(132)*(5/7)+aux(135)
s(1902) =< s(1615)*s(1898)
s(1903) =< s(1615)*s(1899)
s(1904) =< s(1892)*s(1899)
s(1905) =< s(1893)*s(1900)
s(1906) =< s(1895)*s(1899)
s(1907) =< s(1895)*s(1900)
s(1908) =< s(1896)*s(1900)
s(1909) =< s(1897)*aux(131)
s(1910) =< s(1903)*(5/2)
s(1911) =< s(1903)*(1/2)
s(1912) =< s(1903)*2
s(1913) =< s(1901)
s(1914) =< s(1903)
s(1915) =< s(1903)
s(1916) =< s(1903)
s(1914) =< s(1911)
s(1915) =< s(1911)
s(1916) =< s(1911)
s(1916) =< s(1902)
s(1917) =< s(1902)
s(1916) =< s(1910)
s(1917) =< s(1910)
s(1918) =< s(1914)*2
s(1919) =< s(1912)
s(1920) =< s(1917)
s(1921) =< s(1918)
s(1922) =< s(1904)
s(1923) =< s(1905)
s(1924) =< s(1906)
s(1925) =< s(1894)
s(1926) =< s(1908)
s(1927) =< s(1909)
s(1681) =< aux(122)
s(1682) =< aux(122)
s(1684) =< aux(126)
s(1685) =< aux(126)
s(1682) =< aux(126)
s(1686) =< aux(127)
s(1687) =< aux(128)
s(1688) =< aux(122)*(5/2)+16
s(1689) =< aux(122)+1
s(1690) =< aux(122)
s(1691) =< aux(122)*2
s(1685) =< aux(123)*(1/2)+aux(126)
s(1681) =< aux(123)*(1/2)+aux(126)
s(1682) =< aux(123)*(1/2)+aux(126)
s(1684) =< aux(123)*(1/2)+aux(126)
s(1686) =< aux(123)*(2/3)+aux(127)
s(1685) =< aux(123)*(2/3)+aux(127)
s(1681) =< aux(123)*(2/3)+aux(127)
s(1682) =< aux(123)*(2/3)+aux(127)
s(1687) =< aux(123)*(5/7)+aux(128)
s(1686) =< aux(123)*(5/7)+aux(128)
s(1685) =< aux(123)*(5/7)+aux(128)
s(1681) =< aux(123)*(5/7)+aux(128)
s(1682) =< aux(123)*(5/7)+aux(128)
s(1692) =< s(1619)*s(1688)
s(1693) =< s(1619)*s(1689)
s(1694) =< s(1681)*s(1689)
s(1695) =< s(1682)*s(1690)
s(1696) =< s(1685)*s(1689)
s(1697) =< s(1685)*s(1690)
s(1698) =< s(1686)*s(1690)
s(1699) =< s(1687)*aux(122)
s(1700) =< s(1693)*(5/2)
s(1701) =< s(1693)*(1/2)
s(1702) =< s(1693)*2
s(1703) =< s(1691)
s(1704) =< s(1693)
s(1705) =< s(1693)
s(1706) =< s(1693)
s(1704) =< s(1701)
s(1705) =< s(1701)
s(1706) =< s(1701)
s(1706) =< s(1692)
s(1707) =< s(1692)
s(1706) =< s(1700)
s(1707) =< s(1700)
s(1708) =< s(1704)*2
s(1709) =< s(1702)
s(1710) =< s(1707)
s(1711) =< s(1708)
s(1712) =< s(1694)
s(1713) =< s(1695)
s(1714) =< s(1696)
s(1715) =< s(1684)
s(1716) =< s(1698)
s(1717) =< s(1699)
s(2182) =< aux(123)
s(2183) =< aux(123)
s(2182) =< aux(122)
s(1632) =< aux(121)
s(1638) =< aux(122)
s(1639) =< aux(122)
s(1640) =< aux(122)
s(1638) =< aux(126)
s(1639) =< aux(126)
s(1640) =< aux(126)
s(1640) =< aux(130)
s(1641) =< aux(130)
s(1640) =< aux(129)
s(1641) =< aux(129)
s(1642) =< s(1638)*2
s(1643) =< aux(125)
s(1644) =< s(1641)
s(1645) =< s(1642)
s(1623) =< aux(122)

  with precondition: [] 


Closed-form bounds of start(V,V2): 
-------------------------------------
* Chain [65] with precondition: [] 
    - Upper bound: nat(V)*9155+85+nat(V)*3066*nat(V)+nat(V)*189*nat(2/7*V)+nat(V)*126*nat(V/2)+nat(V)*84*nat(V/3)+nat(V2)*4753+nat(V2)*1606*nat(V2)+nat(V2)*99*nat(2/7*V2)+nat(V2)*66*nat(V2/2)+nat(V2)*44*nat(V2/3)+nat(2*V)*32+nat(2/7*V)*63+nat(2/7*V2)*33+nat(V+1)*7+nat(5/2*V+27/2)*32+nat(V-V2+1)*4+nat(V/2)*273+nat(V/3)*21+nat(V2/2)*143+nat(V2/3)*11 
    - Complexity: n^2 

### Maximum cost of start(V,V2): nat(V)*9155+85+nat(V)*3066*nat(V)+nat(V)*189*nat(2/7*V)+nat(V)*126*nat(V/2)+nat(V)*84*nat(V/3)+nat(V2)*4753+nat(V2)*1606*nat(V2)+nat(V2)*99*nat(2/7*V2)+nat(V2)*66*nat(V2/2)+nat(V2)*44*nat(V2/3)+nat(2*V)*32+nat(2/7*V)*63+nat(2/7*V2)*33+nat(V+1)*7+nat(5/2*V+27/2)*32+nat(V-V2+1)*4+nat(V/2)*273+nat(V/3)*21+nat(V2/2)*143+nat(V2/3)*11 
Asymptotic class: n^2 
* Total analysis performed in 6394 ms.


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

(14)
BOUNDS(1, n^2)

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

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

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   pred(s(x)) -> x
   minus(x, 0') -> x
   minus(x, s(y)) -> pred(minus(x, y))
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0')) -> 0'
   log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0') -> 0'
   encArg(cons_pred(x_1)) -> pred(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_pred(x_1) -> pred(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_log(x_1) -> log(encArg(x_1))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
pred(s(x)) -> x
minus(x, 0') -> x
minus(x, s(y)) -> pred(minus(x, y))
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_pred(x_1)) -> pred(encArg(x_1))
encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
encArg(cons_log(x_1)) -> log(encArg(x_1))
encode_pred(x_1) -> pred(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
encode_log(x_1) -> log(encArg(x_1))

Types:
pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
s :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
0' :: s:0':cons_pred:cons_minus:cons_quot:cons_log
quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encArg :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_s :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_0 :: s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
hole_s:0':cons_pred:cons_minus:cons_quot:cons_log1_3 :: s:0':cons_pred:cons_minus:cons_quot:cons_log
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3 :: Nat -> s:0':cons_pred:cons_minus:cons_quot:cons_log

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, quot, log, encArg

They will be analysed ascendingly in the following order:
minus < quot
minus < encArg
quot < log
quot < encArg
log < encArg

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

(20)
Obligation:
Innermost TRS:
Rules:
pred(s(x)) -> x
minus(x, 0') -> x
minus(x, s(y)) -> pred(minus(x, y))
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_pred(x_1)) -> pred(encArg(x_1))
encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
encArg(cons_log(x_1)) -> log(encArg(x_1))
encode_pred(x_1) -> pred(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
encode_log(x_1) -> log(encArg(x_1))

Types:
pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
s :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
0' :: s:0':cons_pred:cons_minus:cons_quot:cons_log
quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encArg :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_s :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_0 :: s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
hole_s:0':cons_pred:cons_minus:cons_quot:cons_log1_3 :: s:0':cons_pred:cons_minus:cons_quot:cons_log
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3 :: Nat -> s:0':cons_pred:cons_minus:cons_quot:cons_log


Generator Equations:
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(0) <=> 0'
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(x, 1)) <=> s(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(x))


The following defined symbols remain to be analysed:
minus, quot, log, encArg

They will be analysed ascendingly in the following order:
minus < quot
minus < encArg
quot < log
quot < encArg
log < encArg

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(a), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3)

Induction Base:
minus(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(a), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(1, 0)))

Induction Step:
minus(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(a), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(1, +(n4_3, 1)))) ->_R^Omega(1)
pred(minus(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(a), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(1, n4_3)))) ->_IH
pred(*3_3)

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

(22)
Complex Obligation (BEST)

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

(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
pred(s(x)) -> x
minus(x, 0') -> x
minus(x, s(y)) -> pred(minus(x, y))
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_pred(x_1)) -> pred(encArg(x_1))
encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
encArg(cons_log(x_1)) -> log(encArg(x_1))
encode_pred(x_1) -> pred(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
encode_log(x_1) -> log(encArg(x_1))

Types:
pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
s :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
0' :: s:0':cons_pred:cons_minus:cons_quot:cons_log
quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encArg :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_s :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_0 :: s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
hole_s:0':cons_pred:cons_minus:cons_quot:cons_log1_3 :: s:0':cons_pred:cons_minus:cons_quot:cons_log
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3 :: Nat -> s:0':cons_pred:cons_minus:cons_quot:cons_log


Generator Equations:
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(0) <=> 0'
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(x, 1)) <=> s(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(x))


The following defined symbols remain to be analysed:
minus, quot, log, encArg

They will be analysed ascendingly in the following order:
minus < quot
minus < encArg
quot < log
quot < encArg
log < encArg

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
pred(s(x)) -> x
minus(x, 0') -> x
minus(x, s(y)) -> pred(minus(x, y))
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_pred(x_1)) -> pred(encArg(x_1))
encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
encArg(cons_log(x_1)) -> log(encArg(x_1))
encode_pred(x_1) -> pred(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
encode_log(x_1) -> log(encArg(x_1))

Types:
pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
s :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
0' :: s:0':cons_pred:cons_minus:cons_quot:cons_log
quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encArg :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_s :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_0 :: s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
hole_s:0':cons_pred:cons_minus:cons_quot:cons_log1_3 :: s:0':cons_pred:cons_minus:cons_quot:cons_log
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3 :: Nat -> s:0':cons_pred:cons_minus:cons_quot:cons_log


Lemmas:
minus(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(a), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3)


Generator Equations:
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(0) <=> 0'
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(x, 1)) <=> s(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(x))


The following defined symbols remain to be analysed:
quot, log, encArg

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

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quot(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(n4813_3), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(1)) -> gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(n4813_3), rt in Omega(1 + n4813_3)

Induction Base:
quot(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(0), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(1)) ->_R^Omega(1)
0'

Induction Step:
quot(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(n4813_3, 1)), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(1)) ->_R^Omega(1)
s(quot(minus(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(n4813_3), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(0)), s(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(0)))) ->_R^Omega(1)
s(quot(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(n4813_3), s(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(0)))) ->_IH
s(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(c4814_3))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(28)
Obligation:
Innermost TRS:
Rules:
pred(s(x)) -> x
minus(x, 0') -> x
minus(x, s(y)) -> pred(minus(x, y))
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_pred(x_1)) -> pred(encArg(x_1))
encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
encArg(cons_log(x_1)) -> log(encArg(x_1))
encode_pred(x_1) -> pred(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
encode_log(x_1) -> log(encArg(x_1))

Types:
pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
s :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
0' :: s:0':cons_pred:cons_minus:cons_quot:cons_log
quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encArg :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
cons_log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_pred :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_s :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_minus :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_0 :: s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_quot :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
encode_log :: s:0':cons_pred:cons_minus:cons_quot:cons_log -> s:0':cons_pred:cons_minus:cons_quot:cons_log
hole_s:0':cons_pred:cons_minus:cons_quot:cons_log1_3 :: s:0':cons_pred:cons_minus:cons_quot:cons_log
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3 :: Nat -> s:0':cons_pred:cons_minus:cons_quot:cons_log


Lemmas:
minus(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(a), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3)
quot(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(n4813_3), gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(1)) -> gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(n4813_3), rt in Omega(1 + n4813_3)


Generator Equations:
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(0) <=> 0'
gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(x, 1)) <=> s(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(x))


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

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

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(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(n9996_3)) -> gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(n9996_3), rt in Omega(0)

Induction Base:
encArg(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(+(n9996_3, 1))) ->_R^Omega(0)
s(encArg(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(n9996_3))) ->_IH
s(gen_s:0':cons_pred:cons_minus:cons_quot:cons_log2_3(c9997_3))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(30)
BOUNDS(1, INF)