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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 38 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
    (7) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 11 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 543 ms]
    (18) BOUNDS(1, n^2)
(19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 185 ms]
    (26) proven lower bound
    (27) LowerBoundPropagationProof [FINISHED, 0 ms]
    (28) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   s(log(0)) -> s(0)
   log(s(x)) -> s(log(half(s(x))))

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

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

   encArg(0) -> 0
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(encArg(x_1))


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

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


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   s(log(0)) -> s(0)
   log(s(x)) -> s(log(half(s(x))))

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

   encArg(0) -> 0
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   s(log(0)) -> s(0)
   log(s(x)) -> s(log(half(s(x))))

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

   encArg(0) -> 0
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   half(0) -> 0
   half(c_s(0)) -> 0
   half(c_s(c_s(x))) -> s(half(x))
   log(c_s(x)) -> s(log(half(s(x))))
   s(c_log(0)) -> s(0)

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

   encArg(0) -> 0
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(encArg(x_1))
   s(x0) -> c_s(x0)
   log(x0) -> c_log(x0)

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

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

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

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


The TRS R consists of the following rules:

   half(0) -> 0
   half(c_s(0)) -> 0
   half(c_s(c_s(x))) -> s(half(x))
   log(c_s(x)) -> s(log(half(s(x))))
   s(c_log(0)) -> s(0)

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

   encArg(0) -> 0
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(encArg(x_1))
   s(x0) -> c_s(x0)
   log(x0) -> c_log(x0)

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

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

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


The TRS R consists of the following rules:

   half(0) -> 0 [1]
   half(c_s(0)) -> 0 [1]
   half(c_s(c_s(x))) -> s(half(x)) [1]
   log(c_s(x)) -> s(log(half(s(x)))) [1]
   s(c_log(0)) -> s(0) [1]
   encArg(0) -> 0 [0]
   encArg(cons_half(x_1)) -> half(encArg(x_1)) [0]
   encArg(cons_s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [0]
   encode_half(x_1) -> half(encArg(x_1)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [0]
   s(x0) -> c_s(x0) [0]
   log(x0) -> c_log(x0) [0]

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

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

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

   half(0) -> 0 [1]
   half(c_s(0)) -> 0 [1]
   half(c_s(c_s(x))) -> s(half(x)) [1]
   log(c_s(x)) -> s(log(half(s(x)))) [1]
   s(c_log(0)) -> s(0) [1]
   encArg(0) -> 0 [0]
   encArg(cons_half(x_1)) -> half(encArg(x_1)) [0]
   encArg(cons_s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [0]
   encode_half(x_1) -> half(encArg(x_1)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [0]
   s(x0) -> c_s(x0) [0]
   log(x0) -> c_log(x0) [0]

The TRS has the following type information:
half :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
0 :: 0:c_s:c_log:cons_half:cons_s:cons_log
c_s :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
s :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
log :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
c_log :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
encArg :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
cons_half :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
cons_s :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
cons_log :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
encode_half :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
encode_0 :: 0:c_s:c_log:cons_half:cons_s:cons_log
encode_s :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log
encode_log :: 0:c_s:c_log:cons_half:cons_s:cons_log -> 0:c_s:c_log:cons_half:cons_s:cons_log

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

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

   encArg(v0) -> null_encArg [0]
   encode_half(v0) -> null_encode_half [0]
   encode_0 -> null_encode_0 [0]
   encode_s(v0) -> null_encode_s [0]
   encode_log(v0) -> null_encode_log [0]
   s(v0) -> null_s [0]
   log(v0) -> null_log [0]
   half(v0) -> null_half [0]

And the following fresh constants:    null_encArg, null_encode_half, null_encode_0, null_encode_s, null_encode_log, null_s, null_log, null_half

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

(14)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   half(0) -> 0 [1]
   half(c_s(0)) -> 0 [1]
   half(c_s(c_s(x))) -> s(half(x)) [1]
   log(c_s(x)) -> s(log(half(s(x)))) [1]
   s(c_log(0)) -> s(0) [1]
   encArg(0) -> 0 [0]
   encArg(cons_half(x_1)) -> half(encArg(x_1)) [0]
   encArg(cons_s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [0]
   encode_half(x_1) -> half(encArg(x_1)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [0]
   s(x0) -> c_s(x0) [0]
   log(x0) -> c_log(x0) [0]
   encArg(v0) -> null_encArg [0]
   encode_half(v0) -> null_encode_half [0]
   encode_0 -> null_encode_0 [0]
   encode_s(v0) -> null_encode_s [0]
   encode_log(v0) -> null_encode_log [0]
   s(v0) -> null_s [0]
   log(v0) -> null_log [0]
   half(v0) -> null_half [0]

The TRS has the following type information:
half :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
0 :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
c_s :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
s :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
log :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
c_log :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
encArg :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
cons_half :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
cons_s :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
cons_log :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
encode_half :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
encode_0 :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
encode_s :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
encode_log :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half -> 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
null_encArg :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
null_encode_half :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
null_encode_0 :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
null_encode_s :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
null_encode_log :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
null_s :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
null_log :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half
null_half :: 0:c_s:c_log:cons_half:cons_s:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_s:null_log:null_half

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

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

   0 => 0
   null_encArg => 0
   null_encode_half => 0
   null_encode_0 => 0
   null_encode_s => 0
   null_encode_log => 0
   null_s => 0
   null_log => 0
   null_half => 0

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

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

   encArg(z) -{ 0 }-> s(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> log(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> half(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_0 -{ 0 }-> 0 :|: 
   encode_half(z) -{ 0 }-> half(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_log(z) -{ 0 }-> log(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> s(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   half(z) -{ 1 }-> s(half(x)) :|: x >= 0, z = 1 + (1 + x)
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   log(z) -{ 1 }-> s(log(half(s(x)))) :|: x >= 0, z = 1 + x
   log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   log(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   s(z) -{ 1 }-> s(0) :|: z = 1 + 0
   s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   s(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V),0,[half(V, Out)],[V >= 0]).
eq(start(V),0,[log(V, Out)],[V >= 0]).
eq(start(V),0,[s(V, Out)],[V >= 0]).
eq(start(V),0,[encArg(V, Out)],[V >= 0]).
eq(start(V),0,[fun(V, Out)],[V >= 0]).
eq(start(V),0,[fun1(Out)],[]).
eq(start(V),0,[fun2(V, Out)],[V >= 0]).
eq(start(V),0,[fun3(V, Out)],[V >= 0]).
eq(half(V, Out),1,[],[Out = 0,V = 0]).
eq(half(V, Out),1,[],[Out = 0,V = 1]).
eq(half(V, Out),1,[half(V1, Ret0),s(Ret0, Ret)],[Out = Ret,V1 >= 0,V = 2 + V1]).
eq(log(V, Out),1,[s(V2, Ret000),half(Ret000, Ret00),log(Ret00, Ret01),s(Ret01, Ret1)],[Out = Ret1,V2 >= 0,V = 1 + V2]).
eq(s(V, Out),1,[s(0, Ret2)],[Out = Ret2,V = 1]).
eq(encArg(V, Out),0,[],[Out = 0,V = 0]).
eq(encArg(V, Out),0,[encArg(V3, Ret02),half(Ret02, Ret3)],[Out = Ret3,V = 1 + V3,V3 >= 0]).
eq(encArg(V, Out),0,[encArg(V4, Ret03),s(Ret03, Ret4)],[Out = Ret4,V = 1 + V4,V4 >= 0]).
eq(encArg(V, Out),0,[encArg(V5, Ret04),log(Ret04, Ret5)],[Out = Ret5,V = 1 + V5,V5 >= 0]).
eq(fun(V, Out),0,[encArg(V6, Ret05),half(Ret05, Ret6)],[Out = Ret6,V6 >= 0,V = V6]).
eq(fun1(Out),0,[],[Out = 0]).
eq(fun2(V, Out),0,[encArg(V7, Ret06),s(Ret06, Ret7)],[Out = Ret7,V7 >= 0,V = V7]).
eq(fun3(V, Out),0,[encArg(V8, Ret07),log(Ret07, Ret8)],[Out = Ret8,V8 >= 0,V = V8]).
eq(s(V, Out),0,[],[Out = 1 + V9,V = V9,V9 >= 0]).
eq(log(V, Out),0,[],[Out = 1 + V10,V = V10,V10 >= 0]).
eq(encArg(V, Out),0,[],[Out = 0,V11 >= 0,V = V11]).
eq(fun(V, Out),0,[],[Out = 0,V12 >= 0,V = V12]).
eq(fun2(V, Out),0,[],[Out = 0,V13 >= 0,V = V13]).
eq(fun3(V, Out),0,[],[Out = 0,V14 >= 0,V = V14]).
eq(s(V, Out),0,[],[Out = 0,V15 >= 0,V = V15]).
eq(log(V, Out),0,[],[Out = 0,V16 >= 0,V = V16]).
eq(half(V, Out),0,[],[Out = 0,V17 >= 0,V = V17]).
input_output_vars(half(V,Out),[V],[Out]).
input_output_vars(log(V,Out),[V],[Out]).
input_output_vars(s(V,Out),[V],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(fun1(Out),[],[Out]).
input_output_vars(fun2(V,Out),[V],[Out]).
input_output_vars(fun3(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [s/2]
1. recursive [non_tail] : [half/2]
2. recursive [non_tail] : [log/2]
3. recursive [non_tail] : [encArg/2]
4. non_recursive  : [fun/2]
5. non_recursive  : [fun1/1]
6. non_recursive  : [fun2/2]
7. non_recursive  : [fun3/2]
8. non_recursive  : [start/1]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into s/2
1. SCC is partially evaluated into half/2
2. SCC is partially evaluated into log/2
3. SCC is partially evaluated into encArg/2
4. SCC is partially evaluated into fun/2
5. SCC is completely evaluated into other SCCs
6. SCC is partially evaluated into fun2/2
7. SCC is partially evaluated into fun3/2
8. SCC is partially evaluated into start/1

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

### Specialization of cost equations s/2 
* CE 17 is refined into CE [29] 
* CE 18 is refined into CE [30] 
* CE 16 is refined into CE [31] 


### Cost equations --> "Loop" of s/2 
* CEs [31] --> Loop 16 
* CEs [29] --> Loop 17 
* CEs [30] --> Loop 18 

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

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


### Specialization of cost equations half/2 
* CE 10 is refined into CE [32] 
* CE 9 is refined into CE [33] 
* CE 12 is refined into CE [34] 
* CE 11 is refined into CE [35,36,37] 


### Cost equations --> "Loop" of half/2 
* CEs [37] --> Loop 19 
* CEs [36] --> Loop 20 
* CEs [35] --> Loop 21 
* CEs [32] --> Loop 22 
* CEs [33,34] --> Loop 23 

### Ranking functions of CR half(V,Out) 
* RF of phase [19,20,21]: [V-1]

#### Partial ranking functions of CR half(V,Out) 
* Partial RF of phase [19,20,21]:
  - RF of loop [19:1,20:1,21:1]:
    V-1


### Specialization of cost equations log/2 
* CE 14 is refined into CE [38] 
* CE 15 is refined into CE [39] 
* CE 13 is refined into CE [40,41,42,43,44,45,46,47,48,49,50,51] 


### Cost equations --> "Loop" of log/2 
* CEs [51] --> Loop 24 
* CEs [50] --> Loop 25 
* CEs [49] --> Loop 26 
* CEs [42,45,48] --> Loop 27 
* CEs [41,44,47] --> Loop 28 
* CEs [40,43,46] --> Loop 29 
* CEs [38] --> Loop 30 
* CEs [39] --> Loop 31 

### Ranking functions of CR log(V,Out) 
* RF of phase [24,25,26]: [2*V-3]

#### Partial ranking functions of CR log(V,Out) 
* Partial RF of phase [24,25,26]:
  - RF of loop [24:1,25:1,26:1]:
    2*V-3


### Specialization of cost equations encArg/2 
* CE 19 is refined into CE [52] 
* CE 20 is refined into CE [53,54] 
* CE 21 is refined into CE [55,56,57] 
* CE 22 is refined into CE [58,59,60,61,62] 


### Cost equations --> "Loop" of encArg/2 
* CEs [54,62] --> Loop 32 
* CEs [57,61] --> Loop 33 
* CEs [60] --> Loop 34 
* CEs [56,59] --> Loop 35 
* CEs [53,55,58] --> Loop 36 
* CEs [52] --> Loop 37 

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

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


### Specialization of cost equations fun/2 
* CE 23 is refined into CE [63,64,65] 
* CE 24 is refined into CE [66] 


### Cost equations --> "Loop" of fun/2 
* CEs [65] --> Loop 38 
* CEs [63,64,66] --> Loop 39 

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

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


### Specialization of cost equations fun2/2 
* CE 25 is refined into CE [67,68,69,70,71] 
* CE 26 is refined into CE [72] 


### Cost equations --> "Loop" of fun2/2 
* CEs [71] --> Loop 40 
* CEs [68,70] --> Loop 41 
* CEs [67,69,72] --> Loop 42 

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

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


### Specialization of cost equations fun3/2 
* CE 27 is refined into CE [73,74,75,76,77,78,79] 
* CE 28 is refined into CE [80] 


### Cost equations --> "Loop" of fun3/2 
* CEs [79] --> Loop 43 
* CEs [77,78] --> Loop 44 
* CEs [74,76] --> Loop 45 
* CEs [73,75,80] --> Loop 46 

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

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


### Specialization of cost equations start/1 
* CE 1 is refined into CE [81,82] 
* CE 2 is refined into CE [83,84,85,86,87] 
* CE 3 is refined into CE [88,89,90] 
* CE 4 is refined into CE [91,92] 
* CE 5 is refined into CE [93,94] 
* CE 6 is refined into CE [95] 
* CE 7 is refined into CE [96,97,98] 
* CE 8 is refined into CE [99,100,101,102] 


### Cost equations --> "Loop" of start/1 
* CEs [81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102] --> Loop 47 

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of s(V,Out):
* Chain [18]: 0
  with precondition: [Out=0,V>=0] 

* Chain [17]: 0
  with precondition: [V+1=Out,V>=0] 

* Chain [16,18]: 1
  with precondition: [V=1,Out=0] 

* Chain [16,17]: 1
  with precondition: [V=1,Out=1] 


#### Cost of chains of half(V,Out):
* Chain [[19,20,21],23]: 5*it(19)+1
  Such that:aux(3) =< V
it(19) =< aux(3)

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

* Chain [[19,20,21],22]: 5*it(19)+1
  Such that:aux(4) =< V
it(19) =< aux(4)

  with precondition: [V>=3,Out>=0,V>=2*Out+1] 

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

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


#### Cost of chains of log(V,Out):
* Chain [[24,25,26],31]: 38*it(24)+0
  Such that:aux(8) =< 2*V
it(24) =< aux(8)

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

* Chain [[24,25,26],30]: 38*it(24)+0
  Such that:aux(9) =< 2*V
it(24) =< aux(9)

  with precondition: [V>=2,Out>=0,V+4>=2*Out] 

* Chain [[24,25,26],29,31]: 38*it(24)+4
  Such that:aux(10) =< 2*V
it(24) =< aux(10)

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

* Chain [[24,25,26],29,30]: 38*it(24)+4
  Such that:aux(11) =< 2*V
it(24) =< aux(11)

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

* Chain [[24,25,26],28,30]: 38*it(24)+4
  Such that:aux(12) =< 2*V
it(24) =< aux(12)

  with precondition: [V>=2,Out>=0,V+2>=2*Out] 

* Chain [[24,25,26],27,31]: 38*it(24)+3
  Such that:aux(13) =< 2*V
it(24) =< aux(13)

  with precondition: [V>=2,Out>=0,V+2>=2*Out] 

* Chain [[24,25,26],27,30]: 38*it(24)+3
  Such that:aux(14) =< 2*V
it(24) =< aux(14)

  with precondition: [V>=2,Out>=0,V+4>=2*Out] 

* Chain [31]: 0
  with precondition: [Out=0,V>=0] 

* Chain [30]: 0
  with precondition: [V+1=Out,V>=0] 

* Chain [29,31]: 4
  with precondition: [Out=0,V>=1] 

* Chain [29,30]: 4
  with precondition: [Out=0,V>=1] 

* Chain [28,30]: 4
  with precondition: [Out=1,V>=1] 

* Chain [27,31]: 3
  with precondition: [Out=1,V>=1] 

* Chain [27,30]: 3
  with precondition: [Out=2,V>=1] 


#### Cost of chains of encArg(V,Out):
* Chain [[32,33,34,35,36],37]: 15*it(32)+10*s(45)+266*s(46)+0
  Such that:aux(20) =< V
it(32) =< aux(20)
aux(17) =< it(32)*aux(20)
s(47) =< aux(17)*2
s(45) =< aux(17)
s(46) =< s(47)

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

* Chain [37]: 0
  with precondition: [Out=0,V>=0] 


#### Cost of chains of fun(V,Out):
* Chain [39]: 15*s(50)+10*s(53)+266*s(54)+1
  Such that:s(49) =< V
s(50) =< s(49)
s(51) =< s(50)*s(49)
s(52) =< s(51)*2
s(53) =< s(51)
s(54) =< s(52)

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

* Chain [38]: 25*s(56)+10*s(59)+266*s(60)+1
  Such that:aux(21) =< V
s(56) =< aux(21)
s(57) =< s(56)*aux(21)
s(58) =< s(57)*2
s(59) =< s(57)
s(60) =< s(58)

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


#### Cost of chains of fun2(V,Out):
* Chain [42]: 15*s(64)+10*s(67)+266*s(68)+1
  Such that:s(63) =< V
s(64) =< s(63)
s(65) =< s(64)*s(63)
s(66) =< s(65)*2
s(67) =< s(65)
s(68) =< s(66)

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

* Chain [41]: 15*s(70)+10*s(73)+266*s(74)+1
  Such that:s(69) =< V
s(70) =< s(69)
s(71) =< s(70)*s(69)
s(72) =< s(71)*2
s(73) =< s(71)
s(74) =< s(72)

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

* Chain [40]: 15*s(76)+10*s(79)+266*s(80)+0
  Such that:s(75) =< V
s(76) =< s(75)
s(77) =< s(76)*s(75)
s(78) =< s(77)*2
s(79) =< s(77)
s(80) =< s(78)

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


#### Cost of chains of fun3(V,Out):
* Chain [46]: 15*s(82)+10*s(85)+266*s(86)+4
  Such that:s(81) =< V
s(82) =< s(81)
s(83) =< s(82)*s(81)
s(84) =< s(83)*2
s(85) =< s(83)
s(86) =< s(84)

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

* Chain [45]: 15*s(88)+10*s(91)+266*s(92)+4
  Such that:s(87) =< V
s(88) =< s(87)
s(89) =< s(88)*s(87)
s(90) =< s(89)*2
s(91) =< s(89)
s(92) =< s(90)

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

* Chain [44]: 30*s(94)+20*s(97)+532*s(98)+3
  Such that:aux(22) =< V
s(94) =< aux(22)
s(95) =< s(94)*aux(22)
s(96) =< s(95)*2
s(97) =< s(95)
s(98) =< s(96)

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

* Chain [43]: 15*s(106)+10*s(109)+266*s(110)+266*s(112)+4
  Such that:s(105) =< V
s(111) =< 2*V
s(112) =< s(111)
s(106) =< s(105)
s(107) =< s(106)*s(105)
s(108) =< s(107)*2
s(109) =< s(107)
s(110) =< s(108)

  with precondition: [V>=2,Out>=0,V+4>=2*Out] 


#### Cost of chains of start(V):
* Chain [47]: 185*s(114)+532*s(116)+110*s(121)+2926*s(122)+4
  Such that:aux(23) =< V
aux(24) =< 2*V
s(114) =< aux(23)
s(119) =< s(114)*aux(23)
s(120) =< s(119)*2
s(121) =< s(119)
s(122) =< s(120)
s(116) =< aux(24)

  with precondition: [] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [47] with precondition: [] 
    - Upper bound: nat(V)*185+4+nat(V)*5962*nat(V)+nat(2*V)*532 
    - Complexity: n^2 

### Maximum cost of start(V): nat(V)*185+4+nat(V)*5962*nat(V)+nat(2*V)*532 
Asymptotic class: n^2 
* Total analysis performed in 454 ms.


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

(18)
BOUNDS(1, n^2)

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

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

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


The TRS R consists of the following rules:

   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   s(log(0')) -> s(0')
   log(s(x)) -> s(log(half(s(x))))

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

   encArg(0') -> 0'
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(encArg(x_1))

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

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

(22)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
s(log(0')) -> s(0')
log(s(x)) -> s(log(half(s(x))))
encArg(0') -> 0'
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_log(x_1)) -> log(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_log(x_1) -> log(encArg(x_1))

Types:
half :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
0' :: 0':cons_half:cons_s:cons_log
s :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
log :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encArg :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
cons_half :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
cons_s :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
cons_log :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encode_half :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encode_0 :: 0':cons_half:cons_s:cons_log
encode_s :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encode_log :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
hole_0':cons_half:cons_s:cons_log1_2 :: 0':cons_half:cons_s:cons_log
gen_0':cons_half:cons_s:cons_log2_2 :: Nat -> 0':cons_half:cons_s:cons_log

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

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

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

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

(24)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
s(log(0')) -> s(0')
log(s(x)) -> s(log(half(s(x))))
encArg(0') -> 0'
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_log(x_1)) -> log(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_log(x_1) -> log(encArg(x_1))

Types:
half :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
0' :: 0':cons_half:cons_s:cons_log
s :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
log :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encArg :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
cons_half :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
cons_s :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
cons_log :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encode_half :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encode_0 :: 0':cons_half:cons_s:cons_log
encode_s :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encode_log :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
hole_0':cons_half:cons_s:cons_log1_2 :: 0':cons_half:cons_s:cons_log
gen_0':cons_half:cons_s:cons_log2_2 :: Nat -> 0':cons_half:cons_s:cons_log


Generator Equations:
gen_0':cons_half:cons_s:cons_log2_2(0) <=> 0'
gen_0':cons_half:cons_s:cons_log2_2(+(x, 1)) <=> cons_half(gen_0':cons_half:cons_s:cons_log2_2(x))


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

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

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':cons_half:cons_s:cons_log2_2(n21_2)) -> gen_0':cons_half:cons_s:cons_log2_2(0), rt in Omega(n21_2)

Induction Base:
encArg(gen_0':cons_half:cons_s:cons_log2_2(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':cons_half:cons_s:cons_log2_2(+(n21_2, 1))) ->_R^Omega(0)
half(encArg(gen_0':cons_half:cons_s:cons_log2_2(n21_2))) ->_IH
half(gen_0':cons_half:cons_s:cons_log2_2(0)) ->_R^Omega(1)
0'

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

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

TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
s(log(0')) -> s(0')
log(s(x)) -> s(log(half(s(x))))
encArg(0') -> 0'
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_log(x_1)) -> log(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_log(x_1) -> log(encArg(x_1))

Types:
half :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
0' :: 0':cons_half:cons_s:cons_log
s :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
log :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encArg :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
cons_half :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
cons_s :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
cons_log :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encode_half :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encode_0 :: 0':cons_half:cons_s:cons_log
encode_s :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
encode_log :: 0':cons_half:cons_s:cons_log -> 0':cons_half:cons_s:cons_log
hole_0':cons_half:cons_s:cons_log1_2 :: 0':cons_half:cons_s:cons_log
gen_0':cons_half:cons_s:cons_log2_2 :: Nat -> 0':cons_half:cons_s:cons_log


Generator Equations:
gen_0':cons_half:cons_s:cons_log2_2(0) <=> 0'
gen_0':cons_half:cons_s:cons_log2_2(+(x, 1)) <=> cons_half(gen_0':cons_half:cons_s:cons_log2_2(x))


The following defined symbols remain to be analysed:
encArg
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(27) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(28)
BOUNDS(n^1, INF)