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


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


KILLED
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 36 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 8 ms]
    (8) typed CpxTrs
    (9) OrderProof [LOWER BOUND(ID), 0 ms]
    (10) typed CpxTrs
    (11) RewriteLemmaProof [LOWER BOUND(ID), 464 ms]
    (12) BOUNDS(1, INF)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(18) CpxRelTRS
    (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxWeightedTrs
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxTypedWeightedTrs
    (23) CompletionProof [UPPER BOUND(ID), 0 ms]
    (24) CpxTypedWeightedCompleteTrs
        (25) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) SimplificationProof [BOTH BOUNDS(ID, ID), 4 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (40) CdtProblem
    (41) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 5 ms]
    (42) CdtProblem
    (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem


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

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


The TRS R consists of the following rules:

   a(a(a(a(x1)))) -> a(c(a(c(c(x1)))))
   c(c(c(x1))) -> a(a(a(x1)))

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(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))


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

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


The TRS R consists of the following rules:

   a(a(a(a(x1)))) -> a(c(a(c(c(x1)))))
   c(c(c(x1))) -> a(a(a(x1)))

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

   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(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(1, INF).


The TRS R consists of the following rules:

   a(a(a(a(x1)))) -> a(c(a(c(c(x1)))))
   c(c(c(x1))) -> a(a(a(x1)))

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

   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   a(a(a(a(x1)))) -> a(c(a(c(c(x1)))))
   c(c(c(x1))) -> a(a(a(x1)))

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

   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
a(a(a(a(x1)))) -> a(c(a(c(c(x1)))))
c(c(c(x1))) -> a(a(a(x1)))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_c(x_1)) -> c(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
a :: a:c:encArg:encode_a:encode_c -> a:c:encArg:encode_a:encode_c
c :: a:c:encArg:encode_a:encode_c -> a:c:encArg:encode_a:encode_c
encArg :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c
cons_a :: cons_a:cons_c -> cons_a:cons_c
cons_c :: cons_a:cons_c -> cons_a:cons_c
encode_a :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c
encode_c :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c
hole_a:c:encArg:encode_a:encode_c1_0 :: a:c:encArg:encode_a:encode_c
hole_cons_a:cons_c2_0 :: cons_a:cons_c
gen_cons_a:cons_c3_0 :: Nat -> cons_a:cons_c

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a, c, encArg

They will be analysed ascendingly in the following order:
a = c
a < encArg
c < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
a(a(a(a(x1)))) -> a(c(a(c(c(x1)))))
c(c(c(x1))) -> a(a(a(x1)))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_c(x_1)) -> c(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
a :: a:c:encArg:encode_a:encode_c -> a:c:encArg:encode_a:encode_c
c :: a:c:encArg:encode_a:encode_c -> a:c:encArg:encode_a:encode_c
encArg :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c
cons_a :: cons_a:cons_c -> cons_a:cons_c
cons_c :: cons_a:cons_c -> cons_a:cons_c
encode_a :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c
encode_c :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c
hole_a:c:encArg:encode_a:encode_c1_0 :: a:c:encArg:encode_a:encode_c
hole_cons_a:cons_c2_0 :: cons_a:cons_c
gen_cons_a:cons_c3_0 :: Nat -> cons_a:cons_c


Generator Equations:
gen_cons_a:cons_c3_0(0) <=> hole_cons_a:cons_c2_0
gen_cons_a:cons_c3_0(+(x, 1)) <=> cons_a(gen_cons_a:cons_c3_0(x))


The following defined symbols remain to be analysed:
c, a, encArg

They will be analysed ascendingly in the following order:
a = c
a < encArg
c < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_cons_a:cons_c3_0(+(1, n11_0))) -> *4_0, rt in Omega(0)

Induction Base:
encArg(gen_cons_a:cons_c3_0(+(1, 0)))

Induction Step:
encArg(gen_cons_a:cons_c3_0(+(1, +(n11_0, 1)))) ->_R^Omega(0)
a(encArg(gen_cons_a:cons_c3_0(+(1, n11_0)))) ->_IH
a(*4_0)

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

(12)
BOUNDS(1, INF)

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

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   a(a(a(a(x1)))) -> a(c(a(c(c(x1)))))
   c(c(c(x1))) -> a(a(a(x1)))

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

   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

(15) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

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


The TRS R consists of the following rules:

   a(a(a(a(x1)))) -> a(c(a(c(c(x1)))))
   c(c(c(x1))) -> a(a(a(x1)))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

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

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   c(c_c(c_c(x1))) -> a(a(a(x1)))
   a(c_a(c_a(c_a(x1)))) -> a(c(a(c(c(x1)))))

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

   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   a(x0) -> c_a(x0)
   c(x0) -> c_c(x0)

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

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

(20)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   c(c_c(c_c(x1))) -> a(a(a(x1))) [1]
   a(c_a(c_a(c_a(x1)))) -> a(c(a(c(c(x1))))) [1]
   encArg(cons_a(x_1)) -> a(encArg(x_1)) [0]
   encArg(cons_c(x_1)) -> c(encArg(x_1)) [0]
   encode_a(x_1) -> a(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]
   a(x0) -> c_a(x0) [0]
   c(x0) -> c_c(x0) [0]

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

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

(22)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   c(c_c(c_c(x1))) -> a(a(a(x1))) [1]
   a(c_a(c_a(c_a(x1)))) -> a(c(a(c(c(x1))))) [1]
   encArg(cons_a(x_1)) -> a(encArg(x_1)) [0]
   encArg(cons_c(x_1)) -> c(encArg(x_1)) [0]
   encode_a(x_1) -> a(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]
   a(x0) -> c_a(x0) [0]
   c(x0) -> c_c(x0) [0]

The TRS has the following type information:
c :: c_c:c_a -> c_c:c_a
c_c :: c_c:c_a -> c_c:c_a
a :: c_c:c_a -> c_c:c_a
c_a :: c_c:c_a -> c_c:c_a
encArg :: cons_a:cons_c -> c_c:c_a
cons_a :: cons_a:cons_c -> cons_a:cons_c
cons_c :: cons_a:cons_c -> cons_a:cons_c
encode_a :: cons_a:cons_c -> c_c:c_a
encode_c :: cons_a:cons_c -> c_c:c_a

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

(23) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   encArg_1
   encode_a_1
   encode_c_1
   a_1
   c_1

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_a(v0) -> const [0]
   encode_c(v0) -> const [0]
   a(v0) -> const [0]
   c(v0) -> const [0]

And the following fresh constants:    const, const1

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

(24)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   c(c_c(c_c(x1))) -> a(a(a(x1))) [1]
   a(c_a(c_a(c_a(x1)))) -> a(c(a(c(c(x1))))) [1]
   encArg(cons_a(x_1)) -> a(encArg(x_1)) [0]
   encArg(cons_c(x_1)) -> c(encArg(x_1)) [0]
   encode_a(x_1) -> a(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]
   a(x0) -> c_a(x0) [0]
   c(x0) -> c_c(x0) [0]
   encArg(v0) -> const [0]
   encode_a(v0) -> const [0]
   encode_c(v0) -> const [0]
   a(v0) -> const [0]
   c(v0) -> const [0]

The TRS has the following type information:
c :: c_c:c_a:const -> c_c:c_a:const
c_c :: c_c:c_a:const -> c_c:c_a:const
a :: c_c:c_a:const -> c_c:c_a:const
c_a :: c_c:c_a:const -> c_c:c_a:const
encArg :: cons_a:cons_c -> c_c:c_a:const
cons_a :: cons_a:cons_c -> cons_a:cons_c
cons_c :: cons_a:cons_c -> cons_a:cons_c
encode_a :: cons_a:cons_c -> c_c:c_a:const
encode_c :: cons_a:cons_c -> c_c:c_a:const
const :: c_c:c_a:const
const1 :: cons_a:cons_c

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

(25) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(26)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   c(c_c(c_c(c_a(c_a(c_a(x1')))))) -> a(a(a(c(a(c(c(x1'))))))) [2]
   c(c_c(c_c(x1))) -> a(a(c_a(x1))) [1]
   c(c_c(c_c(x1))) -> a(a(const)) [1]
   a(c_a(c_a(c_a(c_c(c_c(x1'')))))) -> a(c(a(c(a(a(a(x1''))))))) [2]
   a(c_a(c_a(c_a(x1)))) -> a(c(a(c(c_c(x1))))) [1]
   a(c_a(c_a(c_a(x1)))) -> a(c(a(c(const)))) [1]
   encArg(cons_a(cons_a(x_1'))) -> a(a(encArg(x_1'))) [0]
   encArg(cons_a(cons_c(x_1''))) -> a(c(encArg(x_1''))) [0]
   encArg(cons_a(x_1)) -> a(const) [0]
   encArg(cons_c(cons_a(x_11))) -> c(a(encArg(x_11))) [0]
   encArg(cons_c(cons_c(x_12))) -> c(c(encArg(x_12))) [0]
   encArg(cons_c(x_1)) -> c(const) [0]
   encode_a(cons_a(x_13)) -> a(a(encArg(x_13))) [0]
   encode_a(cons_c(x_14)) -> a(c(encArg(x_14))) [0]
   encode_a(x_1) -> a(const) [0]
   encode_c(cons_a(x_15)) -> c(a(encArg(x_15))) [0]
   encode_c(cons_c(x_16)) -> c(c(encArg(x_16))) [0]
   encode_c(x_1) -> c(const) [0]
   a(x0) -> c_a(x0) [0]
   c(x0) -> c_c(x0) [0]
   encArg(v0) -> const [0]
   encode_a(v0) -> const [0]
   encode_c(v0) -> const [0]
   a(v0) -> const [0]
   c(v0) -> const [0]

The TRS has the following type information:
c :: c_c:c_a:const -> c_c:c_a:const
c_c :: c_c:c_a:const -> c_c:c_a:const
a :: c_c:c_a:const -> c_c:c_a:const
c_a :: c_c:c_a:const -> c_c:c_a:const
encArg :: cons_a:cons_c -> c_c:c_a:const
cons_a :: cons_a:cons_c -> cons_a:cons_c
cons_c :: cons_a:cons_c -> cons_a:cons_c
encode_a :: cons_a:cons_c -> c_c:c_a:const
encode_c :: cons_a:cons_c -> c_c:c_a:const
const :: c_c:c_a:const
const1 :: cons_a:cons_c

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

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

   const => 0
   const1 => 0

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   a(z) -{ 2 }-> a(c(a(c(a(a(a(x1''))))))) :|: x1'' >= 0, z = 1 + (1 + (1 + (1 + (1 + x1''))))
   a(z) -{ 1 }-> a(c(a(c(0)))) :|: x1 >= 0, z = 1 + (1 + (1 + x1))
   a(z) -{ 1 }-> a(c(a(c(1 + x1)))) :|: x1 >= 0, z = 1 + (1 + (1 + x1))
   a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   a(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   c(z) -{ 2 }-> a(a(a(c(a(c(c(x1'))))))) :|: x1' >= 0, z = 1 + (1 + (1 + (1 + (1 + x1'))))
   c(z) -{ 1 }-> a(a(0)) :|: z = 1 + (1 + x1), x1 >= 0
   c(z) -{ 1 }-> a(a(1 + x1)) :|: z = 1 + (1 + x1), x1 >= 0
   c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   c(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   encArg(z) -{ 0 }-> c(c(encArg(x_12))) :|: z = 1 + (1 + x_12), x_12 >= 0
   encArg(z) -{ 0 }-> c(a(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11)
   encArg(z) -{ 0 }-> c(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> a(c(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0
   encArg(z) -{ 0 }-> a(a(encArg(x_1'))) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> a(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a(z) -{ 0 }-> a(c(encArg(x_14))) :|: x_14 >= 0, z = 1 + x_14
   encode_a(z) -{ 0 }-> a(a(encArg(x_13))) :|: z = 1 + x_13, x_13 >= 0
   encode_a(z) -{ 0 }-> a(0) :|: x_1 >= 0, z = x_1
   encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_c(z) -{ 0 }-> c(c(encArg(x_16))) :|: z = 1 + x_16, x_16 >= 0
   encode_c(z) -{ 0 }-> c(a(encArg(x_15))) :|: x_15 >= 0, z = 1 + x_15
   encode_c(z) -{ 0 }-> c(0) :|: x_1 >= 0, z = x_1
   encode_c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

(29) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   a(z) -{ 2 }-> a(c(a(c(a(a(a(z - 5))))))) :|: z - 5 >= 0
   a(z) -{ 1 }-> a(c(a(c(0)))) :|: z - 3 >= 0
   a(z) -{ 1 }-> a(c(a(c(1 + (z - 3))))) :|: z - 3 >= 0
   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 0 }-> 1 + z :|: z >= 0
   c(z) -{ 2 }-> a(a(a(c(a(c(c(z - 5))))))) :|: z - 5 >= 0
   c(z) -{ 1 }-> a(a(0)) :|: z - 2 >= 0
   c(z) -{ 1 }-> a(a(1 + (z - 2))) :|: z - 2 >= 0
   c(z) -{ 0 }-> 0 :|: z >= 0
   c(z) -{ 0 }-> 1 + z :|: z >= 0
   encArg(z) -{ 0 }-> c(c(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> c(a(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> c(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(c(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a(z) -{ 0 }-> a(c(encArg(z - 1))) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(0) :|: z >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 0 }-> c(c(encArg(z - 1))) :|: z - 1 >= 0
   encode_c(z) -{ 0 }-> c(a(encArg(z - 1))) :|: z - 1 >= 0
   encode_c(z) -{ 0 }-> c(0) :|: z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0


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

(31) 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_a(v0) -> null_encode_a [0]
   encode_c(v0) -> null_encode_c [0]
   a(v0) -> null_a [0]
   c(v0) -> null_c [0]

And the following fresh constants:    null_encArg, null_encode_a, null_encode_c, null_a, null_c, const

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

(32)
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:

   c(c_c(c_c(x1))) -> a(a(a(x1))) [1]
   a(c_a(c_a(c_a(x1)))) -> a(c(a(c(c(x1))))) [1]
   encArg(cons_a(x_1)) -> a(encArg(x_1)) [0]
   encArg(cons_c(x_1)) -> c(encArg(x_1)) [0]
   encode_a(x_1) -> a(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]
   a(x0) -> c_a(x0) [0]
   c(x0) -> c_c(x0) [0]
   encArg(v0) -> null_encArg [0]
   encode_a(v0) -> null_encode_a [0]
   encode_c(v0) -> null_encode_c [0]
   a(v0) -> null_a [0]
   c(v0) -> null_c [0]

The TRS has the following type information:
c :: c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c -> c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
c_c :: c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c -> c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
a :: c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c -> c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
c_a :: c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c -> c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
encArg :: cons_a:cons_c -> c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
cons_a :: cons_a:cons_c -> cons_a:cons_c
cons_c :: cons_a:cons_c -> cons_a:cons_c
encode_a :: cons_a:cons_c -> c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
encode_c :: cons_a:cons_c -> c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
null_encArg :: c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
null_encode_a :: c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
null_encode_c :: c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
null_a :: c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
null_c :: c_c:c_a:null_encArg:null_encode_a:null_encode_c:null_a:null_c
const :: cons_a:cons_c

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

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

   null_encArg => 0
   null_encode_a => 0
   null_encode_c => 0
   null_a => 0
   null_c => 0
   const => 0

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   a(z) -{ 1 }-> a(c(a(c(c(x1))))) :|: x1 >= 0, z = 1 + (1 + (1 + x1))
   a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   a(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   c(z) -{ 1 }-> a(a(a(x1))) :|: z = 1 + (1 + x1), x1 >= 0
   c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   c(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   encArg(z) -{ 0 }-> c(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> a(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a(z) -{ 0 }-> a(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_c(z) -{ 0 }-> c(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

(35) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c3(A(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c4(C(encArg(z0)), ENCARG(z0))
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
S tuples:
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   a_1, c_1, encArg_1, encode_a_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_C_1, A_1, C_1

Compound Symbols:   c1_2, c2_2, c3_2, c4_2, c5_5, c6_3


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

(37) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_A(z0) -> c7(ENCARG(z0))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
   ENCODE_C(z0) -> c7(ENCARG(z0))
S tuples:
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   a_1, c_1, encArg_1, encode_a_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c1_2, c2_2, c5_5, c6_3, c7_1


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

(39) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_A(z0) -> c7(ENCARG(z0))
   ENCODE_C(z0) -> c7(ENCARG(z0))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
S tuples:
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   a_1, c_1, encArg_1, encode_a_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c1_2, c2_2, c5_5, c6_3, c7_1


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

(41) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_a(z0) -> a(encArg(z0))
   encode_c(z0) -> c(encArg(z0))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
S tuples:
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c1_2, c2_2, c5_5, c6_3, c7_1


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

(43) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_a(cons_c(z0))) -> c1(A(c(encArg(z0))), ENCARG(cons_c(z0)))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0))
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
   ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_a(cons_c(z0))) -> c1(A(c(encArg(z0))), ENCARG(cons_c(z0)))
S tuples:
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c2_2, c5_5, c6_3, c7_1, c1_2


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

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0)))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
   ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_a(cons_c(z0))) -> c1(A(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0)))
S tuples:
   A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   A_1, C_1, ENCODE_A_1, ENCODE_C_1, ENCARG_1

Compound Symbols:   c5_5, c6_3, c7_1, c1_2, c2_2


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A(a(a(a(z0)))) -> c5(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0)) by 
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_A(z0) -> c7(A(encArg(z0)))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
   ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_a(cons_c(z0))) -> c1(A(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCODE_A_1, ENCODE_C_1, ENCARG_1, A_1

Compound Symbols:   c6_3, c7_1, c1_2, c2_2, c5_5


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

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_A(z0) -> c7(A(encArg(z0))) by 
   ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0))))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_C(z0) -> c7(C(encArg(z0)))
   ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_a(cons_c(z0))) -> c1(A(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
   ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0))))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCODE_C_1, ENCARG_1, A_1, ENCODE_A_1

Compound Symbols:   c6_3, c7_1, c1_2, c2_2, c5_5


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_C(z0) -> c7(C(encArg(z0))) by 
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_a(cons_c(z0))) -> c1(A(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
   ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0))))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCARG_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c6_3, c1_2, c2_2, c5_5, c7_1


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) by 
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCARG(cons_a(cons_c(z0))) -> c1(A(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
   ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0))))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCARG_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c6_3, c1_2, c2_2, c5_5, c7_1


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_a(cons_c(z0))) -> c1(A(c(encArg(z0))), ENCARG(cons_c(z0))) by 
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
   ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0))))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCARG_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c6_3, c2_2, c5_5, c7_1, c1_2


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0))) by 
   ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
   ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0))))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCARG_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c6_3, c2_2, c5_5, c7_1, c1_2


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0))) by 
   ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
   ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0))))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, A_1, ENCODE_A_1, ENCODE_C_1, ENCARG_1

Compound Symbols:   c6_3, c5_5, c7_1, c1_2, c2_2


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A(a(a(a(c(z0))))) -> c5(A(c(a(a(a(a(z0)))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0))) by 
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))), C(c(a(z0))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))), C(c(a(a(z0)))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))), C(c(a(a(a(z0))))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
   ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0))))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))), C(c(a(z0))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))), C(c(a(a(z0)))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))), C(c(a(a(a(z0))))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))), C(c(a(z0))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))), C(c(a(a(z0)))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))), C(c(a(a(a(z0))))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, A_1, ENCODE_A_1, ENCODE_C_1, ENCARG_1

Compound Symbols:   c6_3, c5_5, c7_1, c1_2, c2_2


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

(63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
   ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0))))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0))))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, A_1, ENCODE_A_1, ENCODE_C_1, ENCARG_1

Compound Symbols:   c6_3, c5_5, c7_1, c1_2, c2_2, c5_4


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A(a(a(a(c(c(z0)))))) -> c5(A(c(a(c(a(a(a(z0))))))), C(a(c(c(c(c(z0)))))), A(c(c(c(c(z0))))), C(c(c(c(z0)))), C(c(c(z0)))) by 
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0))))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCODE_A_1, ENCODE_C_1, ENCARG_1, A_1

Compound Symbols:   c6_3, c7_1, c1_2, c2_2, c5_5, c5_4


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_A(cons_a(z0)) -> c7(A(a(encArg(z0)))) by 
   ENCODE_A(cons_a(cons_a(z0))) -> c7(A(a(a(encArg(z0)))))
   ENCODE_A(cons_a(cons_c(z0))) -> c7(A(a(c(encArg(z0)))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0))))
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))
   ENCODE_A(cons_a(cons_a(z0))) -> c7(A(a(a(encArg(z0)))))
   ENCODE_A(cons_a(cons_c(z0))) -> c7(A(a(c(encArg(z0)))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCODE_A_1, ENCODE_C_1, ENCARG_1, A_1

Compound Symbols:   c6_3, c7_1, c1_2, c2_2, c5_5, c5_4


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_A(cons_c(z0)) -> c7(A(c(encArg(z0)))) by 
   ENCODE_A(cons_c(cons_a(z0))) -> c7(A(c(a(encArg(z0)))))
   ENCODE_A(cons_c(cons_c(z0))) -> c7(A(c(c(encArg(z0)))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0))))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))
   ENCODE_A(cons_a(cons_a(z0))) -> c7(A(a(a(encArg(z0)))))
   ENCODE_A(cons_a(cons_c(z0))) -> c7(A(a(c(encArg(z0)))))
   ENCODE_A(cons_c(cons_a(z0))) -> c7(A(c(a(encArg(z0)))))
   ENCODE_A(cons_c(cons_c(z0))) -> c7(A(c(c(encArg(z0)))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCODE_C_1, ENCARG_1, A_1, ENCODE_A_1

Compound Symbols:   c6_3, c7_1, c1_2, c2_2, c5_5, c5_4


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_C(cons_a(z0)) -> c7(C(a(encArg(z0)))) by 
   ENCODE_C(cons_a(cons_a(z0))) -> c7(C(a(a(encArg(z0)))))
   ENCODE_C(cons_a(cons_c(z0))) -> c7(C(a(c(encArg(z0)))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0))))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))
   ENCODE_A(cons_a(cons_a(z0))) -> c7(A(a(a(encArg(z0)))))
   ENCODE_A(cons_a(cons_c(z0))) -> c7(A(a(c(encArg(z0)))))
   ENCODE_A(cons_c(cons_a(z0))) -> c7(A(c(a(encArg(z0)))))
   ENCODE_A(cons_c(cons_c(z0))) -> c7(A(c(c(encArg(z0)))))
   ENCODE_C(cons_a(cons_a(z0))) -> c7(C(a(a(encArg(z0)))))
   ENCODE_C(cons_a(cons_c(z0))) -> c7(C(a(c(encArg(z0)))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCODE_C_1, ENCARG_1, A_1, ENCODE_A_1

Compound Symbols:   c6_3, c7_1, c1_2, c2_2, c5_5, c5_4


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_C(cons_c(z0)) -> c7(C(c(encArg(z0)))) by 
   ENCODE_C(cons_c(cons_a(z0))) -> c7(C(c(a(encArg(z0)))))
   ENCODE_C(cons_c(cons_c(z0))) -> c7(C(c(c(encArg(z0)))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(a(a(a(z0)))) -> a(c(a(c(c(z0)))))
   c(c(c(z0))) -> a(a(a(z0)))
Tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_a(cons_a(cons_c(z0)))) -> c1(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_a(cons_c(cons_a(z0)))) -> c1(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_a(cons_c(cons_c(z0)))) -> c1(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0))))
   ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0))))
   ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0))))
   ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0))))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))
   ENCODE_A(cons_a(cons_a(z0))) -> c7(A(a(a(encArg(z0)))))
   ENCODE_A(cons_a(cons_c(z0))) -> c7(A(a(c(encArg(z0)))))
   ENCODE_A(cons_c(cons_a(z0))) -> c7(A(c(a(encArg(z0)))))
   ENCODE_A(cons_c(cons_c(z0))) -> c7(A(c(c(encArg(z0)))))
   ENCODE_C(cons_a(cons_a(z0))) -> c7(C(a(a(encArg(z0)))))
   ENCODE_C(cons_a(cons_c(z0))) -> c7(C(a(c(encArg(z0)))))
   ENCODE_C(cons_c(cons_a(z0))) -> c7(C(c(a(encArg(z0)))))
   ENCODE_C(cons_c(cons_c(z0))) -> c7(C(c(c(encArg(z0)))))
S tuples:
   C(c(c(z0))) -> c6(A(a(a(z0))), A(a(z0)), A(z0))
   A(a(a(a(c(z0))))) -> c5(A(c(a(c(a(c(c(z0))))))), C(a(c(c(c(z0))))), A(c(c(c(z0)))), C(c(c(z0))), C(c(z0)))
   A(a(a(a(c(a(z0)))))) -> c5(A(c(a(a(c(a(c(c(z0)))))))), C(a(c(c(c(a(z0)))))), A(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(a(a(z0))))))) -> c5(A(c(a(a(a(c(a(c(c(z0))))))))), C(a(c(c(c(a(a(z0))))))), A(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(a(a(a(z0)))))))) -> c5(A(c(a(a(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(a(a(a(z0)))))))), A(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(a(z0))))))) -> c5(A(c(a(c(a(c(a(c(c(z0))))))))), C(a(c(c(c(c(a(z0))))))), A(c(c(c(c(a(z0)))))), C(c(c(c(a(z0))))), C(c(c(a(z0)))))
   A(a(a(a(c(c(a(a(z0)))))))) -> c5(A(c(a(c(a(a(c(a(c(c(z0)))))))))), C(a(c(c(c(c(a(a(z0)))))))), A(c(c(c(c(a(a(z0))))))), C(c(c(c(a(a(z0)))))), C(c(c(a(a(z0))))))
   A(a(a(a(c(c(a(a(a(z0))))))))) -> c5(A(c(a(c(a(a(a(c(a(c(c(z0))))))))))), C(a(c(c(c(c(a(a(a(z0))))))))), A(c(c(c(c(a(a(a(z0)))))))), C(c(c(c(a(a(a(z0))))))), C(c(c(a(a(a(z0)))))))
   A(a(a(a(c(c(x0)))))) -> c5(C(a(c(c(c(c(x0)))))), A(c(c(c(c(x0))))), C(c(c(c(x0)))), C(c(c(x0))))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   C_1, ENCARG_1, A_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c6_3, c1_2, c2_2, c5_5, c5_4, c7_1