KILLED
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 52 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 472 ms]
    (14) BOUNDS(1, INF)
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(18) CpxRelTRS
    (19) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRelTRS
    (21) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxWeightedTrs
        (23) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (24) CpxTypedWeightedTrs
        (25) CompletionProof [UPPER BOUND(ID), 0 ms]
        (26) CpxTypedWeightedCompleteTrs
            (27) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (28) CpxTypedWeightedCompleteTrs
            (29) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (30) CpxRNTS
            (31) InliningProof [UPPER BOUND(ID), 191 ms]
            (32) CpxRNTS
            (33) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
            (34) CpxRNTS
        (35) CompletionProof [UPPER BOUND(ID), 0 ms]
        (36) CpxTypedWeightedCompleteTrs
            (37) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (38) CpxRNTS
    (39) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (40) CdtProblem
        (41) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (42) CdtProblem
        (43) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (44) CdtProblem
        (45) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (46) CdtProblem
        (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (48) CdtProblem
        (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (50) CdtProblem
        (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (52) CdtProblem
        (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (54) CdtProblem
        (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (56) CdtProblem
        (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
        (58) CdtProblem
        (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (60) CdtProblem
        (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CdtProblem
        (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CdtProblem
        (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CdtProblem
        (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CdtProblem
        (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (70) CdtProblem
        (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
        (72) CdtProblem
        (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (74) CdtProblem
        (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (76) CdtProblem
        (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (78) CdtProblem
        (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (80) CdtProblem
        (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (82) CdtProblem
        (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (84) CdtProblem
        (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (86) CdtProblem
        (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (88) CdtProblem
        (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (90) CdtProblem
        (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (92) CdtProblem
        (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (94) CdtProblem
        (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 11 ms]
        (96) CdtProblem
        (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 12 ms]
        (98) CdtProblem
        (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (100) CdtProblem
        (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 26 ms]
        (102) CdtProblem
        (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (104) CdtProblem
        (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (106) CdtProblem
        (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 14 ms]
        (108) CdtProblem
        (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 50 ms]
        (110) CdtProblem
        (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (112) CdtProblem


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

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


The TRS R consists of the following rules:

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

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(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))


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

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


The TRS R consists of the following rules:

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

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(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(1, INF).


The TRS R consists of the following rules:

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

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

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

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

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

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

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

(8)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

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

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

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

(10)
Obligation:
TRS:
Rules:
a(x1) -> x1
a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
b(c(x1)) -> x1
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
a :: c:cons_a:cons_b -> c:cons_a:cons_b
b :: c:cons_a:cons_b -> c:cons_a:cons_b
c :: c:cons_a:cons_b -> c:cons_a:cons_b
encArg :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_a :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_a :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_c :: c:cons_a:cons_b -> c:cons_a:cons_b
hole_c:cons_a:cons_b1_0 :: c:cons_a:cons_b
gen_c:cons_a:cons_b2_0 :: Nat -> c:cons_a:cons_b

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

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

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

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

(12)
Obligation:
TRS:
Rules:
a(x1) -> x1
a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
b(c(x1)) -> x1
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
a :: c:cons_a:cons_b -> c:cons_a:cons_b
b :: c:cons_a:cons_b -> c:cons_a:cons_b
c :: c:cons_a:cons_b -> c:cons_a:cons_b
encArg :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_a :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_a :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_c :: c:cons_a:cons_b -> c:cons_a:cons_b
hole_c:cons_a:cons_b1_0 :: c:cons_a:cons_b
gen_c:cons_a:cons_b2_0 :: Nat -> c:cons_a:cons_b


Generator Equations:
gen_c:cons_a:cons_b2_0(0) <=> hole_c:cons_a:cons_b1_0
gen_c:cons_a:cons_b2_0(+(x, 1)) <=> c(gen_c:cons_a:cons_b2_0(x))


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_c:cons_a:cons_b2_0(+(1, n10_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_c:cons_a:cons_b2_0(+(1, 0)))

Induction Step:
encArg(gen_c:cons_a:cons_b2_0(+(1, +(n10_0, 1)))) ->_R^Omega(0)
c(encArg(gen_c:cons_a:cons_b2_0(+(1, n10_0)))) ->_IH
c(*3_0)

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

(14)
BOUNDS(1, INF)

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

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

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


The TRS R consists of the following rules:

   a(x1) -> x1
   a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
   b(c(x1)) -> x1
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   a(x1) -> x1
   b(c(x1)) -> x1
   a(c_b(c_b(x1))) -> b(b(a(b(c(a(x1))))))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   b(x0) -> c_b(x0)

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

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

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

(20)
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(x1) -> x1
   b(c(x1)) -> x1
   a(c_b(c_b(x1))) -> b(b(a(b(c(a(x1))))))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   b(x0) -> c_b(x0)

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

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

(22)
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:

   a(x1) -> x1 [1]
   b(c(x1)) -> x1 [1]
   a(c_b(c_b(x1))) -> b(b(a(b(c(a(x1)))))) [1]
   encArg(c(x_1)) -> c(encArg(x_1)) [0]
   encArg(cons_a(x_1)) -> a(encArg(x_1)) [0]
   encArg(cons_b(x_1)) -> b(encArg(x_1)) [0]
   encode_a(x_1) -> a(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]
   b(x0) -> c_b(x0) [0]

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

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

(24)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   a(x1) -> x1 [1]
   b(c(x1)) -> x1 [1]
   a(c_b(c_b(x1))) -> b(b(a(b(c(a(x1)))))) [1]
   encArg(c(x_1)) -> c(encArg(x_1)) [0]
   encArg(cons_a(x_1)) -> a(encArg(x_1)) [0]
   encArg(cons_b(x_1)) -> b(encArg(x_1)) [0]
   encode_a(x_1) -> a(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]
   b(x0) -> c_b(x0) [0]

The TRS has the following type information:
a :: c:c_b:cons_a:cons_b -> c:c_b:cons_a:cons_b
b :: c:c_b:cons_a:cons_b -> c:c_b:cons_a:cons_b
c :: c:c_b:cons_a:cons_b -> c:c_b:cons_a:cons_b
c_b :: c:c_b:cons_a:cons_b -> c:c_b:cons_a:cons_b
encArg :: c:c_b:cons_a:cons_b -> c:c_b:cons_a:cons_b
cons_a :: c:c_b:cons_a:cons_b -> c:c_b:cons_a:cons_b
cons_b :: c:c_b:cons_a:cons_b -> c:c_b:cons_a:cons_b
encode_a :: c:c_b:cons_a:cons_b -> c:c_b:cons_a:cons_b
encode_b :: c:c_b:cons_a:cons_b -> c:c_b:cons_a:cons_b
encode_c :: c:c_b:cons_a:cons_b -> c:c_b:cons_a:cons_b

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

(25) 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:

   a_1
   encArg_1
   encode_a_1
   encode_b_1
   encode_c_1
   b_1

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_a(v0) -> const [0]
   encode_b(v0) -> const [0]
   encode_c(v0) -> const [0]
   b(v0) -> const [0]

And the following fresh constants:    const

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

(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:

   a(x1) -> x1 [1]
   b(c(x1)) -> x1 [1]
   a(c_b(c_b(x1))) -> b(b(a(b(c(a(x1)))))) [1]
   encArg(c(x_1)) -> c(encArg(x_1)) [0]
   encArg(cons_a(x_1)) -> a(encArg(x_1)) [0]
   encArg(cons_b(x_1)) -> b(encArg(x_1)) [0]
   encode_a(x_1) -> a(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]
   b(x0) -> c_b(x0) [0]
   encArg(v0) -> const [0]
   encode_a(v0) -> const [0]
   encode_b(v0) -> const [0]
   encode_c(v0) -> const [0]
   b(v0) -> const [0]

The TRS has the following type information:
a :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
b :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
c :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
c_b :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
encArg :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
cons_a :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
cons_b :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
encode_a :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
encode_b :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
encode_c :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
const :: c:c_b:cons_a:cons_b:const

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

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

(28)
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:

   a(x1) -> x1 [1]
   b(c(x1)) -> x1 [1]
   a(c_b(c_b(x1))) -> b(b(a(b(c(x1))))) [2]
   a(c_b(c_b(c_b(c_b(x1'))))) -> b(b(a(b(c(b(b(a(b(c(a(x1'))))))))))) [2]
   encArg(c(x_1)) -> c(encArg(x_1)) [0]
   encArg(cons_a(c(x_1'))) -> a(c(encArg(x_1'))) [0]
   encArg(cons_a(cons_a(x_1''))) -> a(a(encArg(x_1''))) [0]
   encArg(cons_a(cons_b(x_11))) -> a(b(encArg(x_11))) [0]
   encArg(cons_a(x_1)) -> a(const) [0]
   encArg(cons_b(c(x_12))) -> b(c(encArg(x_12))) [0]
   encArg(cons_b(cons_a(x_13))) -> b(a(encArg(x_13))) [0]
   encArg(cons_b(cons_b(x_14))) -> b(b(encArg(x_14))) [0]
   encArg(cons_b(x_1)) -> b(const) [0]
   encode_a(c(x_15)) -> a(c(encArg(x_15))) [0]
   encode_a(cons_a(x_16)) -> a(a(encArg(x_16))) [0]
   encode_a(cons_b(x_17)) -> a(b(encArg(x_17))) [0]
   encode_a(x_1) -> a(const) [0]
   encode_b(c(x_18)) -> b(c(encArg(x_18))) [0]
   encode_b(cons_a(x_19)) -> b(a(encArg(x_19))) [0]
   encode_b(cons_b(x_110)) -> b(b(encArg(x_110))) [0]
   encode_b(x_1) -> b(const) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]
   b(x0) -> c_b(x0) [0]
   encArg(v0) -> const [0]
   encode_a(v0) -> const [0]
   encode_b(v0) -> const [0]
   encode_c(v0) -> const [0]
   b(v0) -> const [0]

The TRS has the following type information:
a :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
b :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
c :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
c_b :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
encArg :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
cons_a :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
cons_b :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
encode_a :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
encode_b :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
encode_c :: c:c_b:cons_a:cons_b:const -> c:c_b:cons_a:cons_b:const
const :: c:c_b:cons_a:cons_b:const

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

(29) 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

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

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

   a(z) -{ 1 }-> x1 :|: x1 >= 0, z = x1
   a(z) -{ 2 }-> b(b(a(b(1 + x1)))) :|: z = 1 + (1 + x1), x1 >= 0
   a(z) -{ 2 }-> b(b(a(b(1 + b(b(a(b(1 + a(x1'))))))))) :|: x1' >= 0, z = 1 + (1 + (1 + (1 + x1')))
   b(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(x_14))) :|: x_14 >= 0, z = 1 + (1 + x_14)
   encArg(z) -{ 0 }-> b(a(encArg(x_13))) :|: z = 1 + (1 + x_13), x_13 >= 0
   encArg(z) -{ 0 }-> b(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(x_12)) :|: z = 1 + (1 + x_12), x_12 >= 0
   encArg(z) -{ 0 }-> a(b(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11)
   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 }-> a(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_a(z) -{ 0 }-> a(b(encArg(x_17))) :|: x_17 >= 0, z = 1 + x_17
   encode_a(z) -{ 0 }-> a(a(encArg(x_16))) :|: z = 1 + x_16, x_16 >= 0
   encode_a(z) -{ 0 }-> a(0) :|: x_1 >= 0, z = x_1
   encode_a(z) -{ 0 }-> a(1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + x_15
   encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_b(z) -{ 0 }-> b(b(encArg(x_110))) :|: z = 1 + x_110, x_110 >= 0
   encode_b(z) -{ 0 }-> b(a(encArg(x_19))) :|: z = 1 + x_19, x_19 >= 0
   encode_b(z) -{ 0 }-> b(0) :|: x_1 >= 0, z = x_1
   encode_b(z) -{ 0 }-> b(1 + encArg(x_18)) :|: z = 1 + x_18, x_18 >= 0
   encode_b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_c(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1


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

(31) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   b(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   a(z) -{ 1 }-> x1 :|: x1 >= 0, z = x1
   a(z) -{ 3 }-> b(b(a(x1'))) :|: z = 1 + (1 + x1), x1 >= 0, x1' >= 0, 1 + x1 = 1 + x1'
   a(z) -{ 2 }-> b(b(a(b(1 + b(b(a(b(1 + a(x1'))))))))) :|: x1' >= 0, z = 1 + (1 + (1 + (1 + x1')))
   a(z) -{ 2 }-> b(b(a(0))) :|: z = 1 + (1 + x1), x1 >= 0, v0 >= 0, 1 + x1 = v0
   a(z) -{ 2 }-> b(b(a(1 + x0))) :|: z = 1 + (1 + x1), x1 >= 0, 1 + x1 = x0, x0 >= 0
   b(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(x_14))) :|: x_14 >= 0, z = 1 + (1 + x_14)
   encArg(z) -{ 0 }-> b(a(encArg(x_13))) :|: z = 1 + (1 + x_13), x_13 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(x_12)) :|: z = 1 + (1 + x_12), x_12 >= 0
   encArg(z) -{ 0 }-> a(b(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11)
   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 }-> a(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + x_1, x_1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + x_1, x_1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_a(z) -{ 0 }-> a(b(encArg(x_17))) :|: x_17 >= 0, z = 1 + x_17
   encode_a(z) -{ 0 }-> a(a(encArg(x_16))) :|: z = 1 + x_16, x_16 >= 0
   encode_a(z) -{ 0 }-> a(0) :|: x_1 >= 0, z = x_1
   encode_a(z) -{ 0 }-> a(1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + x_15
   encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_b(z) -{ 0 }-> b(b(encArg(x_110))) :|: z = 1 + x_110, x_110 >= 0
   encode_b(z) -{ 0 }-> b(a(encArg(x_19))) :|: z = 1 + x_19, x_19 >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(x_18)) :|: z = 1 + x_18, x_18 >= 0
   encode_b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_b(z) -{ 0 }-> 0 :|: x_1 >= 0, z = x_1, v0 >= 0, 0 = v0
   encode_b(z) -{ 0 }-> 1 + x0 :|