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


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


WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 38 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(12) CdtProblem
(13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
(15) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CdtProblem
(19) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CdtProblem
(21) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CdtProblem
(23) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CdtProblem
(25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CdtProblem
(27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(28) CdtProblem
(29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(30) CdtProblem
(31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(32) CdtProblem
(33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(34) CdtProblem
(35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CdtProblem
(37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(38) CdtProblem
(39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(40) CdtProblem
(41) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(42) CdtProblem
(43) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(44) CdtProblem
(45) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms]
(46) CdtProblem
(47) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) BOUNDS(1, 1)


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

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


The TRS R consists of the following rules:

   0(0(0(0(x1)))) -> 1(0(1(1(x1))))
   1(0(0(1(x1)))) -> 0(0(0(0(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_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))


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

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


The TRS R consists of the following rules:

   0(0(0(0(x1)))) -> 1(0(1(1(x1))))
   1(0(0(1(x1)))) -> 0(0(0(0(x1))))

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

   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(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, n^1).


The TRS R consists of the following rules:

   0(0(0(0(x1)))) -> 1(0(1(1(x1))))
   1(0(0(1(x1)))) -> 0(0(0(0(x1))))

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

   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encode_0(z0) -> 0(encArg(z0))
   encode_1(z0) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0))
   ENCODE_0(z0) -> c2(0'(encArg(z0)), ENCARG(z0))
   ENCODE_1(z0) -> c3(1'(encArg(z0)), ENCARG(z0))
   0'(0(0(0(z0)))) -> c4(1'(0(1(1(z0)))), 0'(1(1(z0))), 1'(1(z0)), 1'(z0))
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
S tuples:
   0'(0(0(0(z0)))) -> c4(1'(0(1(1(z0)))), 0'(1(1(z0))), 1'(1(z0)), 1'(z0))
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
K tuples:none
Defined Rule Symbols:   0_1, 1_1, encArg_1, encode_0_1, encode_1_1

Defined Pair Symbols:   ENCARG_1, ENCODE_0_1, ENCODE_1_1, 0'_1, 1'_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4_4, c5_4


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

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encode_0(z0) -> 0(encArg(z0))
   encode_1(z0) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0))
   ENCODE_0(z0) -> c2(0'(encArg(z0)), ENCARG(z0))
   ENCODE_1(z0) -> c3(1'(encArg(z0)), ENCARG(z0))
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   0_1, 1_1, encArg_1, encode_0_1, encode_1_1

Defined Pair Symbols:   ENCARG_1, ENCODE_0_1, ENCODE_1_1, 1'_1, 0'_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c5_4, c4_1


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encode_0(z0) -> 0(encArg(z0))
   encode_1(z0) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0))
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCODE_0(z0) -> c6(0'(encArg(z0)))
   ENCODE_0(z0) -> c6(ENCARG(z0))
   ENCODE_1(z0) -> c6(1'(encArg(z0)))
   ENCODE_1(z0) -> c6(ENCARG(z0))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   0_1, 1_1, encArg_1, encode_0_1, encode_1_1

Defined Pair Symbols:   ENCARG_1, 1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1

Compound Symbols:   c_2, c1_2, c5_4, c4_1, c6_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_0(z0) -> c6(ENCARG(z0))
   ENCODE_1(z0) -> c6(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encode_0(z0) -> 0(encArg(z0))
   encode_1(z0) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0))
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCODE_0(z0) -> c6(0'(encArg(z0)))
   ENCODE_1(z0) -> c6(1'(encArg(z0)))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   0_1, 1_1, encArg_1, encode_0_1, encode_1_1

Defined Pair Symbols:   ENCARG_1, 1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1

Compound Symbols:   c_2, c1_2, c5_4, c4_1, c6_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_0(z0) -> 0(encArg(z0))
   encode_1(z0) -> 1(encArg(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0))
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCODE_0(z0) -> c6(0'(encArg(z0)))
   ENCODE_1(z0) -> c6(1'(encArg(z0)))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   ENCARG_1, 1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1

Compound Symbols:   c_2, c1_2, c5_4, c4_1, c6_1


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

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0)))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0))
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCODE_0(z0) -> c6(0'(encArg(z0)))
   ENCODE_1(z0) -> c6(1'(encArg(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   ENCARG_1, 1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1

Compound Symbols:   c1_2, c5_4, c4_1, c6_1, c_2


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

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0)))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCODE_0(z0) -> c6(0'(encArg(z0)))
   ENCODE_1(z0) -> c6(1'(encArg(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1

Compound Symbols:   c5_4, c4_1, c6_1, c_2, c1_2


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

(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_0(z0) -> c6(0'(encArg(z0))) by 
   ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0))))

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCODE_1(z0) -> c6(1'(encArg(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCODE_1_1, ENCARG_1, ENCODE_0_1

Compound Symbols:   c5_4, c4_1, c6_1, c_2, c1_2


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

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_1(z0) -> c6(1'(encArg(z0))) by 
   ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0))))
   ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0))))

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0))))
   ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCARG_1, ENCODE_0_1, ENCODE_1_1

Compound Symbols:   c5_4, c4_1, c_2, c1_2, c6_1


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

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0))) by 
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0))))
   ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0))))
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCARG_1, ENCODE_0_1, ENCODE_1_1

Compound Symbols:   c5_4, c4_1, c_2, c1_2, c6_1


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

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0))) by 
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0))))
   ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0))))
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCARG_1, ENCODE_0_1, ENCODE_1_1

Compound Symbols:   c5_4, c4_1, c1_2, c6_1, c_2


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

(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0))) by 
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0))))
   ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0))))
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCARG_1, ENCODE_0_1, ENCODE_1_1

Compound Symbols:   c5_4, c4_1, c1_2, c6_1, c_2


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

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) by 
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0))))
   ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0))))
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1

Compound Symbols:   c5_4, c4_1, c6_1, c_2, c1_2


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

(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) by 
   ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0)))))
   ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0)))))

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0))))
   ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0))))
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0)))))
   ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0)))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1

Compound Symbols:   c5_4, c4_1, c6_1, c_2, c1_2


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

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) by 
   ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0)))))
   ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0)))))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0))))
   ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0))))
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0)))))
   ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0)))))
   ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0)))))
   ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0)))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCODE_1_1, ENCARG_1, ENCODE_0_1

Compound Symbols:   c5_4, c4_1, c6_1, c_2, c1_2


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

(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) by 
   ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0)))))
   ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0)))))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0))))
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0)))))
   ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0)))))
   ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0)))))
   ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0)))))
   ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0)))))
   ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0)))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCODE_1_1, ENCARG_1, ENCODE_0_1

Compound Symbols:   c5_4, c4_1, c6_1, c_2, c1_2


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

(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) by 
   ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0)))))
   ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0)))))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0)))))
   ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0)))))
   ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0)))))
   ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0)))))
   ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0)))))
   ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0)))))
   ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0)))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(z0)))) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, 0'_1, ENCARG_1, ENCODE_0_1, ENCODE_1_1

Compound Symbols:   c5_4, c4_1, c_2, c1_2, c6_1


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

(39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 0'(0(0(0(z0)))) -> c4(1'(z0)) by 
   0'(0(0(0(0(0(1(y0))))))) -> c4(1'(0(0(1(y0)))))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0)))))
   ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0)))))
   ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0)))))
   ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0)))))
   ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0)))))
   ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0)))))
   ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0)))))
   0'(0(0(0(0(0(1(y0))))))) -> c4(1'(0(0(1(y0)))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(0(0(0(0(0(1(y0))))))) -> c4(1'(0(0(1(y0)))))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, ENCARG_1, ENCODE_0_1, ENCODE_1_1, 0'_1

Compound Symbols:   c5_4, c_2, c1_2, c6_1, c4_1


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

(41) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   0'(0(0(0(0(0(1(y0))))))) -> c4(1'(0(0(1(y0)))))
Removed 4 trailing nodes:
   ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0)))))
   ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0)))))
   ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0)))))
   ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0)))))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0)))))
   ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0)))))
   ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0)))))
S tuples:
   1'(0(0(1(z0)))) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   1'_1, ENCARG_1, ENCODE_1_1

Compound Symbols:   c5_4, c_2, c1_2, c6_1


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

(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing tuple parts
----------------------------------------

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0)))))
   ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0)))))
   ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0)))))
   1'(0(0(1(z0)))) -> c5
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0))))
S tuples:
   1'(0(0(1(z0)))) -> c5
K tuples:none
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   ENCARG_1, ENCODE_1_1, 1'_1

Compound Symbols:   c1_2, c6_1, c5, c_1


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

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   1'(0(0(1(z0)))) -> c5
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0)))))
   ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0)))))
   ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0)))))
   1'(0(0(1(z0)))) -> c5
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0(x_1)) = [1]
   POL(1(x_1)) = 0
   POL(1'(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_1(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c5) = 0
   POL(c6(x_1)) = x_1
   POL(cons_0(x_1)) = [1] + x_1
   POL(cons_1(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1]

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encArg(cons_1(z0)) -> 1(encArg(z0))
   0(0(0(0(z0)))) -> 1(0(1(1(z0))))
   1(0(0(1(z0)))) -> 0(0(0(0(z0))))
Tuples:
   ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0)))))
   ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0)))))
   ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0)))))
   1'(0(0(1(z0)))) -> c5
   ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0))))
   ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0))))
S tuples:none
K tuples:
   1'(0(0(1(z0)))) -> c5
Defined Rule Symbols:   encArg_1, 0_1, 1_1

Defined Pair Symbols:   ENCARG_1, ENCODE_1_1, 1'_1

Compound Symbols:   c1_2, c6_1, c5, c_1


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

(47) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(48)
BOUNDS(1, 1)