/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 60 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(10) CdtProblem
(11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtNarrowingProof [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) CdtLeafRemovalProof [ComplexityIfPolyImplication, 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) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(32) CdtProblem
(33) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(34) CdtProblem
(35) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(36) CdtProblem
(37) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(38) CdtProblem
(39) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(40) CdtProblem
(41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms]
(42) CdtProblem
(43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(44) CdtProblem
(45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(46) CdtProblem
(47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
(49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(50) CdtProblem
(51) CdtRhsSimplificationProcessorProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(58) CdtProblem
(59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(60) CdtProblem
(61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(62) CdtProblem
(63) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(64) 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:

   1(0(x1)) -> 0(0(0(1(x1))))
   0(1(x1)) -> 1(x1)
   1(1(x1)) -> 0(0(0(0(x1))))
   0(0(x1)) -> 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_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_0(x_1) -> 0(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:

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

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

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

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

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

   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_0(x_1) -> 0(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_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encode_1(z0) -> 1(encArg(z0))
   encode_0(z0) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(z0)) -> c(1'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0))
   ENCODE_1(z0) -> c2(1'(encArg(z0)), ENCARG(z0))
   ENCODE_0(z0) -> c3(0'(encArg(z0)), ENCARG(z0))
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
S tuples:
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
K tuples:none
Defined Rule Symbols:   1_1, 0_1, encArg_1, encode_1_1, encode_0_1

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

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4_4, c5_4, c6_1, c7_1


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

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encode_1(z0) -> 1(encArg(z0))
   encode_0(z0) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(z0)) -> c(1'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0))
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCODE_1(z0) -> c8(1'(encArg(z0)))
   ENCODE_1(z0) -> c8(ENCARG(z0))
   ENCODE_0(z0) -> c8(0'(encArg(z0)))
   ENCODE_0(z0) -> c8(ENCARG(z0))
S tuples:
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
K tuples:none
Defined Rule Symbols:   1_1, 0_1, encArg_1, encode_1_1, encode_0_1

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

Compound Symbols:   c_2, c1_2, c4_4, c5_4, c6_1, c7_1, c8_1


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

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_1(z0) -> c8(ENCARG(z0))
   ENCODE_0(z0) -> c8(ENCARG(z0))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   encode_1(z0) -> 1(encArg(z0))
   encode_0(z0) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(z0)) -> c(1'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0))
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCODE_1(z0) -> c8(1'(encArg(z0)))
   ENCODE_0(z0) -> c8(0'(encArg(z0)))
S tuples:
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
K tuples:none
Defined Rule Symbols:   1_1, 0_1, encArg_1, encode_1_1, encode_0_1

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

Compound Symbols:   c_2, c1_2, c4_4, c5_4, c6_1, c7_1, c8_1


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

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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(z0)) -> c(1'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0))
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCODE_1(z0) -> c8(1'(encArg(z0)))
   ENCODE_0(z0) -> c8(0'(encArg(z0)))
S tuples:
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c_2, c1_2, c4_4, c5_4, c6_1, c7_1, c8_1


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

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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0))
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCODE_1(z0) -> c8(1'(encArg(z0)))
   ENCODE_0(z0) -> c8(0'(encArg(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
S tuples:
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c1_2, c4_4, c5_4, c6_1, c7_1, c8_1, c_2


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

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

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCODE_1(z0) -> c8(1'(encArg(z0)))
   ENCODE_0(z0) -> c8(0'(encArg(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
S tuples:
   1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c4_4, c5_4, c6_1, c7_1, c8_1, c_2, c1_2


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

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace 1'(0(z0)) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0)) by 
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCODE_1(z0) -> c8(1'(encArg(z0)))
   ENCODE_0(z0) -> c8(0'(encArg(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))
S tuples:
   1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c5_4, c6_1, c7_1, c8_1, c_2, c1_2, c4_4


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

(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace 1'(1(z0)) -> c5(0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)), 0'(z0)) by 
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
   1'(1(1(z0))) -> c5(0'(0(0(1(z0)))), 0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)))
   1'(1(0(z0))) -> c5(0'(0(0(0(z0)))), 0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)))

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCODE_1(z0) -> c8(1'(encArg(z0)))
   ENCODE_0(z0) -> c8(0'(encArg(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
   1'(1(1(z0))) -> c5(0'(0(0(1(z0)))), 0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)))
   1'(1(0(z0))) -> c5(0'(0(0(0(z0)))), 0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)))
S tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
   1'(1(1(z0))) -> c5(0'(0(0(1(z0)))), 0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)))
   1'(1(0(z0))) -> c5(0'(0(0(0(z0)))), 0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c6_1, c7_1, c8_1, c_2, c1_2, c4_4, c5_4


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

(21) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   1'(1(0(z0))) -> c5(0'(0(0(0(z0)))), 0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)))
   1'(1(1(z0))) -> c5(0'(0(0(1(z0)))), 0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)))

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCODE_1(z0) -> c8(1'(encArg(z0)))
   ENCODE_0(z0) -> c8(0'(encArg(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
S tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c6_1, c7_1, c8_1, c_2, c1_2, c4_4, c5_4


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

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_1(z0) -> c8(1'(encArg(z0))) by 
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCODE_0(z0) -> c8(0'(encArg(z0)))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
S tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c6_1, c7_1, c8_1, c_2, c1_2, c4_4, c5_4


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

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_0(z0) -> c8(0'(encArg(z0))) by 
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
S tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0))
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c6_1, c7_1, c_2, c1_2, c4_4, c5_4, c8_1


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

(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace 1'(0(x0)) -> c4(0'(0(1(x0))), 0'(0(1(x0))), 0'(1(x0)), 1'(x0)) by 
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
S tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0))
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c6_1, c7_1, c_2, c1_2, c5_4, c8_1, c4_4, c4_1


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

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace 1'(1(x0)) -> c5(0'(0(0(x0))), 0'(0(0(x0))), 0'(0(x0)), 0'(x0)) by 
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(0(z0))) -> c5(0'(0(0(z0))), 0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)))
   1'(1(x0)) -> c5(0'(0(0(x0))))

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(0(z0))) -> c5(0'(0(0(z0))), 0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)))
   1'(1(x0)) -> c5(0'(0(0(x0))))
S tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(0(z0))) -> c5(0'(0(0(z0))), 0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)))
   1'(1(x0)) -> c5(0'(0(0(x0))))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c6_1, c7_1, c_2, c1_2, c8_1, c4_4, c4_1, c5_4, c5_1


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

(31) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   1'(1(0(z0))) -> c5(0'(0(0(z0))), 0'(0(0(0(z0)))), 0'(0(0(z0))), 0'(0(z0)))

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(x0)) -> c5(0'(0(0(x0))))
S tuples:
   0'(1(z0)) -> c6(1'(z0))
   0'(0(z0)) -> c7(0'(z0))
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(x0)) -> c5(0'(0(0(x0))))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c6_1, c7_1, c_2, c1_2, c8_1, c4_4, c4_1, c5_4, c5_1


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

(33) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 0'(1(z0)) -> c6(1'(z0)) by 
   0'(1(0(y0))) -> c6(1'(0(y0)))
   0'(1(1(y0))) -> c6(1'(1(y0)))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   0'(1(0(y0))) -> c6(1'(0(y0)))
   0'(1(1(y0))) -> c6(1'(1(y0)))
S tuples:
   0'(0(z0)) -> c7(0'(z0))
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   0'(1(0(y0))) -> c6(1'(0(y0)))
   0'(1(1(y0))) -> c6(1'(1(y0)))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c7_1, c_2, c1_2, c8_1, c4_4, c4_1, c5_4, c5_1, c6_1


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

(35) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   0'(1(0(y0))) -> c6(1'(0(y0)))
   0'(1(1(y0))) -> c6(1'(1(y0)))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(x0)) -> c5(0'(0(0(x0))))
S tuples:
   0'(0(z0)) -> c7(0'(z0))
   1'(0(z0)) -> c4(0'(1(z0)), 0'(0(1(z0))), 0'(1(z0)), 1'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(x0)) -> c5(0'(0(0(x0))))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c7_1, c_2, c1_2, c8_1, c4_4, c4_1, c5_4, c5_1


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

(37) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
S tuples:
   0'(0(z0)) -> c7(0'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c5(0'(0(z0)), 0'(0(0(z0))), 0'(0(z0)), 0'(z0))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c7_1, c_2, c1_2, c8_1, c4_1, c5_4, c5_1, c4_2


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

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
S tuples:
   0'(0(z0)) -> c7(0'(z0))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c7_1, c_2, c1_2, c8_1, c4_1, c5_1, c4_2, c2_1


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

(41) 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(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
We considered the (Usable) Rules:none
And the Tuples:
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0(x_1)) = 0
   POL(0'(x_1)) = 0
   POL(1(x_1)) = [2] + [2]x_1
   POL(1'(x_1)) = [2]
   POL(ENCARG(x_1)) = [2]x_1
   POL(ENCODE_0(x_1)) = [3] + [3]x_1
   POL(ENCODE_1(x_1)) = [2]x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(cons_0(x_1)) = [1] + x_1
   POL(cons_1(x_1)) = [2] + x_1
   POL(encArg(x_1)) = 0

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
S tuples:
   0'(0(z0)) -> c7(0'(z0))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
K tuples:
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c7_1, c_2, c1_2, c8_1, c4_1, c5_1, c4_2, c2_1


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

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

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c1(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
S tuples:
   0'(0(z0)) -> c7(0'(z0))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
K tuples:
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c7_1, c_2, c1_2, c8_1, c4_1, c5_1, c4_2, c2_1


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

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_0(cons_1(z0)) -> c8(0'(1(encArg(z0)))) by 
   ENCODE_0(cons_1(cons_1(z0))) -> c8(0'(1(1(encArg(z0)))))
   ENCODE_0(cons_1(cons_0(z0))) -> c8(0'(1(0(encArg(z0)))))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   0'(0(z0)) -> c7(0'(z0))
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c1(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCODE_0(cons_1(cons_1(z0))) -> c8(0'(1(1(encArg(z0)))))
   ENCODE_0(cons_1(cons_0(z0))) -> c8(0'(1(0(encArg(z0)))))
S tuples:
   0'(0(z0)) -> c7(0'(z0))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
K tuples:
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c7_1, c_2, c1_2, c8_1, c4_1, c5_1, c4_2, c2_1


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

(47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 0'(0(z0)) -> c7(0'(z0)) by 
   0'(0(0(y0))) -> c7(0'(0(y0)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c1(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCODE_0(cons_1(cons_1(z0))) -> c8(0'(1(1(encArg(z0)))))
   ENCODE_0(cons_1(cons_0(z0))) -> c8(0'(1(0(encArg(z0)))))
   0'(0(0(y0))) -> c7(0'(0(y0)))
S tuples:
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
   0'(0(0(y0))) -> c7(0'(0(y0)))
K tuples:
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(x0)) -> c5(0'(0(0(x0))))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   1'(1(z0)) -> c2(0'(z0))
Defined Rule Symbols:   encArg_1, 1_1, 0_1

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

Compound Symbols:   c_2, c1_2, c8_1, c4_1, c5_1, c4_2, c2_1, c7_1


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

(49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 9 trailing nodes:
   1'(0(x0)) -> c4(0'(0(1(x0))))
   1'(1(z0)) -> c2(0'(0(z0)))
   1'(1(z0)) -> c2(0'(0(0(z0))))
   ENCODE_0(cons_0(z0)) -> c8(0'(0(encArg(z0))))
   ENCODE_0(cons_1(cons_1(z0))) -> c8(0'(1(1(encArg(z0)))))
   ENCODE_0(cons_1(cons_0(z0))) -> c8(0'(1(0(encArg(z0)))))
   1'(1(z0)) -> c2(0'(z0))
   0'(0(0(y0))) -> c7(0'(0(y0)))
   1'(1(x0)) -> c5(0'(0(0(x0))))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c1(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
S tuples:
   1'(0(z0)) -> c4(0'(0(1(z0))), 1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

Defined Pair Symbols:   ENCARG_1, ENCODE_1_1, 1'_1

Compound Symbols:   c_2, c1_2, c8_1, c4_2


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

(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(cons_1(z0))) -> c(1'(1(encArg(z0))), ENCARG(cons_1(z0)))
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCARG(cons_0(cons_0(z0))) -> c1(ENCARG(cons_0(z0)))
   1'(0(z0)) -> c4(1'(z0))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c1(ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c1(ENCARG(cons_1(cons_0(z0))))
S tuples:
   1'(0(z0)) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

Defined Pair Symbols:   ENCARG_1, ENCODE_1_1, 1'_1

Compound Symbols:   c_2, c8_1, c1_1, c4_1


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

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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCARG(cons_0(cons_0(z0))) -> c1(ENCARG(cons_0(z0)))
   1'(0(z0)) -> c4(1'(z0))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c1(ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c1(ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
S tuples:
   1'(0(z0)) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

Defined Pair Symbols:   ENCARG_1, ENCODE_1_1, 1'_1

Compound Symbols:   c_2, c8_1, c1_1, c4_1


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_1(cons_1(z0)) -> c8(1'(1(encArg(z0)))) by 
   ENCODE_1(cons_1(cons_1(z0))) -> c8(1'(1(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c8(1'(1(0(encArg(z0)))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCARG(cons_0(cons_0(z0))) -> c1(ENCARG(cons_0(z0)))
   1'(0(z0)) -> c4(1'(z0))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c1(ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c1(ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCODE_1(cons_1(cons_1(z0))) -> c8(1'(1(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c8(1'(1(0(encArg(z0)))))
S tuples:
   1'(0(z0)) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

Defined Pair Symbols:   ENCARG_1, ENCODE_1_1, 1'_1

Compound Symbols:   c_2, c8_1, c1_1, c4_1


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

(57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENCARG(cons_0(cons_0(z0))) -> c1(ENCARG(cons_0(z0))) by 
   ENCARG(cons_0(cons_0(cons_0(y0)))) -> c1(ENCARG(cons_0(cons_0(y0))))
   ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c1(ENCARG(cons_0(cons_1(cons_1(y0)))))
   ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c1(ENCARG(cons_0(cons_1(cons_0(y0)))))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   1'(0(z0)) -> c4(1'(z0))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c1(ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c1(ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCODE_1(cons_1(cons_1(z0))) -> c8(1'(1(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c8(1'(1(0(encArg(z0)))))
   ENCARG(cons_0(cons_0(cons_0(y0)))) -> c1(ENCARG(cons_0(cons_0(y0))))
   ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c1(ENCARG(cons_0(cons_1(cons_1(y0)))))
   ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c1(ENCARG(cons_0(cons_1(cons_0(y0)))))
S tuples:
   1'(0(z0)) -> c4(1'(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

Defined Pair Symbols:   ENCARG_1, ENCODE_1_1, 1'_1

Compound Symbols:   c_2, c8_1, c4_1, c1_1


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

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

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_1(z0)) -> 1(encArg(z0))
   encArg(cons_0(z0)) -> 0(encArg(z0))
   1(0(z0)) -> 0(0(0(1(z0))))
   1(1(z0)) -> 0(0(0(0(z0))))
   0(1(z0)) -> 1(z0)
   0(0(z0)) -> 0(z0)
Tuples:
   ENCARG(cons_1(cons_0(z0))) -> c(1'(0(encArg(z0))), ENCARG(cons_0(z0)))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))
   ENCARG(cons_0(cons_1(cons_1(z0)))) -> c1(ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_0(cons_1(cons_0(z0)))) -> c1(ENCARG(cons_1(cons_0(z0))))
   ENCARG(cons_1(cons_1(cons_1(z0)))) -> c(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0))))
   ENCARG(cons_1(cons_1(cons_0(z0)))) -> c(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0))))
   ENCODE_1(cons_1(cons_1(z0))) -> c8(1'(1(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c8(1'(1(0(encArg(z0)))))
   ENCARG(cons_0(cons_0(cons_0(y0)))) -> c1(ENCARG(cons_0(cons_0(y0))))
   ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c1(ENCARG(cons_0(cons_1(cons_1(y0)))))
   ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c1(ENCARG(cons_0(cons_1(cons_0(y0)))))
   1'(0(0(y0))) -> c4(1'(0(y0)))
S tuples:
   1'(0(0(y0))) -> c4(1'(0(y0)))
K tuples:none
Defined Rule Symbols:   encArg_1, 1_1, 0_1

Defined Pair Symbols:   ENCARG_1, ENCODE_1_1, 1'_1

Compound Symbols:   c_2, c8_1, c1_1, c4_1


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

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   1'(0(0(y0))) -> c4(1'(0(y0)))
   ENCODE_1(cons_1(cons_1(z0))) -> c8(1'(1(1(encArg(z0)))))
   ENCODE_1(cons_1(cons_0(z0))) -> c8(1'(1(0(encArg(z0)))))
   ENCODE_1(cons_0(z0)) -> c8(1'(0(encArg(z0))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   ENCARG_1

Compound Symbols:   c_2, c1_1


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

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

(64)
BOUNDS(1, 1)