/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 44 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 132 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 42 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 43 ms]
(20) CdtProblem
(21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) 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:

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


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

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


The TRS R consists of the following rules:

   a(d(x1)) -> d(b(x1))
   a(x1) -> b(b(b(x1)))
   d(x1) -> x1
   a(x1) -> x1
   b(d(b(x1))) -> a(d(x1))
   b(c(x1)) -> c(d(d(x1)))
   a(c(x1)) -> b(b(c(d(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_d(x_1)) -> d(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   a(d(x1)) -> d(b(x1))
   a(x1) -> b(b(b(x1)))
   d(x1) -> x1
   a(x1) -> x1
   b(d(b(x1))) -> a(d(x1))
   b(c(x1)) -> c(d(d(x1)))
   a(c(x1)) -> b(b(c(d(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_d(x_1)) -> d(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(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(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(d(z0)) -> d(b(z0))
   a(z0) -> b(b(b(z0)))
   a(z0) -> z0
   a(c(z0)) -> b(b(c(d(z0))))
   d(z0) -> z0
   b(d(b(z0))) -> a(d(z0))
   b(c(z0)) -> c(d(d(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)), ENCARG(z0))
   ENCODE_D(z0) -> c6(D(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c7(B(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c8(ENCARG(z0))
   A(d(z0)) -> c9(D(b(z0)), B(z0))
   A(z0) -> c10(B(b(b(z0))), B(b(z0)), B(z0))
   A(z0) -> c11
   A(c(z0)) -> c12(B(b(c(d(z0)))), B(c(d(z0))), D(z0))
   D(z0) -> c13
   B(d(b(z0))) -> c14(A(d(z0)), D(z0))
   B(c(z0)) -> c15(D(d(z0)), D(z0))
S tuples:
   A(d(z0)) -> c9(D(b(z0)), B(z0))
   A(z0) -> c10(B(b(b(z0))), B(b(z0)), B(z0))
   A(z0) -> c11
   A(c(z0)) -> c12(B(b(c(d(z0)))), B(c(d(z0))), D(z0))
   D(z0) -> c13
   B(d(b(z0))) -> c14(A(d(z0)), D(z0))
   B(c(z0)) -> c15(D(d(z0)), D(z0))
K tuples:none
Defined Rule Symbols:   a_1, d_1, b_1, encArg_1, encode_a_1, encode_d_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_D_1, ENCODE_B_1, ENCODE_C_1, A_1, D_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c6_2, c7_2, c8_1, c9_2, c10_3, c11, c12_3, c13, c14_2, c15_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_C(z0) -> c8(ENCARG(z0))
   A(d(z0)) -> c9(D(b(z0)), B(z0))
   B(d(b(z0))) -> c14(A(d(z0)), D(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(d(z0)) -> d(b(z0))
   a(z0) -> b(b(b(z0)))
   a(z0) -> z0
   a(c(z0)) -> b(b(c(d(z0))))
   d(z0) -> z0
   b(d(b(z0))) -> a(d(z0))
   b(c(z0)) -> c(d(d(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)), ENCARG(z0))
   ENCODE_D(z0) -> c6(D(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c7(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c10(B(b(b(z0))), B(b(z0)), B(z0))
   A(z0) -> c11
   A(c(z0)) -> c12(B(b(c(d(z0)))), B(c(d(z0))), D(z0))
   D(z0) -> c13
   B(c(z0)) -> c15(D(d(z0)), D(z0))
S tuples:
   A(z0) -> c10(B(b(b(z0))), B(b(z0)), B(z0))
   A(z0) -> c11
   A(c(z0)) -> c12(B(b(c(d(z0)))), B(c(d(z0))), D(z0))
   D(z0) -> c13
   B(c(z0)) -> c15(D(d(z0)), D(z0))
K tuples:none
Defined Rule Symbols:   a_1, d_1, b_1, encArg_1, encode_a_1, encode_d_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_D_1, ENCODE_B_1, A_1, D_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c6_2, c7_2, c10_3, c11, c12_3, c13, c15_2


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(d(z0)) -> d(b(z0))
   a(z0) -> b(b(b(z0)))
   a(z0) -> z0
   a(c(z0)) -> b(b(c(d(z0))))
   d(z0) -> z0
   b(d(b(z0))) -> a(d(z0))
   b(c(z0)) -> c(d(d(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c11
   D(z0) -> c13
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_A(z0) -> c8(ENCARG(z0))
   ENCODE_D(z0) -> c8(D(encArg(z0)))
   ENCODE_D(z0) -> c8(ENCARG(z0))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_B(z0) -> c8(ENCARG(z0))
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
S tuples:
   A(z0) -> c11
   D(z0) -> c13
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
K tuples:none
Defined Rule Symbols:   a_1, d_1, b_1, encArg_1, encode_a_1, encode_d_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, A_1, D_1, ENCODE_A_1, ENCODE_D_1, ENCODE_B_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c11, c13, c8_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_A(z0) -> c8(ENCARG(z0))
   ENCODE_D(z0) -> c8(ENCARG(z0))
   ENCODE_B(z0) -> c8(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(d(z0)) -> d(b(z0))
   a(z0) -> b(b(b(z0)))
   a(z0) -> z0
   a(c(z0)) -> b(b(c(d(z0))))
   d(z0) -> z0
   b(d(b(z0))) -> a(d(z0))
   b(c(z0)) -> c(d(d(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c11
   D(z0) -> c13
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_D(z0) -> c8(D(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
S tuples:
   A(z0) -> c11
   D(z0) -> c13
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
K tuples:none
Defined Rule Symbols:   a_1, d_1, b_1, encArg_1, encode_a_1, encode_d_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, A_1, D_1, ENCODE_A_1, ENCODE_D_1, ENCODE_B_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c11, c13, c8_1


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

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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> b(b(b(z0)))
   a(z0) -> z0
   a(c(z0)) -> b(b(c(d(z0))))
   d(z0) -> z0
   b(c(z0)) -> c(d(d(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c11
   D(z0) -> c13
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_D(z0) -> c8(D(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
S tuples:
   A(z0) -> c11
   D(z0) -> c13
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, d_1, b_1

Defined Pair Symbols:   ENCARG_1, A_1, D_1, ENCODE_A_1, ENCODE_D_1, ENCODE_B_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c11, c13, c8_1


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A(z0) -> c11
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c11
   D(z0) -> c13
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_D(z0) -> c8(D(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [1]
   POL(B(x_1)) = 0
   POL(D(x_1)) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A(x_1)) = [1]
   POL(ENCODE_B(x_1)) = 0
   POL(ENCODE_D(x_1)) = x_1
   POL(a(x_1)) = [1] + x_1
   POL(b(x_1)) = x_1
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c11) = 0
   POL(c13) = 0
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(cons_a(x_1)) = [1] + x_1
   POL(cons_b(x_1)) = [1] + x_1
   POL(cons_d(x_1)) = [1] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = [1] + x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> b(b(b(z0)))
   a(z0) -> z0
   a(c(z0)) -> b(b(c(d(z0))))
   d(z0) -> z0
   b(c(z0)) -> c(d(d(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c11
   D(z0) -> c13
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_D(z0) -> c8(D(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
S tuples:
   D(z0) -> c13
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
K tuples:
   A(z0) -> c11
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
Defined Rule Symbols:   encArg_1, a_1, d_1, b_1

Defined Pair Symbols:   ENCARG_1, A_1, D_1, ENCODE_A_1, ENCODE_D_1, ENCODE_B_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c11, c13, c8_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   D(z0) -> c13
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c11
   D(z0) -> c13
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_D(z0) -> c8(D(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [1]
   POL(B(x_1)) = [1]
   POL(D(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A(x_1)) = [1] + x_1
   POL(ENCODE_B(x_1)) = [1]
   POL(ENCODE_D(x_1)) = [1] + x_1
   POL(a(x_1)) = [1] + x_1
   POL(b(x_1)) = x_1
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c11) = 0
   POL(c13) = 0
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(cons_a(x_1)) = [1] + x_1
   POL(cons_b(x_1)) = [1] + x_1
   POL(cons_d(x_1)) = [1] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = [1] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> b(b(b(z0)))
   a(z0) -> z0
   a(c(z0)) -> b(b(c(d(z0))))
   d(z0) -> z0
   b(c(z0)) -> c(d(d(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c11
   D(z0) -> c13
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_D(z0) -> c8(D(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
S tuples:
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
K tuples:
   A(z0) -> c11
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   D(z0) -> c13
Defined Rule Symbols:   encArg_1, a_1, d_1, b_1

Defined Pair Symbols:   ENCARG_1, A_1, D_1, ENCODE_A_1, ENCODE_D_1, ENCODE_B_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c11, c13, c8_1


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c11
   D(z0) -> c13
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_D(z0) -> c8(D(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [1]
   POL(B(x_1)) = [1]
   POL(D(x_1)) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A(x_1)) = [1] + x_1
   POL(ENCODE_B(x_1)) = [1] + x_1
   POL(ENCODE_D(x_1)) = 0
   POL(a(x_1)) = [1] + x_1
   POL(b(x_1)) = [1]
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c11) = 0
   POL(c13) = 0
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(cons_a(x_1)) = [1] + x_1
   POL(cons_b(x_1)) = [1] + x_1
   POL(cons_d(x_1)) = [1] + x_1
   POL(d(x_1)) = [1]
   POL(encArg(x_1)) = [1] + x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   a(z0) -> b(b(b(z0)))
   a(z0) -> z0
   a(c(z0)) -> b(b(c(d(z0))))
   d(z0) -> z0
   b(c(z0)) -> c(d(d(z0)))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c3(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c4(B(encArg(z0)), ENCARG(z0))
   A(z0) -> c11
   D(z0) -> c13
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_D(z0) -> c8(D(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
S tuples:none
K tuples:
   A(z0) -> c11
   A(z0) -> c8(B(b(b(z0))))
   A(z0) -> c8(B(b(z0)))
   A(z0) -> c8(B(z0))
   A(c(z0)) -> c8(B(b(c(d(z0)))))
   A(c(z0)) -> c8(B(c(d(z0))))
   A(c(z0)) -> c8(D(z0))
   D(z0) -> c13
   B(c(z0)) -> c8(D(d(z0)))
   B(c(z0)) -> c8(D(z0))
Defined Rule Symbols:   encArg_1, a_1, d_1, b_1

Defined Pair Symbols:   ENCARG_1, A_1, D_1, ENCODE_A_1, ENCODE_D_1, ENCODE_B_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c11, c13, c8_1


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

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

(22)
BOUNDS(1, 1)