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


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


WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 65 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)), 157 ms]
(16) CdtProblem
(17) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CdtProblem
(19) CdtRhsSimplificationProcessorProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CdtProblem
(27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(28) CdtProblem
(29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(30) CdtProblem
(31) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 45 ms]
(32) CdtProblem
(33) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 356 ms]
(34) CdtProblem
(35) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 392 ms]
(36) CdtProblem
(37) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 359 ms]
(38) CdtProblem
(39) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(40) BOUNDS(1, 1)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

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

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))


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

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


The TRS R consists of the following rules:

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

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

   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_f(x_1) -> f(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^2).


The TRS R consists of the following rules:

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

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

   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_f(x_1) -> f(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(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   a(a(z0)) -> b(b(b(z0)))
   a(z0) -> d(c(d(z0)))
   b(b(b(z0))) -> a(f(z0))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c4(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c7(B(encArg(z0)), ENCARG(z0))
   ENCODE_D(z0) -> c8(ENCARG(z0))
   ENCODE_C(z0) -> c9(C(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c10(F(encArg(z0)), ENCARG(z0))
   A(a(z0)) -> c11(B(b(b(z0))), B(b(z0)), B(z0))
   A(z0) -> c12(C(d(z0)))
   B(b(b(z0))) -> c13(A(f(z0)), F(z0))
   B(b(z0)) -> c14(C(c(c(z0))), C(c(z0)), C(z0))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
S tuples:
   A(a(z0)) -> c11(B(b(b(z0))), B(b(z0)), B(z0))
   A(z0) -> c12(C(d(z0)))
   B(b(b(z0))) -> c13(A(f(z0)), F(z0))
   B(b(z0)) -> c14(C(c(c(z0))), C(c(z0)), C(z0))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, c_1, f_1, encArg_1, encode_a_1, encode_b_1, encode_d_1, encode_c_1, encode_f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_B_1, ENCODE_D_1, ENCODE_C_1, ENCODE_F_1, A_1, B_1, C_1, F_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c6_2, c7_2, c8_1, c9_2, c10_2, c11_3, c12_1, c13_2, c14_3, c15, c16_1, c17_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_D(z0) -> c8(ENCARG(z0))
   A(a(z0)) -> c11(B(b(b(z0))), B(b(z0)), B(z0))
   B(b(b(z0))) -> c13(A(f(z0)), F(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   a(a(z0)) -> b(b(b(z0)))
   a(z0) -> d(c(d(z0)))
   b(b(b(z0))) -> a(f(z0))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c4(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c6(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c7(B(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c9(C(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c10(F(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   B(b(z0)) -> c14(C(c(c(z0))), C(c(z0)), C(z0))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
S tuples:
   A(z0) -> c12(C(d(z0)))
   B(b(z0)) -> c14(C(c(c(z0))), C(c(z0)), C(z0))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, c_1, f_1, encArg_1, encode_a_1, encode_b_1, encode_d_1, encode_c_1, encode_f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, A_1, B_1, C_1, F_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c6_2, c7_2, c9_2, c10_2, c12_1, c14_3, c15, c16_1, c17_2


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   a(a(z0)) -> b(b(b(z0)))
   a(z0) -> d(c(d(z0)))
   b(b(b(z0))) -> a(f(z0))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c4(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_A(z0) -> c8(ENCARG(z0))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_B(z0) -> c8(ENCARG(z0))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_C(z0) -> c8(ENCARG(z0))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   ENCODE_F(z0) -> c8(ENCARG(z0))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, c_1, f_1, encArg_1, encode_a_1, encode_b_1, encode_d_1, encode_c_1, encode_f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, F_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c12_1, c15, c16_1, c17_2, c8_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 4 leading nodes:
   ENCODE_A(z0) -> c8(ENCARG(z0))
   ENCODE_B(z0) -> c8(ENCARG(z0))
   ENCODE_C(z0) -> c8(ENCARG(z0))
   ENCODE_F(z0) -> c8(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   a(a(z0)) -> b(b(b(z0)))
   a(z0) -> d(c(d(z0)))
   b(b(b(z0))) -> a(f(z0))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c4(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
K tuples:none
Defined Rule Symbols:   a_1, b_1, c_1, f_1, encArg_1, encode_a_1, encode_b_1, encode_d_1, encode_c_1, encode_f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, F_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c12_1, c15, c16_1, c17_2, c8_1


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

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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c4(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, F_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c12_1, c15, c16_1, c17_2, 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.
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c4(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = 0
   POL(B(x_1)) = [1]
   POL(C(x_1)) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A(x_1)) = 0
   POL(ENCODE_B(x_1)) = [1] + x_1
   POL(ENCODE_C(x_1)) = x_1
   POL(ENCODE_F(x_1)) = x_1
   POL(F(x_1)) = 0
   POL(a(x_1)) = x_1
   POL(b(x_1)) = [1] + x_1
   POL(c(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c15) = 0
   POL(c16(x_1)) = x_1
   POL(c17(x_1, x_2)) = x_1 + x_2
   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(c5(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_c(x_1)) = [1] + x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = 0

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c4(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, F_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_2, c12_1, c15, c16_1, c17_2, c8_1


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

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_b(d(z0))) -> c3(B(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c4(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(d(z0))) -> c3(B(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, F_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1

Compound Symbols:   c1_1, c2_2, c4_2, c5_2, c12_1, c15, c16_1, c17_2, c8_1, c3_2


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

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c4(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, F_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1

Compound Symbols:   c1_1, c2_2, c4_2, c5_2, c12_1, c15, c16_1, c17_2, c8_1, c3_2, c3_1


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

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_c(z0)) -> c4(C(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, F_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1

Compound Symbols:   c1_1, c2_2, c5_2, c12_1, c15, c16_1, c17_2, c8_1, c3_2, c3_1, c4_2


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

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(z0)) -> c5(F(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_f(d(z0))) -> c5(F(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(F(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, F_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1

Compound Symbols:   c1_1, c2_2, c12_1, c15, c16_1, c17_2, c8_1, c3_2, c3_1, c4_2, c5_2


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

(25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(a(z0)), A(z0))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, F_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1

Compound Symbols:   c1_1, c2_2, c12_1, c15, c16_1, c17_2, c8_1, c3_2, c3_1, c4_2, c5_2, c5_1


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

(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(f(z0)) -> c17(F(a(z0)), A(z0)) by 
   F(f(z0)) -> c17(F(d(c(d(z0)))), A(z0))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
   F(f(z0)) -> c17(F(d(c(d(z0)))), A(z0))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(F(d(c(d(z0)))), A(z0))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1, F_1

Compound Symbols:   c1_1, c2_2, c12_1, c15, c16_1, c8_1, c3_2, c3_1, c4_2, c5_2, c5_1, c17_2


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

(29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
   F(f(z0)) -> c17(A(z0))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(A(z0))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1, F_1

Compound Symbols:   c1_1, c2_2, c12_1, c15, c16_1, c8_1, c3_2, c3_1, c4_2, c5_2, c5_1, c17_1


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

(31) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   C(c(z0)) -> c15
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
   F(f(z0)) -> c17(A(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [1]
   POL(B(x_1)) = [1]
   POL(C(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A(x_1)) = [1]
   POL(ENCODE_B(x_1)) = [1]
   POL(ENCODE_C(x_1)) = [1]
   POL(ENCODE_F(x_1)) = [1]
   POL(F(x_1)) = [1]
   POL(a(x_1)) = 0
   POL(b(x_1)) = 0
   POL(c(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c15) = 0
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c5(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_c(x_1)) = [1] + x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(d(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = 0

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
   F(f(z0)) -> c17(A(z0))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(d(d(z0))) -> c16(F(z0))
   F(f(z0)) -> c17(A(z0))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   C(c(z0)) -> c15
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1, F_1

Compound Symbols:   c1_1, c2_2, c12_1, c15, c16_1, c8_1, c3_2, c3_1, c4_2, c5_2, c5_1, c17_1


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

(33) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(f(z0)) -> c17(A(z0))
We considered the (Usable) Rules:
   a(z0) -> d(c(d(z0)))
   f(f(z0)) -> f(a(z0))
   encArg(d(z0)) -> d(encArg(z0))
   c(d(d(z0))) -> f(z0)
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   c(c(z0)) -> d(d(d(z0)))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   b(b(z0)) -> c(c(c(z0)))
And the Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
   F(f(z0)) -> c17(A(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = x_1
   POL(B(x_1)) = [2] + x_1
   POL(C(x_1)) = x_1
   POL(ENCARG(x_1)) = [2]x_1^2
   POL(ENCODE_A(x_1)) = [2] + x_1 + [2]x_1^2
   POL(ENCODE_B(x_1)) = [2] + [2]x_1 + x_1^2
   POL(ENCODE_C(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_F(x_1)) = [1] + x_1 + [2]x_1^2
   POL(F(x_1)) = x_1
   POL(a(x_1)) = [1] + x_1
   POL(b(x_1)) = [2] + x_1
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c15) = 0
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(cons_a(x_1)) = [2] + x_1
   POL(cons_b(x_1)) = [2] + x_1
   POL(cons_c(x_1)) = [2] + x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [1] + x_1

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
   F(f(z0)) -> c17(A(z0))
S tuples:
   A(z0) -> c12(C(d(z0)))
   C(d(d(z0))) -> c16(F(z0))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   C(c(z0)) -> c15
   F(f(z0)) -> c17(A(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1, F_1

Compound Symbols:   c1_1, c2_2, c12_1, c15, c16_1, c8_1, c3_2, c3_1, c4_2, c5_2, c5_1, c17_1


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

(35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   C(d(d(z0))) -> c16(F(z0))
We considered the (Usable) Rules:
   a(z0) -> d(c(d(z0)))
   f(f(z0)) -> f(a(z0))
   encArg(d(z0)) -> d(encArg(z0))
   c(d(d(z0))) -> f(z0)
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   c(c(z0)) -> d(d(d(z0)))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   b(b(z0)) -> c(c(c(z0)))
And the Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
   F(f(z0)) -> c17(A(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [1] + x_1
   POL(B(x_1)) = [1] + x_1
   POL(C(x_1)) = [1] + x_1
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A(x_1)) = [1] + x_1 + [2]x_1^2
   POL(ENCODE_B(x_1)) = [2] + x_1 + [2]x_1^2
   POL(ENCODE_C(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_F(x_1)) = [2] + x_1 + [2]x_1^2
   POL(F(x_1)) = x_1
   POL(a(x_1)) = [1] + x_1
   POL(b(x_1)) = [2] + x_1
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c15) = 0
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c5(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)) = [2] + x_1
   POL(cons_c(x_1)) = [1] + x_1
   POL(cons_f(x_1)) = [2] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [1] + x_1

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
   F(f(z0)) -> c17(A(z0))
S tuples:
   A(z0) -> c12(C(d(z0)))
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   C(c(z0)) -> c15
   F(f(z0)) -> c17(A(z0))
   C(d(d(z0))) -> c16(F(z0))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1, F_1

Compound Symbols:   c1_1, c2_2, c12_1, c15, c16_1, c8_1, c3_2, c3_1, c4_2, c5_2, c5_1, c17_1


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

(37) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A(z0) -> c12(C(d(z0)))
We considered the (Usable) Rules:
   a(z0) -> d(c(d(z0)))
   f(f(z0)) -> f(a(z0))
   encArg(d(z0)) -> d(encArg(z0))
   c(d(d(z0))) -> f(z0)
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   c(c(z0)) -> d(d(d(z0)))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   b(b(z0)) -> c(c(c(z0)))
And the Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
   F(f(z0)) -> c17(A(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [1] + x_1
   POL(B(x_1)) = x_1
   POL(C(x_1)) = x_1
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A(x_1)) = [1] + x_1 + [2]x_1^2
   POL(ENCODE_B(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_C(x_1)) = [1] + x_1 + [2]x_1^2
   POL(ENCODE_F(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(F(x_1)) = x_1
   POL(a(x_1)) = [1] + x_1
   POL(b(x_1)) = [2] + x_1
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c15) = 0
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c5(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)) = [2] + x_1
   POL(cons_c(x_1)) = [1] + x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [1] + x_1

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(d(z0)) -> d(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   a(z0) -> d(c(d(z0)))
   b(b(z0)) -> c(c(c(z0)))
   c(c(z0)) -> d(d(d(z0)))
   c(d(d(z0))) -> f(z0)
   f(f(z0)) -> f(a(z0))
Tuples:
   ENCARG(d(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   A(z0) -> c12(C(d(z0)))
   C(c(z0)) -> c15
   C(d(d(z0))) -> c16(F(z0))
   ENCODE_A(z0) -> c8(A(encArg(z0)))
   ENCODE_B(z0) -> c8(B(encArg(z0)))
   ENCODE_C(z0) -> c8(C(encArg(z0)))
   ENCODE_F(z0) -> c8(F(encArg(z0)))
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_b(cons_c(z0))) -> c3(B(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_b(cons_f(z0))) -> c3(B(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_b(d(z0))) -> c3(ENCARG(d(z0)))
   ENCARG(cons_c(d(z0))) -> c4(C(d(encArg(z0))), ENCARG(d(z0)))
   ENCARG(cons_c(cons_a(z0))) -> c4(C(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_c(cons_b(z0))) -> c4(C(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_c(cons_c(z0))) -> c4(C(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_c(cons_f(z0))) -> c4(C(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a(z0))) -> c5(F(a(encArg(z0))), ENCARG(cons_a(z0)))
   ENCARG(cons_f(cons_b(z0))) -> c5(F(b(encArg(z0))), ENCARG(cons_b(z0)))
   ENCARG(cons_f(cons_c(z0))) -> c5(F(c(encArg(z0))), ENCARG(cons_c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c5(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(d(z0))) -> c5(ENCARG(d(z0)))
   F(f(z0)) -> c17(A(z0))
S tuples:none
K tuples:
   B(b(z0)) -> c8(C(c(c(z0))))
   B(b(z0)) -> c8(C(c(z0)))
   B(b(z0)) -> c8(C(z0))
   C(c(z0)) -> c15
   F(f(z0)) -> c17(A(z0))
   C(d(d(z0))) -> c16(F(z0))
   A(z0) -> c12(C(d(z0)))
Defined Rule Symbols:   encArg_1, a_1, b_1, c_1, f_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, ENCODE_F_1, B_1, F_1

Compound Symbols:   c1_1, c2_2, c12_1, c15, c16_1, c8_1, c3_2, c3_1, c4_2, c5_2, c5_1, c17_1


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

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

(40)
BOUNDS(1, 1)