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


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


WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 91 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
(9) CpxTrsToCdtProof [UPPER BOUND(ID), 1 ms]
(10) CdtProblem
(11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(12) CdtProblem
(13) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
(15) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(16) CdtProblem
(17) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 188 ms]
(20) CdtProblem
(21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 102 ms]
(22) CdtProblem
(23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 182 ms]
(24) CdtProblem
(25) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 127 ms]
(26) CdtProblem
(27) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 99 ms]
(28) CdtProblem
(29) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 110 ms]
(30) CdtProblem
(31) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(32) BOUNDS(1, 1)


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

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


The TRS R consists of the following rules:

   f(f(X)) -> c(n__f(g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X

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

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

   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(n__d(x_1)) -> n__d(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_d(x_1)) -> d(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_n__d(x_1) -> n__d(encArg(x_1))


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

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


The TRS R consists of the following rules:

   f(f(X)) -> c(n__f(g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X

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

   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(n__d(x_1)) -> n__d(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_d(x_1)) -> d(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_n__d(x_1) -> n__d(encArg(x_1))

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

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   f(f(X)) -> c(n__f(g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X

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

   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(n__d(x_1)) -> n__d(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_d(x_1)) -> d(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_n__d(x_1) -> n__d(encArg(x_1))

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

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

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X
   f(c_f(X)) -> c(n__f(g(n__f(X))))

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

   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(n__d(x_1)) -> n__d(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_d(x_1)) -> d(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_n__d(x_1) -> n__d(encArg(x_1))
   f(x0) -> c_f(x0)

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

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

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

(8)
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:

   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X
   f(c_f(X)) -> c(n__f(g(n__f(X))))

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

   encArg(n__f(x_1)) -> n__f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(n__d(x_1)) -> n__d(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_d(x_1)) -> d(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_n__f(x_1) -> n__f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_n__d(x_1) -> n__d(encArg(x_1))
   f(x0) -> c_f(x0)

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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_n__f(z0) -> n__f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_activate(z0) -> activate(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   encode_n__d(z0) -> n__d(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c9(F(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c10(C(encArg(z0)), ENCARG(z0))
   ENCODE_N__F(z0) -> c11(ENCARG(z0))
   ENCODE_G(z0) -> c12(ENCARG(z0))
   ENCODE_D(z0) -> c13(D(encArg(z0)), ENCARG(z0))
   ENCODE_ACTIVATE(z0) -> c14(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_H(z0) -> c15(H(encArg(z0)), ENCARG(z0))
   ENCODE_N__D(z0) -> c16(ENCARG(z0))
   F(z0) -> c17
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
S tuples:
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
K tuples:none
Defined Rule Symbols:   c_1, h_1, f_1, d_1, activate_1, encArg_1, encode_f_1, encode_c_1, encode_n__f_1, encode_g_1, encode_d_1, encode_activate_1, encode_h_1, encode_n__d_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_C_1, ENCODE_N__F_1, ENCODE_G_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1, ENCODE_N__D_1, F_1, C_1, H_1, D_1, ACTIVATE_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c9_2, c10_2, c11_1, c12_1, c13_2, c14_2, c15_2, c16_1, c17, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_N__F(z0) -> c11(ENCARG(z0))
   ENCODE_G(z0) -> c12(ENCARG(z0))
   ENCODE_N__D(z0) -> c16(ENCARG(z0))
Removed 1 trailing nodes:
   F(z0) -> c17

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_n__f(z0) -> n__f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_activate(z0) -> activate(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   encode_n__d(z0) -> n__d(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c9(F(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c10(C(encArg(z0)), ENCARG(z0))
   ENCODE_D(z0) -> c13(D(encArg(z0)), ENCARG(z0))
   ENCODE_ACTIVATE(z0) -> c14(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_H(z0) -> c15(H(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
S tuples:
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
K tuples:none
Defined Rule Symbols:   c_1, h_1, f_1, d_1, activate_1, encArg_1, encode_f_1, encode_c_1, encode_n__f_1, encode_g_1, encode_d_1, encode_activate_1, encode_h_1, encode_n__d_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_C_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1, F_1, C_1, H_1, D_1, ACTIVATE_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c9_2, c10_2, c13_2, c14_2, c15_2, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25


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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_n__f(z0) -> n__f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_activate(z0) -> activate(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   encode_n__d(z0) -> n__d(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_F(z0) -> c11(ENCARG(z0))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_C(z0) -> c11(ENCARG(z0))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_D(z0) -> c11(ENCARG(z0))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ENCARG(z0))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
   ENCODE_H(z0) -> c11(ENCARG(z0))
S tuples:
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
K tuples:none
Defined Rule Symbols:   c_1, h_1, f_1, d_1, activate_1, encArg_1, encode_f_1, encode_c_1, encode_n__f_1, encode_g_1, encode_d_1, encode_activate_1, encode_h_1, encode_n__d_1

Defined Pair Symbols:   ENCARG_1, F_1, C_1, H_1, D_1, ACTIVATE_1, ENCODE_F_1, ENCODE_C_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25, c11_1


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

(15) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 5 leading nodes:
   ENCODE_F(z0) -> c11(ENCARG(z0))
   ENCODE_C(z0) -> c11(ENCARG(z0))
   ENCODE_D(z0) -> c11(ENCARG(z0))
   ENCODE_ACTIVATE(z0) -> c11(ENCARG(z0))
   ENCODE_H(z0) -> c11(ENCARG(z0))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_n__f(z0) -> n__f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_activate(z0) -> activate(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   encode_n__d(z0) -> n__d(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
S tuples:
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
K tuples:none
Defined Rule Symbols:   c_1, h_1, f_1, d_1, activate_1, encArg_1, encode_f_1, encode_c_1, encode_n__f_1, encode_g_1, encode_d_1, encode_activate_1, encode_h_1, encode_n__d_1

Defined Pair Symbols:   ENCARG_1, F_1, C_1, H_1, D_1, ACTIVATE_1, ENCODE_F_1, ENCODE_C_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25, c11_1


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

(17) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0) -> f(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_n__f(z0) -> n__f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_activate(z0) -> activate(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   encode_n__d(z0) -> n__d(encArg(z0))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
S tuples:
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, c_1, h_1, d_1, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, C_1, H_1, D_1, ACTIVATE_1, ENCODE_F_1, ENCODE_C_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25, c11_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.
   H(z0) -> c21(C(n__d(z0)))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ACTIVATE(x_1)) = 0
   POL(C(x_1)) = 0
   POL(D(x_1)) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_ACTIVATE(x_1)) = x_1
   POL(ENCODE_C(x_1)) = x_1
   POL(ENCODE_D(x_1)) = 0
   POL(ENCODE_F(x_1)) = 0
   POL(ENCODE_H(x_1)) = [1] + x_1
   POL(F(x_1)) = 0
   POL(H(x_1)) = [1]
   POL(activate(x_1)) = [1] + x_1
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c18) = 0
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c_f(x_1)) = [1] + x_1
   POL(cons_activate(x_1)) = [1] + x_1
   POL(cons_c(x_1)) = [1] + x_1
   POL(cons_d(x_1)) = [1] + x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(cons_h(x_1)) = [1] + x_1
   POL(d(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = [1] + x_1
   POL(h(x_1)) = [1] + x_1
   POL(n__d(x_1)) = [1] + x_1
   POL(n__f(x_1)) = x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
S tuples:
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
K tuples:
   H(z0) -> c21(C(n__d(z0)))
Defined Rule Symbols:   encArg_1, f_1, c_1, h_1, d_1, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, C_1, H_1, D_1, ACTIVATE_1, ENCODE_F_1, ENCODE_C_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25, c11_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(z0) -> c18
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ACTIVATE(x_1)) = [1]
   POL(C(x_1)) = [1]
   POL(D(x_1)) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_ACTIVATE(x_1)) = [1] + x_1
   POL(ENCODE_C(x_1)) = [1]
   POL(ENCODE_D(x_1)) = 0
   POL(ENCODE_F(x_1)) = [1]
   POL(ENCODE_H(x_1)) = [1] + x_1
   POL(F(x_1)) = [1]
   POL(H(x_1)) = [1]
   POL(activate(x_1)) = [1] + x_1
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c18) = 0
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c_f(x_1)) = [1] + x_1
   POL(cons_activate(x_1)) = [1] + x_1
   POL(cons_c(x_1)) = [1] + x_1
   POL(cons_d(x_1)) = [1] + x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(cons_h(x_1)) = [1] + x_1
   POL(d(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = [1] + x_1
   POL(h(x_1)) = [1] + x_1
   POL(n__d(x_1)) = [1] + x_1
   POL(n__f(x_1)) = x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
S tuples:
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
K tuples:
   H(z0) -> c21(C(n__d(z0)))
   F(z0) -> c18
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
Defined Rule Symbols:   encArg_1, f_1, c_1, h_1, d_1, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, C_1, H_1, D_1, ACTIVATE_1, ENCODE_F_1, ENCODE_C_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25, c11_1


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

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
We considered the (Usable) Rules:
   activate(n__d(z0)) -> d(z0)
   f(z0) -> c_f(z0)
   activate(n__f(z0)) -> f(z0)
   f(z0) -> n__f(z0)
   d(z0) -> n__d(z0)
   h(z0) -> c(n__d(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   activate(z0) -> z0
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   encArg(g(z0)) -> g(encArg(z0))
   c(z0) -> d(activate(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
And the Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ACTIVATE(x_1)) = x_1
   POL(C(x_1)) = x_1
   POL(D(x_1)) = 0
   POL(ENCARG(x_1)) = [2]x_1 + x_1^2
   POL(ENCODE_ACTIVATE(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_C(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_D(x_1)) = [1] + x_1^2
   POL(ENCODE_F(x_1)) = [1] + [2]x_1 + x_1^2
   POL(ENCODE_H(x_1)) = [2] + [2]x_1 + x_1^2
   POL(F(x_1)) = x_1
   POL(H(x_1)) = x_1
   POL(activate(x_1)) = [2] + x_1
   POL(c(x_1)) = [2] + x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c18) = 0
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c_f(x_1)) = [1] + x_1
   POL(cons_activate(x_1)) = [2] + x_1
   POL(cons_c(x_1)) = [2] + x_1
   POL(cons_d(x_1)) = x_1
   POL(cons_f(x_1)) = [2] + x_1
   POL(cons_h(x_1)) = [1] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = [2]x_1
   POL(f(x_1)) = [2] + x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = [2] + x_1
   POL(n__d(x_1)) = x_1
   POL(n__f(x_1)) = x_1

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
S tuples:
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
K tuples:
   H(z0) -> c21(C(n__d(z0)))
   F(z0) -> c18
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
Defined Rule Symbols:   encArg_1, f_1, c_1, h_1, d_1, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, C_1, H_1, D_1, ACTIVATE_1, ENCODE_F_1, ENCODE_C_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25, c11_1


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

(25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   ACTIVATE(n__f(z0)) -> c23(F(z0))
We considered the (Usable) Rules:
   activate(n__d(z0)) -> d(z0)
   f(z0) -> c_f(z0)
   activate(n__f(z0)) -> f(z0)
   f(z0) -> n__f(z0)
   d(z0) -> n__d(z0)
   h(z0) -> c(n__d(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   activate(z0) -> z0
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   encArg(g(z0)) -> g(encArg(z0))
   c(z0) -> d(activate(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
And the Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ACTIVATE(x_1)) = x_1
   POL(C(x_1)) = x_1
   POL(D(x_1)) = 0
   POL(ENCARG(x_1)) = [2]x_1 + x_1^2
   POL(ENCODE_ACTIVATE(x_1)) = [2] + [2]x_1 + x_1^2
   POL(ENCODE_C(x_1)) = [1] + x_1 + [2]x_1^2
   POL(ENCODE_D(x_1)) = [1] + [2]x_1^2
   POL(ENCODE_F(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_H(x_1)) = [2] + x_1 + [2]x_1^2
   POL(F(x_1)) = x_1
   POL(H(x_1)) = x_1
   POL(activate(x_1)) = [2] + x_1
   POL(c(x_1)) = [2] + x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c18) = 0
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c_f(x_1)) = [2] + x_1
   POL(cons_activate(x_1)) = [2] + x_1
   POL(cons_c(x_1)) = [2] + x_1
   POL(cons_d(x_1)) = x_1
   POL(cons_f(x_1)) = [2] + x_1
   POL(cons_h(x_1)) = [2] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [2] + x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = [2] + x_1
   POL(n__d(x_1)) = x_1
   POL(n__f(x_1)) = [1] + x_1

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
S tuples:
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   D(z0) -> c22
K tuples:
   H(z0) -> c21(C(n__d(z0)))
   F(z0) -> c18
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   ACTIVATE(n__f(z0)) -> c23(F(z0))
Defined Rule Symbols:   encArg_1, f_1, c_1, h_1, d_1, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, C_1, H_1, D_1, ACTIVATE_1, ENCODE_F_1, ENCODE_C_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25, c11_1


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

(27) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:
   activate(n__d(z0)) -> d(z0)
   f(z0) -> c_f(z0)
   activate(n__f(z0)) -> f(z0)
   f(z0) -> n__f(z0)
   d(z0) -> n__d(z0)
   h(z0) -> c(n__d(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   activate(z0) -> z0
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   encArg(g(z0)) -> g(encArg(z0))
   c(z0) -> d(activate(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
And the Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ACTIVATE(x_1)) = x_1
   POL(C(x_1)) = [1] + x_1
   POL(D(x_1)) = 0
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_ACTIVATE(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_C(x_1)) = [2] + [2]x_1 + x_1^2
   POL(ENCODE_D(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_F(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_H(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(F(x_1)) = x_1
   POL(H(x_1)) = [1] + x_1
   POL(activate(x_1)) = [1] + x_1
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c18) = 0
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c_f(x_1)) = [1] + x_1
   POL(cons_activate(x_1)) = [2] + x_1
   POL(cons_c(x_1)) = [1] + x_1
   POL(cons_d(x_1)) = x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(cons_h(x_1)) = [1] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = [2]x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = [1] + x_1
   POL(n__d(x_1)) = x_1
   POL(n__f(x_1)) = x_1

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
S tuples:
   D(z0) -> c22
K tuples:
   H(z0) -> c21(C(n__d(z0)))
   F(z0) -> c18
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:   encArg_1, f_1, c_1, h_1, d_1, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, C_1, H_1, D_1, ACTIVATE_1, ENCODE_F_1, ENCODE_C_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25, c11_1


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

(29) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   D(z0) -> c22
We considered the (Usable) Rules:
   activate(n__d(z0)) -> d(z0)
   f(z0) -> c_f(z0)
   activate(n__f(z0)) -> f(z0)
   f(z0) -> n__f(z0)
   d(z0) -> n__d(z0)
   h(z0) -> c(n__d(z0))
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   activate(z0) -> z0
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   encArg(g(z0)) -> g(encArg(z0))
   c(z0) -> d(activate(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
And the Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ACTIVATE(x_1)) = [1] + [2]x_1
   POL(C(x_1)) = [2] + [2]x_1
   POL(D(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_ACTIVATE(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_C(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_D(x_1)) = [2] + [2]x_1 + x_1^2
   POL(ENCODE_F(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_H(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(F(x_1)) = [1] + [2]x_1
   POL(H(x_1)) = [2] + [2]x_1
   POL(activate(x_1)) = [1] + x_1
   POL(c(x_1)) = [2] + x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c18) = 0
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c_f(x_1)) = [1] + x_1
   POL(cons_activate(x_1)) = [1] + x_1
   POL(cons_c(x_1)) = [2] + x_1
   POL(cons_d(x_1)) = [1] + x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(cons_h(x_1)) = [2] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = [2] + x_1
   POL(n__d(x_1)) = x_1
   POL(n__f(x_1)) = x_1

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(n__f(z0)) -> n__f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(n__d(z0)) -> n__d(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_d(z0)) -> d(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   f(z0) -> c_f(z0)
   f(z0) -> n__f(z0)
   f(c_f(z0)) -> c(n__f(g(n__f(z0))))
   c(z0) -> d(activate(z0))
   h(z0) -> c(n__d(z0))
   d(z0) -> n__d(z0)
   activate(n__f(z0)) -> f(z0)
   activate(n__d(z0)) -> d(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(n__f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(n__d(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c5(C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0)) -> c6(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_d(z0)) -> c7(D(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c8(ACTIVATE(encArg(z0)), ENCARG(z0))
   F(z0) -> c18
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   H(z0) -> c21(C(n__d(z0)))
   D(z0) -> c22
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   ENCODE_F(z0) -> c11(F(encArg(z0)))
   ENCODE_C(z0) -> c11(C(encArg(z0)))
   ENCODE_D(z0) -> c11(D(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c11(ACTIVATE(encArg(z0)))
   ENCODE_H(z0) -> c11(H(encArg(z0)))
S tuples:none
K tuples:
   H(z0) -> c21(C(n__d(z0)))
   F(z0) -> c18
   ACTIVATE(n__d(z0)) -> c24(D(z0))
   ACTIVATE(z0) -> c25
   F(c_f(z0)) -> c19(C(n__f(g(n__f(z0)))))
   ACTIVATE(n__f(z0)) -> c23(F(z0))
   C(z0) -> c20(D(activate(z0)), ACTIVATE(z0))
   D(z0) -> c22
Defined Rule Symbols:   encArg_1, f_1, c_1, h_1, d_1, activate_1

Defined Pair Symbols:   ENCARG_1, F_1, C_1, H_1, D_1, ACTIVATE_1, ENCODE_F_1, ENCODE_C_1, ENCODE_D_1, ENCODE_ACTIVATE_1, ENCODE_H_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c18, c19_1, c20_2, c21_1, c22, c23_1, c24_1, c25, c11_1


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

(32)
BOUNDS(1, 1)