/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 53 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(8) CdtProblem
(9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(14) CdtProblem
(15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 354 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 243 ms]
(20) CdtProblem
(21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 139 ms]
(22) CdtProblem
(23) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CdtProblem
(25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CdtProblem
(27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(28) CdtProblem
(29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(30) CdtProblem
(31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(32) CdtProblem
(33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(34) CdtProblem
(35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CdtProblem
(37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(38) CdtProblem
(39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(40) CdtProblem
(41) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(42) CdtProblem
(43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(44) CdtProblem
(45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(46) CdtProblem
(47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
(49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(50) CdtProblem
(51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(52) CdtProblem
(53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(54) CdtProblem
(55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(56) CdtProblem
(57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(58) CdtProblem
(59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(60) CdtProblem
(61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(62) CdtProblem
(63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms]
(64) CdtProblem
(65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(66) CdtProblem
(67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(68) CdtProblem
(69) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(70) 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:

   g(c(x1)) -> g(f(c(x1)))
   g(f(c(x1))) -> g(f(f(c(x1))))
   g(g(x1)) -> g(f(g(x1)))
   f(f(g(x1))) -> g(f(x1))

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

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_g(x_1) -> g(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:

   g(c(x1)) -> g(f(c(x1)))
   g(f(c(x1))) -> g(f(f(c(x1))))
   g(g(x1)) -> g(f(g(x1)))
   f(f(g(x1))) -> g(f(x1))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_g(x_1) -> g(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:

   g(c(x1)) -> g(f(c(x1)))
   g(f(c(x1))) -> g(f(f(c(x1))))
   g(g(x1)) -> g(f(g(x1)))
   f(f(g(x1))) -> g(f(x1))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_g(x_1) -> g(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(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c4(G(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c5(ENCARG(z0))
   ENCODE_F(z0) -> c6(F(encArg(z0)), ENCARG(z0))
   G(c(z0)) -> c7(G(f(c(z0))), F(c(z0)))
   G(f(c(z0))) -> c8(G(f(f(c(z0)))), F(f(c(z0))), F(c(z0)))
   G(g(z0)) -> c9(G(f(g(z0))), F(g(z0)), G(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
S tuples:
   G(c(z0)) -> c7(G(f(c(z0))), F(c(z0)))
   G(f(c(z0))) -> c8(G(f(f(c(z0)))), F(f(c(z0))), F(c(z0)))
   G(g(z0)) -> c9(G(f(g(z0))), F(g(z0)), G(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
K tuples:none
Defined Rule Symbols:   g_1, f_1, encArg_1, encode_g_1, encode_c_1, encode_f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_G_1, ENCODE_C_1, ENCODE_F_1, G_1, F_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_1, c6_2, c7_2, c8_3, c9_3, c10_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_C(z0) -> c5(ENCARG(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c4(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c6(F(encArg(z0)), ENCARG(z0))
   G(c(z0)) -> c7(G(f(c(z0))), F(c(z0)))
   G(f(c(z0))) -> c8(G(f(f(c(z0)))), F(f(c(z0))), F(c(z0)))
   G(g(z0)) -> c9(G(f(g(z0))), F(g(z0)), G(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
S tuples:
   G(c(z0)) -> c7(G(f(c(z0))), F(c(z0)))
   G(f(c(z0))) -> c8(G(f(f(c(z0)))), F(f(c(z0))), F(c(z0)))
   G(g(z0)) -> c9(G(f(g(z0))), F(g(z0)), G(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
K tuples:none
Defined Rule Symbols:   g_1, f_1, encArg_1, encode_g_1, encode_c_1, encode_f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_G_1, ENCODE_F_1, G_1, F_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c6_2, c7_2, c8_3, c9_3, c10_2


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c4(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c6(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
S tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   g_1, f_1, encArg_1, encode_g_1, encode_c_1, encode_f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_G_1, ENCODE_F_1, F_1, G_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c6_2, c10_2, c7_1, c8, c9_1


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_G(z0) -> c5(ENCARG(z0))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
   ENCODE_F(z0) -> c5(ENCARG(z0))
S tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   g_1, f_1, encArg_1, encode_g_1, encode_c_1, encode_f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c2_2, c3_2, c10_2, c7_1, c8, c9_1, c5_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_G(z0) -> c5(ENCARG(z0))
   ENCODE_F(z0) -> c5(ENCARG(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
S tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   g_1, f_1, encArg_1, encode_g_1, encode_c_1, encode_f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c2_2, c3_2, c10_2, c7_1, c8, c9_1, c5_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_g(z0) -> g(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_f(z0) -> f(encArg(z0))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
S tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c2_2, c3_2, c10_2, c7_1, c8, c9_1, c5_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(c(z0)) -> c7(G(f(c(z0))))
We considered the (Usable) Rules:
   f(f(g(z0))) -> g(f(z0))
   g(g(z0)) -> g(f(g(z0)))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
And the Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_F(x_1)) = [2] + [2]x_1 + x_1^2
   POL(ENCODE_G(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(F(x_1)) = 0
   POL(G(x_1)) = [2]x_1
   POL(c(x_1)) = [2] + x_1
   POL(c1(x_1)) = x_1
   POL(c10(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(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8) = 0
   POL(c9(x_1)) = x_1
   POL(cons_f(x_1)) = x_1
   POL(cons_g(x_1)) = [2] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = 0
   POL(g(x_1)) = x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
S tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
K tuples:
   G(c(z0)) -> c7(G(f(c(z0))))
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c2_2, c3_2, c10_2, c7_1, c8, c9_1, c5_1


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

(19) 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(g(z0))) -> c10(G(f(z0)), F(z0))
We considered the (Usable) Rules:
   f(f(g(z0))) -> g(f(z0))
   g(g(z0)) -> g(f(g(z0)))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
And the Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = [2]x_1^2
   POL(ENCODE_F(x_1)) = [2] + x_1 + [2]x_1^2
   POL(ENCODE_G(x_1)) = [2] + x_1 + [2]x_1^2
   POL(F(x_1)) = x_1
   POL(G(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(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(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8) = 0
   POL(c9(x_1)) = x_1
   POL(cons_f(x_1)) = [2] + x_1
   POL(cons_g(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = x_1
   POL(g(x_1)) = [1] + x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
S tuples:
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
K tuples:
   G(c(z0)) -> c7(G(f(c(z0))))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c2_2, c3_2, c10_2, c7_1, c8, c9_1, c5_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(f(c(z0))) -> c8
We considered the (Usable) Rules:
   f(f(g(z0))) -> g(f(z0))
   g(g(z0)) -> g(f(g(z0)))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
And the Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_F(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_G(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(F(x_1)) = [2]x_1
   POL(G(x_1)) = [2]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(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(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8) = 0
   POL(c9(x_1)) = x_1
   POL(cons_f(x_1)) = [2] + x_1
   POL(cons_g(x_1)) = [2] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = x_1
   POL(g(x_1)) = [2] + x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:
   G(c(z0)) -> c7(G(f(c(z0))))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(f(c(z0))) -> c8
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c2_2, c3_2, c10_2, c7_1, c8, c9_1, c5_1


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

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_g(c(z0))) -> c2(G(c(encArg(z0))), ENCARG(c(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
   ENCARG(cons_g(c(z0))) -> c2(G(c(encArg(z0))), ENCARG(c(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:
   G(c(z0)) -> c7(G(f(c(z0))))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(f(c(z0))) -> c8
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c3_2, c10_2, c7_1, c8, c9_1, c5_1, c2_2


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

(25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   G(c(z0)) -> c7(G(f(c(z0))))
   G(f(c(z0))) -> c8

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
   ENCARG(cons_g(c(z0))) -> c2(G(c(encArg(z0))), ENCARG(c(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c3_2, c10_2, c9_1, c5_1, c2_2


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

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c3_2, c10_2, c9_1, c5_1, c2_2, c2_1


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

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_f(c(z0))) -> c3(F(c(encArg(z0))), ENCARG(c(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(F(c(encArg(z0))), ENCARG(c(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c10_2, c9_1, c5_1, c2_2, c2_1, c3_2


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

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c10_2, c9_1, c5_1, c2_2, c2_1, c3_2, c3_1


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

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(f(g(z0))) -> c10(G(f(z0)), F(z0)) by 
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCODE_G(z0) -> c5(G(encArg(z0)))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, ENCODE_G_1, ENCODE_F_1, F_1

Compound Symbols:   c1_1, c9_1, c5_1, c2_2, c2_1, c3_2, c3_1, c10_2


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

(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_G(z0) -> c5(G(encArg(z0))) by 
   ENCODE_G(c(z0)) -> c5(G(c(encArg(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0))))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(c(z0)) -> c5(G(c(encArg(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:
   F(f(g(z0))) -> c10(G(f(z0)), F(z0))
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, ENCODE_F_1, F_1, ENCODE_G_1

Compound Symbols:   c1_1, c9_1, c5_1, c2_2, c2_1, c3_2, c3_1, c10_2


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

(37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCODE_G(c(z0)) -> c5(G(c(encArg(z0))))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCODE_F(z0) -> c5(F(encArg(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, ENCODE_F_1, F_1, ENCODE_G_1

Compound Symbols:   c1_1, c9_1, c5_1, c2_2, c2_1, c3_2, c3_1, c10_2


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

(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_F(z0) -> c5(F(encArg(z0))) by 
   ENCODE_F(c(z0)) -> c5(F(c(encArg(z0))))
   ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0))))
   ENCODE_F(c(z0)) -> c5(F(c(encArg(z0))))
   ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_2, c3_1, c10_2, c5_1


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

(41) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCODE_F(c(z0)) -> c5(F(c(encArg(z0))))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0))))
   ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_2, c3_1, c10_2, c5_1


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

(43) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_g(cons_f(z0))) -> c2(G(f(encArg(z0))), ENCARG(cons_f(z0))) by 
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(G(f(c(encArg(z0)))), ENCARG(cons_f(c(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0))))
   ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(G(f(c(encArg(z0)))), ENCARG(cons_f(c(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_2, c3_1, c10_2, c5_1


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

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0))))
   ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_2, c3_1, c10_2, c5_1


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(cons_g(z0))) -> c3(F(g(encArg(z0))), ENCARG(cons_g(z0))) by 
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0))))
   ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_2, c3_1, c10_2, c5_1


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

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(cons_f(z0))) -> c3(F(f(encArg(z0))), ENCARG(cons_f(z0))) by 
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(F(f(c(encArg(z0)))), ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0))))
   ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(F(f(c(encArg(z0)))), ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_1, c10_2, c5_1, c3_2


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

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0))))
   ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(ENCARG(cons_f(c(z0))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_1, c10_2, c5_1, c3_2


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_G(cons_f(z0)) -> c5(G(f(encArg(z0)))) by 
   ENCODE_G(cons_f(c(z0))) -> c5(G(f(c(encArg(z0)))))
   ENCODE_G(cons_f(cons_g(z0))) -> c5(G(f(g(encArg(z0)))))
   ENCODE_G(cons_f(cons_f(z0))) -> c5(G(f(f(encArg(z0)))))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(ENCARG(cons_f(c(z0))))
   ENCODE_G(cons_f(c(z0))) -> c5(G(f(c(encArg(z0)))))
   ENCODE_G(cons_f(cons_g(z0))) -> c5(G(f(g(encArg(z0)))))
   ENCODE_G(cons_f(cons_f(z0))) -> c5(G(f(f(encArg(z0)))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_1, c10_2, c5_1, c3_2


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

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCODE_G(cons_f(c(z0))) -> c5(G(f(c(encArg(z0)))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(ENCARG(cons_f(c(z0))))
   ENCODE_G(cons_f(cons_g(z0))) -> c5(G(f(g(encArg(z0)))))
   ENCODE_G(cons_f(cons_f(z0))) -> c5(G(f(f(encArg(z0)))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_1, c10_2, c5_1, c3_2


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_F(cons_g(z0)) -> c5(F(g(encArg(z0)))) by 
   ENCODE_F(cons_g(c(z0))) -> c5(F(g(c(encArg(z0)))))
   ENCODE_F(cons_g(cons_g(z0))) -> c5(F(g(g(encArg(z0)))))
   ENCODE_F(cons_g(cons_f(z0))) -> c5(F(g(f(encArg(z0)))))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(ENCARG(cons_f(c(z0))))
   ENCODE_G(cons_f(cons_g(z0))) -> c5(G(f(g(encArg(z0)))))
   ENCODE_G(cons_f(cons_f(z0))) -> c5(G(f(f(encArg(z0)))))
   ENCODE_F(cons_g(c(z0))) -> c5(F(g(c(encArg(z0)))))
   ENCODE_F(cons_g(cons_g(z0))) -> c5(F(g(g(encArg(z0)))))
   ENCODE_F(cons_g(cons_f(z0))) -> c5(F(g(f(encArg(z0)))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_1, c10_2, c5_1, c3_2


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_F(cons_f(z0)) -> c5(F(f(encArg(z0)))) by 
   ENCODE_F(cons_f(c(z0))) -> c5(F(f(c(encArg(z0)))))
   ENCODE_F(cons_f(cons_g(z0))) -> c5(F(f(g(encArg(z0)))))
   ENCODE_F(cons_f(cons_f(z0))) -> c5(F(f(f(encArg(z0)))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(ENCARG(cons_f(c(z0))))
   ENCODE_G(cons_f(cons_g(z0))) -> c5(G(f(g(encArg(z0)))))
   ENCODE_G(cons_f(cons_f(z0))) -> c5(G(f(f(encArg(z0)))))
   ENCODE_F(cons_g(c(z0))) -> c5(F(g(c(encArg(z0)))))
   ENCODE_F(cons_g(cons_g(z0))) -> c5(F(g(g(encArg(z0)))))
   ENCODE_F(cons_g(cons_f(z0))) -> c5(F(g(f(encArg(z0)))))
   ENCODE_F(cons_f(c(z0))) -> c5(F(f(c(encArg(z0)))))
   ENCODE_F(cons_f(cons_g(z0))) -> c5(F(f(g(encArg(z0)))))
   ENCODE_F(cons_f(cons_f(z0))) -> c5(F(f(f(encArg(z0)))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_1, c10_2, c5_1, c3_2


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

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCODE_F(cons_f(c(z0))) -> c5(F(f(c(encArg(z0)))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(c(z0)) -> c1(ENCARG(z0))
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(ENCARG(cons_f(c(z0))))
   ENCODE_G(cons_f(cons_g(z0))) -> c5(G(f(g(encArg(z0)))))
   ENCODE_G(cons_f(cons_f(z0))) -> c5(G(f(f(encArg(z0)))))
   ENCODE_F(cons_g(c(z0))) -> c5(F(g(c(encArg(z0)))))
   ENCODE_F(cons_g(cons_g(z0))) -> c5(F(g(g(encArg(z0)))))
   ENCODE_F(cons_g(cons_f(z0))) -> c5(F(g(f(encArg(z0)))))
   ENCODE_F(cons_f(cons_g(z0))) -> c5(F(f(g(encArg(z0)))))
   ENCODE_F(cons_f(cons_f(z0))) -> c5(F(f(f(encArg(z0)))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, G_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c1_1, c9_1, c2_2, c2_1, c3_1, c10_2, c5_1, c3_2


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENCARG(c(z0)) -> c1(ENCARG(z0)) by 
   ENCARG(c(c(y0))) -> c1(ENCARG(c(y0)))
   ENCARG(c(cons_g(cons_g(y0)))) -> c1(ENCARG(cons_g(cons_g(y0))))
   ENCARG(c(cons_g(c(y0)))) -> c1(ENCARG(cons_g(c(y0))))
   ENCARG(c(cons_f(c(y0)))) -> c1(ENCARG(cons_f(c(y0))))
   ENCARG(c(cons_g(cons_f(cons_g(y0))))) -> c1(ENCARG(cons_g(cons_f(cons_g(y0)))))
   ENCARG(c(cons_g(cons_f(cons_f(y0))))) -> c1(ENCARG(cons_g(cons_f(cons_f(y0)))))
   ENCARG(c(cons_g(cons_f(c(y0))))) -> c1(ENCARG(cons_g(cons_f(c(y0)))))
   ENCARG(c(cons_f(cons_g(c(y0))))) -> c1(ENCARG(cons_f(cons_g(c(y0)))))
   ENCARG(c(cons_f(cons_g(cons_g(y0))))) -> c1(ENCARG(cons_f(cons_g(cons_g(y0)))))
   ENCARG(c(cons_f(cons_g(cons_f(y0))))) -> c1(ENCARG(cons_f(cons_g(cons_f(y0)))))
   ENCARG(c(cons_f(cons_f(cons_g(y0))))) -> c1(ENCARG(cons_f(cons_f(cons_g(y0)))))
   ENCARG(c(cons_f(cons_f(cons_f(y0))))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0)))))
   ENCARG(c(cons_f(cons_f(c(y0))))) -> c1(ENCARG(cons_f(cons_f(c(y0)))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   G(g(z0)) -> c9(G(z0))
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(ENCARG(cons_f(c(z0))))
   ENCODE_G(cons_f(cons_g(z0))) -> c5(G(f(g(encArg(z0)))))
   ENCODE_G(cons_f(cons_f(z0))) -> c5(G(f(f(encArg(z0)))))
   ENCODE_F(cons_g(c(z0))) -> c5(F(g(c(encArg(z0)))))
   ENCODE_F(cons_g(cons_g(z0))) -> c5(F(g(g(encArg(z0)))))
   ENCODE_F(cons_g(cons_f(z0))) -> c5(F(g(f(encArg(z0)))))
   ENCODE_F(cons_f(cons_g(z0))) -> c5(F(f(g(encArg(z0)))))
   ENCODE_F(cons_f(cons_f(z0))) -> c5(F(f(f(encArg(z0)))))
   ENCARG(c(c(y0))) -> c1(ENCARG(c(y0)))
   ENCARG(c(cons_g(cons_g(y0)))) -> c1(ENCARG(cons_g(cons_g(y0))))
   ENCARG(c(cons_g(c(y0)))) -> c1(ENCARG(cons_g(c(y0))))
   ENCARG(c(cons_f(c(y0)))) -> c1(ENCARG(cons_f(c(y0))))
   ENCARG(c(cons_g(cons_f(cons_g(y0))))) -> c1(ENCARG(cons_g(cons_f(cons_g(y0)))))
   ENCARG(c(cons_g(cons_f(cons_f(y0))))) -> c1(ENCARG(cons_g(cons_f(cons_f(y0)))))
   ENCARG(c(cons_g(cons_f(c(y0))))) -> c1(ENCARG(cons_g(cons_f(c(y0)))))
   ENCARG(c(cons_f(cons_g(c(y0))))) -> c1(ENCARG(cons_f(cons_g(c(y0)))))
   ENCARG(c(cons_f(cons_g(cons_g(y0))))) -> c1(ENCARG(cons_f(cons_g(cons_g(y0)))))
   ENCARG(c(cons_f(cons_g(cons_f(y0))))) -> c1(ENCARG(cons_f(cons_g(cons_f(y0)))))
   ENCARG(c(cons_f(cons_f(cons_g(y0))))) -> c1(ENCARG(cons_f(cons_f(cons_g(y0)))))
   ENCARG(c(cons_f(cons_f(cons_f(y0))))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0)))))
   ENCARG(c(cons_f(cons_f(c(y0))))) -> c1(ENCARG(cons_f(cons_f(c(y0)))))
S tuples:
   G(g(z0)) -> c9(G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   G_1, ENCARG_1, F_1, ENCODE_G_1, ENCODE_F_1

Compound Symbols:   c9_1, c2_2, c2_1, c3_1, c10_2, c5_1, c3_2, c1_1


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

(65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(g(z0)) -> c9(G(z0)) by 
   G(g(g(y0))) -> c9(G(g(y0)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(ENCARG(cons_f(c(z0))))
   ENCODE_G(cons_f(cons_g(z0))) -> c5(G(f(g(encArg(z0)))))
   ENCODE_G(cons_f(cons_f(z0))) -> c5(G(f(f(encArg(z0)))))
   ENCODE_F(cons_g(c(z0))) -> c5(F(g(c(encArg(z0)))))
   ENCODE_F(cons_g(cons_g(z0))) -> c5(F(g(g(encArg(z0)))))
   ENCODE_F(cons_g(cons_f(z0))) -> c5(F(g(f(encArg(z0)))))
   ENCODE_F(cons_f(cons_g(z0))) -> c5(F(f(g(encArg(z0)))))
   ENCODE_F(cons_f(cons_f(z0))) -> c5(F(f(f(encArg(z0)))))
   ENCARG(c(c(y0))) -> c1(ENCARG(c(y0)))
   ENCARG(c(cons_g(cons_g(y0)))) -> c1(ENCARG(cons_g(cons_g(y0))))
   ENCARG(c(cons_g(c(y0)))) -> c1(ENCARG(cons_g(c(y0))))
   ENCARG(c(cons_f(c(y0)))) -> c1(ENCARG(cons_f(c(y0))))
   ENCARG(c(cons_g(cons_f(cons_g(y0))))) -> c1(ENCARG(cons_g(cons_f(cons_g(y0)))))
   ENCARG(c(cons_g(cons_f(cons_f(y0))))) -> c1(ENCARG(cons_g(cons_f(cons_f(y0)))))
   ENCARG(c(cons_g(cons_f(c(y0))))) -> c1(ENCARG(cons_g(cons_f(c(y0)))))
   ENCARG(c(cons_f(cons_g(c(y0))))) -> c1(ENCARG(cons_f(cons_g(c(y0)))))
   ENCARG(c(cons_f(cons_g(cons_g(y0))))) -> c1(ENCARG(cons_f(cons_g(cons_g(y0)))))
   ENCARG(c(cons_f(cons_g(cons_f(y0))))) -> c1(ENCARG(cons_f(cons_g(cons_f(y0)))))
   ENCARG(c(cons_f(cons_f(cons_g(y0))))) -> c1(ENCARG(cons_f(cons_f(cons_g(y0)))))
   ENCARG(c(cons_f(cons_f(cons_f(y0))))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0)))))
   ENCARG(c(cons_f(cons_f(c(y0))))) -> c1(ENCARG(cons_f(cons_f(c(y0)))))
   G(g(g(y0))) -> c9(G(g(y0)))
S tuples:
   G(g(g(y0))) -> c9(G(g(y0)))
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_G_1, ENCODE_F_1, G_1

Compound Symbols:   c2_2, c2_1, c3_1, c10_2, c5_1, c3_2, c1_1, c9_1


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

(67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   ENCODE_G(cons_g(z0)) -> c5(G(g(encArg(z0))))
   G(g(g(y0))) -> c9(G(g(y0)))
   ENCODE_G(cons_f(cons_g(z0))) -> c5(G(f(g(encArg(z0)))))
   ENCODE_G(cons_f(cons_f(z0))) -> c5(G(f(f(encArg(z0)))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0)) -> c(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   g(c(z0)) -> g(f(c(z0)))
   g(f(c(z0))) -> g(f(f(c(z0))))
   g(g(z0)) -> g(f(g(z0)))
   f(f(g(z0))) -> g(f(z0))
Tuples:
   ENCARG(cons_g(cons_g(z0))) -> c2(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(c(z0))) -> c2(ENCARG(c(z0)))
   ENCARG(cons_f(c(z0))) -> c3(ENCARG(c(z0)))
   F(f(g(f(g(z0))))) -> c10(G(g(f(z0))), F(f(g(z0))))
   ENCARG(cons_g(cons_f(cons_g(z0)))) -> c2(G(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_g(cons_f(cons_f(z0)))) -> c2(G(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_g(cons_f(c(z0)))) -> c2(ENCARG(cons_f(c(z0))))
   ENCARG(cons_f(cons_g(c(z0)))) -> c3(F(g(c(encArg(z0)))), ENCARG(cons_g(c(z0))))
   ENCARG(cons_f(cons_g(cons_g(z0)))) -> c3(F(g(g(encArg(z0)))), ENCARG(cons_g(cons_g(z0))))
   ENCARG(cons_f(cons_g(cons_f(z0)))) -> c3(F(g(f(encArg(z0)))), ENCARG(cons_g(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_g(z0)))) -> c3(F(f(g(encArg(z0)))), ENCARG(cons_f(cons_g(z0))))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c3(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(c(z0)))) -> c3(ENCARG(cons_f(c(z0))))
   ENCODE_F(cons_g(c(z0))) -> c5(F(g(c(encArg(z0)))))
   ENCODE_F(cons_g(cons_g(z0))) -> c5(F(g(g(encArg(z0)))))
   ENCODE_F(cons_g(cons_f(z0))) -> c5(F(g(f(encArg(z0)))))
   ENCODE_F(cons_f(cons_g(z0))) -> c5(F(f(g(encArg(z0)))))
   ENCODE_F(cons_f(cons_f(z0))) -> c5(F(f(f(encArg(z0)))))
   ENCARG(c(c(y0))) -> c1(ENCARG(c(y0)))
   ENCARG(c(cons_g(cons_g(y0)))) -> c1(ENCARG(cons_g(cons_g(y0))))
   ENCARG(c(cons_g(c(y0)))) -> c1(ENCARG(cons_g(c(y0))))
   ENCARG(c(cons_f(c(y0)))) -> c1(ENCARG(cons_f(c(y0))))
   ENCARG(c(cons_g(cons_f(cons_g(y0))))) -> c1(ENCARG(cons_g(cons_f(cons_g(y0)))))
   ENCARG(c(cons_g(cons_f(cons_f(y0))))) -> c1(ENCARG(cons_g(cons_f(cons_f(y0)))))
   ENCARG(c(cons_g(cons_f(c(y0))))) -> c1(ENCARG(cons_g(cons_f(c(y0)))))
   ENCARG(c(cons_f(cons_g(c(y0))))) -> c1(ENCARG(cons_f(cons_g(c(y0)))))
   ENCARG(c(cons_f(cons_g(cons_g(y0))))) -> c1(ENCARG(cons_f(cons_g(cons_g(y0)))))
   ENCARG(c(cons_f(cons_g(cons_f(y0))))) -> c1(ENCARG(cons_f(cons_g(cons_f(y0)))))
   ENCARG(c(cons_f(cons_f(cons_g(y0))))) -> c1(ENCARG(cons_f(cons_f(cons_g(y0)))))
   ENCARG(c(cons_f(cons_f(cons_f(y0))))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0)))))
   ENCARG(c(cons_f(cons_f(c(y0))))) -> c1(ENCARG(cons_f(cons_f(c(y0)))))
S tuples:none
K tuples:none
Defined Rule Symbols:   encArg_1, g_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c2_2, c2_1, c3_1, c10_2, c3_2, c5_1, c1_1


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

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

(70)
BOUNDS(1, 1)